Crowell benjamin-newtonian-physics-1

Newtonian
Physics
Benjamin Crowell

Book 1 in the Light and Matter series of introductory physics textbooks
www.lightandmatter.com

The Light and Matter series of
introductory physics textbooks:
1 Newtonian Physics
2 Conservation Laws
3 Vibrations and Waves
4 Electricity and Magnetism
5 Optics
6 The Modern Revolution in Physics

Newtonian Physics

Benjamin Crowell


Light and Matter
Fullerton, California

© 1998-2002 by Benjamin Crowell
All rights reserved.

Edition 2.1
rev. 2002-06-03

ISBN 0-9704670-1-X

To Paul Herrschaft and Rich Muller.

Brief Contents
0 Introduction and Review ............................... 15
1 Scaling and Order-of-Magnitude Estimates 35

Motion in One Dimension
2 Velocity and Relative Motion ........................ 54
3 Acceleration and Free Fall ............................ 73
4 Force and Motion ........................................... 97
5 Analysis of Forces ....................................... 113

Motion in Three Dimensions
6 Newton’s Laws in Three Dimensions ........ 135
7 Vectors .......................................................... 145
8 Vectors and Motion ..................................... 155
9 Circular Motion ............................................ 167
10 Gravity ........................................................ 181
Exercises ........................................................... 201
Solutions to Selected Problems ...................... 209
Glossary ............................................................. 215
Mathematical Review ........................................ 217
Trig Tables.......................................................... 218
Index ................................................................... 219

Contents
Preface ......................................................... 13
A Note to the Student Taking Calculus
Concurrently ........................................... 14

0 Introduction
and Review 15
0.1 The Scientific Method .......................... 15
0.2 What Is Physics? ................................. 17
0.3 How to Learn Physics .......................... 20
0.4 Self-Evaluation .................................... 22
0.5 Basics of the Metric System ................ 22
0.6 The Newton, the Metric Unit of Force .. 25
0.7 Less Common Metric Prefixes ............. 26
0.8 Scientific Notation ................................ 27
0.9 Conversions ......................................... 28 Motion in One
0.10 Significant Figures ............................. 30
Summary ...................................................... 32 Dimension 53
Homework Problems .................................... 33
2 Velocity and Relative
Motion 54
2.1 Types of Motion ................................... 54
2.2 Describing Distance and Time ............. 57
2.3 Graphs of Motion; Velocity. .................. 60
2.4 The Principle of Inertia ......................... 64
2.5 Addition of Velocities ........................... 67
2.6 Graphs of Velocity Versus Time ........... 69
2.7 ∫ Applications of Calculus .................... 69
Summary ...................................................... 71
1 Scaling and Order-
of-Magnitude 3 Acceleration
Estimates 35 and Free Fall 73
1.1 Introduction .......................................... 35 3.1 The Motion of Falling Objects .............. 73
1.2 Scaling of Area and Volume ................ 37 3.2 Acceleration ......................................... 76
1.3 Scaling Applied to Biology ................... 44 3.3 Positive and Negative Acceleration ..... 79
1.4 Order-of-Magnitude Estimates ............ 47 3.4 Varying Acceleration ............................ 82
Summary ...................................................... 50 3.5 The Area Under
Homework Problems .................................... 50 the Velocity-Time Graph ................... 85
3.6 Algebraic Results
for Constant Acceleration ................. 87
3.7* Biological Effects of Weightlessness .. 89
3.8 ∫ Applications of Calculus .................... 91
Summary ...................................................... 92

4 Force and Motion 97
4.1 Force ................................................... 97
4.2 Newton’s First Law ............................ 100
4.3 Newton’s Second Law ....................... 104
4.4 What Force Is Not .............................. 106
4.5 Inertial and Noninertial
Frames of Reference ..................... 108
Summary ..................................................... 110
Homework Problems ................................... 111

5 Analysis of Forces 113
5.1 Newton’s Third Law ............................ 113
5.2 Classification and Behavior of Forces 118
5.3 Analysis of Forces ............................. 124
5.4 Transmission of Forces
by Low-Mass Objects ..................... 126
5.5 Objects Under Strain ......................... 128
5.6 Simple Machines: The Pulley ............ 129
Summary .................................................... 130
Homework Problems .................................. 131 7 Vectors 145
7.1 Vector Notation .................................. 145
7.2 Calculations with Magnitude
and Direction .................................. 148
7.3 Techniques for Adding Vectors .......... 151
7.4* Unit Vector Notation ......................... 152
7.5* Rotational Invariance ....................... 152
Summary .................................................... 153
Homework Problems .................................. 154

8 Vectors and Motion 155
8.1 The Velocity Vector ............................ 156
8.2 The Acceleration Vector ..................... 157
8.3 The Force Vector
and Simple Machines ..................... 160
8.4 ∫ Calculus With Vectors ...................... 161
Summary .................................................... 163
Motion in Three Homework Problems .................................. 164

Dimensions 135
6 Newton’s Laws in
Three Dimensions 135
6.1 Forces Have
No Perpendicular Effects ............... 135
6.2 Coordinates and Components ........... 138
6.3 Newton’s Laws in Three Dimensions 140
Summary .................................................... 142

9 Circular Motion 167
9.1 Conceptual Framework
for Circular Motion .......................... 167
9.2 Uniform Circular Motion ..................... 172
9.3 Nonuniform Circular Motion ............... 175
Summary .................................................... 176

10 Gravity 181
10.1 Kepler’s Laws .................................. 182
10.2 Newton’s Law of Gravity .................. 183
10.3 Apparent Weightlessness ................ 187
10.4 Vector Addition
of Gravitational Forces ............... 188
10.5 Weighing the Earth .......................... 190
10.6* Evidence for Repulsive Gravity ...... 194
Summary .................................................... 195
Exercises .................................. 201
Solutions to
Selected Problems ............ 209
Glossary .................................... 215
Mathematical Review ............... 217
Trig Tables................................. 218
Index .......................................... 219

Preface
Why a New Physics Textbook?
We assume that our economic system will always scamper to provide us with the products we want. Special
orders don’t upset us! I want my MTV! The truth is more complicated, especially in our education system, which
is paid for by the students but controlled by the professoriate. Witness the perverse success of the bloated science
textbook. The newspapers continue to compare our system unfavorably to Japanese and European education,
where depth is emphasized over breadth, but we can’t seem to create a physics textbook that covers a manageable
number of topics for a one-year course and gives honest explanations of everything it touches on.
The publishers try to please everybody by including every imaginable topic in the book, but end up pleasing
nobody. There is wide agreement among physics teachers that the traditional one-year introductory textbooks
cannot in fact be taught in one year. One cannot surgically remove enough material and still gracefully navigate
the rest of one of these kitchen-sink textbooks. What is far worse is that the books are so crammed with topics that
nearly all the explanation is cut out in order to keep the page count below 1100. Vital concepts like energy are
introduced abruptly with an equation, like a first-date kiss that comes before “hello.”
The movement to reform physics texts is steaming ahead, but despite excellent books such as Hewitt’s Concep-
tual Physics for non-science majors and Knight’s Physics: A Contemporary Perspective for students who know
calculus, there has been a gap in physics books for life-science majors who haven't learned calculus or are learning
it concurrently with physics. This book is meant to fill that gap.
Learning to Hate Physics?
When you read a mystery novel, you know in advance what structure to expect: a crime, some detective work,
and finally the unmasking of the evildoer. When Charlie Parker plays a blues, your ear expects to hear certain
landmarks of the form regardless of how wild some of his notes are. Surveys of physics students usually show that
they have worse attitudes about the subject after instruction than before, and their comments often boil down to a
complaint that the person who strung the topics together had not learned what Agatha Christie and Charlie Parker
knew intuitively about form and structure: students become bored and demoralized because the “march through
the topics” lacks a coherent story line. You are reading the first volume of the Light and Matter series of introduc-
tory physics textbooks, and as implied by its title, the story line of the series is built around light and matter: how
they behave, how they are different from each other, and, at the end of the story, how they turn out to be similar
in some very bizarre ways. Here is a guide to the structure of the one-year course presented in this series:
1 Newtonian Physics Matter moves at constant speed in a straight line unless a force acts on it. (This seems
intuitively wrong only because we tend to forget the role of friction forces.) Material objects can exert forces on
each other, each changing the other’s motion. A more massive object changes its motion more slowly in re-
sponse to a given force.
2 Conservation Laws Newton’s matter-and-forces picture of the universe is fine as far as it goes, but it doesn’t
apply to light, which is a form of pure energy without mass. A more powerful world-view, applying equally well
to both light and matter, is provided by the conservation laws, for instance the law of conservation of energy,
which states that energy can never be destroyed or created but only changed from one form into another.
3 Vibrations and Waves Light is a wave. We learn how waves travel through space, pass through each other,
speed up, slow down, and are reflected.
4 Electricity and Magnetism Matter is made out of particles such as electrons and protons, which are held
together by electrical forces. Light is a wave that is made out of patterns of electric and magnetic force.
5 Optics Devices such as eyeglasses and searchlights use matter (lenses and mirrors) to manipulate light.
6 The Modern Revolution in Physics Until the twentieth century, physicists thought that matter was made
out of particles and light was purely a wave phenomenon. We now know that both light and matter are made of
building blocks that have both particle and wave properties. In the process of understanding this apparent
contradiction, we find that the universe is a much stranger place than Newton had ever imagined, and also learn
the basis for such devices as lasers and computer chips.

13

A Note to the Student Taking Calculus
Concurrently
Learning calculus and physics concurrently is an excellent idea — it’s not a coincidence that the inventor of
calculus, Isaac Newton, also discovered the laws of motion! If you are worried about taking these two demanding
courses at the same time, let me reassure you. I think you will find that physics helps you with calculus while
calculus deepens and enhances your experience of physics. This book is designed to be used in either an algebra-
based physics course or a calculus-based physics course that has calculus as a corequisite. This note is addressed to
students in the latter type of course.
It has been said that critics discuss art with each other, but artists talk about brushes. Art needs both a “why”
and a “how,” concepts as well as technique. Just as it is easier to enjoy an oil painting than to produce one, it is
easier to understand the concepts of calculus than to learn the techniques of calculus. This book will generally
teach you the concepts of calculus a few weeks before you learn them in your math class, but it does not discuss the
techniques of calculus at all. There will thus be a delay of a few weeks between the time when a calculus application
is first pointed out in this book and the first occurrence of a homework problem that requires the relevant tech-
nique. The following outline shows a typical first-semester calculus curriculum side-by-side with the list of topics
covered in this book, to give you a rough idea of what calculus your physics instructor might expect you to know
at a given point in the semester. The sequence of the calculus topics is the one followed by Calculus of a Single
Variable, 2nd ed., by Swokowski, Olinick, and Pence.

topics typically covered at the same
chapters of this book point in a calculus course
0-1 introduction review

2-3 velocity and acceleration limits

4-5 Newton's laws the derivative concept
techniques for finding derivatives;
6-8 motion in 3 dimensions
derivatives of trigonometric functions
9 circular motion the chain rule

10 gravity local maxima and minima
chapters of
Conservation Laws
1-3 energy concavity and the second derivative

4 momentum

5 angular momentum the indefinite integral
chapters of
Vibrations and Waves
1 vibrations the definite integral

2-3 waves the fundamental theorem of calculus

14

The Mars Climate Orbiter is prepared for its mission.
The laws of physics are the same everywhere, even
on Mars, so the probe could be designed based on
the laws of physics as discovered on earth.
There is unfortunately another reason why this
spacecraft is relevant to the topics of this chapter: it
was destroyed attempting to enter Mars’ atmosphere
because engineers at Lockheed Martin forgot to
convert data on engine thrusts from pounds into the
metric unit of force (newtons) before giving the
information to NASA. Conversions are important!

0 Introduction and Review
If you drop your shoe and a coin side by side, they hit the ground at the
same time. Why doesn’t the shoe get there first, since gravity is pulling
harder on it? How does the lens of your eye work, and why do your eye’s
muscles need to squash its lens into different shapes in order to focus on
objects nearby or far away? These are the kinds of questions that physics
tries to answer about the behavior of light and matter, the two things that
the universe is made of.

0.1 The Scientific Method
Until very recently in history, no progress was made in answering
questions like these. Worse than that, the wrong answers written by thinkers
like the ancient Greek physicist Aristotle were accepted without question for
thousands of years. Why is it that scientific knowledge has progressed more
since the Renaissance than it had in all the preceding millennia since the
beginning of recorded history? Undoubtedly the industrial revolution is part
of the answer. Building its centerpiece, the steam engine, required improved
techniques for precise construction and measurement. (Early on, it was
considered a major advance when English machine shops learned to build
pistons and cylinders that fit together with a gap narrower than the thick-
ness of a penny.) But even before the industrial revolution, the pace of
discovery had picked up, mainly because of the introduction of the modern
scientific method. Although it evolved over time, most scientists today
theory would agree on something like the following list of the basic principles of
the scientific method:
(1)Science is a cycle of theory and experiment. Scientific theories are
created to explain the results of experiments that were created under certain
conditions. A successful theory will also make new predictions about new
experiment experiments under new conditions. Eventually, though, it always seems to
happen that a new experiment comes along, showing that under certain

15

conditions the theory is not a good approximation or is not valid at all. The
ball is then back in the theorists’ court. If an experiment disagrees with the
current theory, the theory has to be changed, not the experiment.
(2)Theories should both predict and explain. The requirement of predic-
tive power means that a theory is only meaningful if it predicts something
that can be checked against experimental measurements that the theorist
did not already have at hand. That is, a theory should be testable. Explana-
tory value means that many phenomena should be accounted for with few
basic principles. If you answer every “why” question with “because that’s the
way it is,” then your theory has no explanatory value. Collecting lots of data
without being able to find any basic underlying principles is not science.
(3)Experiments should be reproducible. An experiment should be treated
with suspicion if it only works for one person, or only in one part of the
world. Anyone with the necessary skills and equipment should be able to
get the same results from the same experiment. This implies that science
transcends national and ethnic boundaries; you can be sure that nobody is
doing actual science who claims that their work is “Aryan, not Jewish,”
“Marxist, not bourgeois,” or “Christian, not atheistic.” An experiment
cannot be reproduced if it is secret, so science is necessarily a public enter-
prise.
As an example of the cycle of theory and experiment, a vital step toward
modern chemistry was the experimental observation that the chemical
elements could not be transformed into each other, e.g. lead could not be
turned into gold. This led to the theory that chemical reactions consisted of
rearrangements of the elements in different combinations, without any
change in the identities of the elements themselves. The theory worked for
hundreds of years, and was confirmed experimentally over a wide range of
pressures and temperatures and with many combinations of elements. Only
in the twentieth century did we learn that one element could be trans-
A satirical drawing of an alchemist’s
formed into one another under the conditions of extremely high pressure
laboratory. H. Cock, after a drawing
by Peter Brueghel the Elder (16th and temperature existing in a nuclear bomb or inside a star. That observa-
century). tion didn’t completely invalidate the original theory of the immutability of
the elements, but it showed that it was only an approximation, valid at
ordinary temperatures and pressures.
Self-Check
A psychic conducts seances in which the spirits of the dead speak to the
participants. He says he has special psychic powers not possessed by other
people, which allow him to “channel” the communications with the spirits.
What part of the scientific method is being violated here? [Answer below.]

The scientific method as described here is an idealization, and should
not be understood as a set procedure for doing science. Scientists have as
many weaknesses and character flaws as any other group, and it is very
common for scientists to try to discredit other people’s experiments when
the results run contrary to their own favored point of view. Successful
science also has more to do with luck, intuition, and creativity than most
people realize, and the restrictions of the scientific method do not stifle
individuality and self-expression any more than the fugue and sonata forms

If only he has the special powers, then his results can never be reproduced.

16 Chapter 0 Introduction and Review

stifled Bach and Haydn. There is a recent tendency among social scientists
to go even further and to deny that the scientific method even exists,
Science is creative. claiming that science is no more than an arbitrary social system that
determines what ideas to accept based on an in-group’s criteria. I think
that’s going too far. If science is an arbitrary social ritual, it would seem
difficult to explain its effectiveness in building such useful items as air-
planes, CD players and sewers. If alchemy and astrology were no less
scientific in their methods than chemistry and astronomy, what was it that
kept them from producing anything useful?
Discussion Questions
Consider whether or not the scientific method is being applied in the following
examples. If the scientific method is not being applied, are the people whose
actions are being described performing a useful human activity, albeit an
unscientific one?
A. Acupuncture is a traditional medical technique of Asian origin in which small
needles are inserted in the patient’s body to relieve pain. Many doctors trained
in the west consider acupuncture unworthy of experimental study because if it
had therapeutic effects, such effects could not be explained by their theories of
the nervous system. Who is being more scientific, the western or eastern
practitioners?
B. Goethe, a famous German poet, is less well known for his theory of color.
He published a book on the subject, in which he argued that scientific
apparatus for measuring and quantifying color, such as prisms, lenses and
colored filters, could not give us full insight into the ultimate meaning of color,
for instance the cold feeling evoked by blue and green or the heroic sentiments
inspired by red. Was his work scientific?
C. A child asks why things fall down, and an adult answers “because of
gravity.” The ancient Greek philosopher Aristotle explained that rocks fell
because it was their nature to seek out their natural place, in contact with the
earth. Are these explanations scientific?
D. Buddhism is partly a psychological explanation of human suffering, and
psychology is of course a science. The Buddha could be said to have
engaged in a cycle of theory and experiment, since he worked by trial and
error, and even late in his life he asked his followers to challenge his ideas.
Buddhism could also be considered reproducible, since the Buddha told his
followers they could find enlightenment for themselves if they followed a
certain course of study and discipline. Is Buddhism a scientific pursuit?

0.2 What Is Physics?
Given for one instant an intelligence which could comprehend all the forces
by which nature is animated and the respective positions of the things which
compose it...nothing would be uncertain, and the future as the past would
be laid out before its eyes.
Pierre Simon de Laplace
Physics is the use of the scientific method to find out the basic prin-
ciples governing light and matter, and to discover the implications of those
laws. Part of what distinguishes the modern outlook from the ancient mind-
Physics is the study set is the assumption that there are rules by which the universe functions,
of light and matter. and that those laws can be at least partially understood by humans. From
the Age of Reason through the nineteenth century, many scientists began to
be convinced that the laws of nature not only could be known but, as
claimed by Laplace, those laws could in principle be used to predict every-

Section 0.2 What Is Physics? 17

thing about the universe’s future if complete information was available
about the present state of all light and matter. In subsequent sections, I’ll
describe two general types of limitations on prediction using the laws of
physics, which were only recognized in the twentieth century.
Weight is what Matter can be defined as anything that is affected by gravity, i.e. that
distinguishes has weight or would have weight if it was near the Earth or another star or
light from matter. planet massive enough to produce measurable gravity. Light can be defined
as anything that can travel from one place to another through empty space
and can influence matter, but has no weight. For example, sunlight can
influence your body by heating it or by damaging your DNA and giving
you skin cancer. The physicist’s definition of light includes a variety of
phenomena that are not visible to the eye, including radio waves, micro-
waves, x-rays, and gamma rays. These are the “colors” of light that do not
happen to fall within the narrow violet-to-red range of the rainbow that we
can see.
Self-check
At the turn of the 20th century, a strange new phenomenon was discovered in
vacuum tubes: mysterious rays of unknown origin and nature. These rays are
the same as the ones that shoot from the back of your TV’s picture tube and hit
the front to make the picture. Physicists in 1895 didn’t have the faintest idea
what the rays were, so they simply named them “cathode rays,” after the name
for the electrical contact from which they sprang. A fierce debate raged,
complete with nationalistic overtones, over whether the rays were a form of
light or of matter. What would they have had to do in order to settle the issue?
Many physical phenomena are not themselves light or matter, but are
properties of light or matter or interactions between light and matter. For
instance, motion is a property of all light and some matter, but it is not
itself light or matter. The pressure that keeps a bicycle tire blown up is an
interaction between the air and the tire. Pressure is not a form of matter in
and of itself. It is as much a property of the tire as of the air. Analogously,
sisterhood and employment are relationships among people but are not
This telescope picture shows two people themselves.
images of the same distant object, an
exotic, very luminous object called a Some things that appear weightless actually do have weight, and so
quasar. This is interpreted as evidence qualify as matter. Air has weight, and is thus a form of matter even though a
that a massive, dark object, possibly cubic inch of air weighs less than a grain of sand. A helium balloon has
a black hole, happens to be between
us and it. Light rays that would
weight, but is kept from falling by the force of the surrounding more dense
otherwise have missed the earth on air, which pushes up on it. Astronauts in orbit around the Earth have
either side have been bent by the dark weight, and are falling along a curved arc, but they are moving so fast that
object’s gravity so that they reach us. the curved arc of their fall is broad enough to carry them all the way around
The actual direction to the quasar is
the Earth in a circle. They perceive themselves as being weightless because
presumably in the center of the image,
but the light along that central line their space capsule is falling along with them, and the floor therefore does
doesn’t get to us because it is not push up on their feet.
absorbed by the dark object. The
quasar is known by its catalog number,
Optional Topic
MG1131+0456, or more informally as Einstein predicted as a consequence of his theory of relativity that
Einstein’s Ring. light would after all be affected by gravity, although the effect would
be extremely weak under normal conditions. His prediction was
borne out by observations of the bending of light rays from stars as
they passed close to the sun on their way to the Earth. Einstein also

They would have had to weigh the rays, or check for a loss of weight in the object from which they were have
emitted. (For technical reasons, this was not a measurement they could actually do, hence the opportunity for
disagreement.)


predicted the existence of black holes, stars so massive and
compact that their intense gravity would not even allow light to
escape. (These days there is strong evidence that black holes
exist.)
Einstein’s interpretation was that light doesn’t really have mass, but
that energy is affected by gravity just like mass is. The energy in a
light beam is equivalent to a certain amount of mass, given by the
famous equation E=mc2, where c is the speed of light. Because the
virus speed of light is such a big number, a large amount of energy is
equivalent to only a very small amount of mass, so the gravitational
molecule force on a light ray can be ignored for most practical purposes.
There is however a more satisfactory and fundamental distinction
between light and matter, which should be understandable to you if
you have had a chemistry course. In chemistry, one learns that
electrons obey the Pauli exclusion principle, which forbids more than
one electron from occupying the same orbital if they have the same
spin. The Pauli exclusion principle is obeyed by the subatomic
particles of which matter is composed, but disobeyed by the
particles, called photons, of which a beam of light is made.
Einstein’s theory of relativity is discussed more fully in book 6 of this
atom series.
The boundary between physics and the other sciences is not always
clear. For instance, chemists study atoms and molecules, which are what
matter is built from, and there are some scientists who would be equally
willing to call themselves physical chemists or chemical physicists. It might
seem that the distinction between physics and biology would be clearer,
since physics seems to deal with inanimate objects. In fact, almost all
physicists would agree that the basic laws of physics that apply to molecules
in a test tube work equally well for the combination of molecules that
neutrons constitutes a bacterium. (Some might believe that something more happens
and protons in the minds of humans, or even those of cats and dogs.) What differenti-
ates physics from biology is that many of the scientific theories that describe
living things, while ultimately resulting from the fundamental laws of
physics, cannot be rigorously derived from physical principles.
Isolated systems and reductionism
To avoid having to study everything at once, scientists isolate the things
they are trying to study. For instance, a physicist who wants to study the
motion of a rotating gyroscope would probably prefer that it be isolated
quarks from vibrations and air currents. Even in biology, where field work is
indispensable for understanding how living things relate to their entire
environment, it is interesting to note the vital historical role played by
Darwin’s study of the Galápagos Islands, which were conveniently isolated
from the rest of the world. Any part of the universe that is considered apart
from the rest can be called a “system.”
Physics has had some of its greatest successes by carrying this process of
isolation to extremes, subdividing the universe into smaller and smaller
parts. Matter can be divided into atoms, and the behavior of individual
? atoms can be studied. Atoms can be split apart into their constituent
neutrons, protons and electrons. Protons and neutrons appear to be made
out of even smaller particles called quarks, and there have even been some
claims of experimental evidence that quarks have smaller parts inside them.

Section 0.2 What Is Physics? 19

This method of splitting things into smaller and smaller parts and studying
how those parts influence each other is called reductionism. The hope is
that the seemingly complex rules governing the larger units can be better
understood in terms of simpler rules governing the smaller units. To
appreciate what reductionism has done for science, it is only necessary to
examine a 19th-century chemistry textbook. At that time, the existence of
atoms was still doubted by some, electrons were not even suspected to exist,
and almost nothing was understood of what basic rules governed the way
atoms interacted with each other in chemical reactions. Students had to
memorize long lists of chemicals and their reactions, and there was no way
to understand any of it systematically. Today, the student only needs to
remember a small set of rules about how atoms interact, for instance that
atoms of one element cannot be converted into another via chemical
reactions, or that atoms from the right side of the periodic table tend to
form strong bonds with atoms from the left side.
Discussion Questions
A. I’ve suggested replacing the ordinary dictionary definition of light with a
more technical, more precise one that involves weightlessness. It’s still
possible, though, that the stuff a lightbulb makes, ordinarily called “light,” does
have some small amount of weight. Suggest an experiment to attempt to
measure whether it does.
B. Heat is weightless (i.e. an object becomes no heavier when heated), and
can travel across an empty room from the fireplace to your skin, where it
influences you by heating you. Should heat therefore be considered a form of
light by our definition? Why or why not?
C. Similarly, should sound be considered a form of light?

0.3 How to Learn Physics
For as knowledges are now delivered, there is a kind of contract of error
between the deliverer and the receiver; for he that delivereth knowledge
desireth to deliver it in such a form as may be best believed, and not as may
be best examined; and he that receiveth knowledge desireth rather present
satisfaction than expectant inquiry.
Sir Francis Bacon
Science is not
about plugging
Many students approach a science course with the idea that they can
into formulas. succeed by memorizing the formulas, so that when a problem is assigned on
the homework or an exam, they will be able to plug numbers in to the
formula and get a numerical result on their calculator. Wrong! That’s not
what learning science is about! There is a big difference between memoriz-
ing formulas and understanding concepts. To start with, different formulas
may apply in different situations. One equation might represent a defini-
tion, which is always true. Another might be a very specific equation for the
speed of an object sliding down an inclined plane, which would not be true
if the object was a rock drifting down to the bottom of the ocean. If you
don’t work to understand physics on a conceptual level, you won’t know
which formulas can be used when.


Most students taking college science courses for the first time also have
very little experience with interpreting the meaning of an equation. Con-
sider the equation w=A/h relating the width of a rectangle to its height and
interpreting area. A student who has not developed skill at interpretation might view
an equation this as yet another equation to memorize and plug in to when needed. A
slightly more savvy student might realize that it is simply the familiar
formula A=wh in a different form. When asked whether a rectangle would
have a greater or smaller width than another with the same area but a
smaller height, the unsophisticated student might be at a loss, not having
Other Books any numbers to plug in on a calculator. The more experienced student
PSSC Physics, Haber-Schaim et would know how to reason about an equation involving division — if h is
al., 7th ed., 1986. Kendall/Hunt, smaller, and A stays the same, then w must be bigger. Often, students fail to
Dubuque, Iowa. recognize a sequence of equations as a derivation leading to a final result, so
A high-school textbook at the they think all the intermediate steps are equally important formulas that
algebra-based level. This book they should memorize.
distinguishes itself by giving a
clear, careful, and honest When learning any subject at all, it is important to become as actively
explanation of every topic, while involved as possible, rather than trying to read through all the information
avoiding unnecessary details. quickly without thinking about it. It is a good idea to read and think about
the questions posed at the end of each section of these notes as you encoun-
Physics for Poets, Robert H. ter them, so that you know you have understood what you were reading.
March, 4th ed., 1996. McGraw-
Hill, New York. Many students’ difficulties in physics boil down mainly to difficulties
As the name implies, this book’s with math. Suppose you feel confident that you have enough mathematical
intended audience is liberal arts preparation to succeed in this course, but you are having trouble with a few
students who want to under- specific things. In some areas, the brief review given in this chapter may be
stand science in a broader sufficient, but in other areas it probably will not. Once you identify the
cultural and historical context. areas of math in which you are having problems, get help in those areas.
Not much math is used, and the Don’t limp along through the whole course with a vague feeling of dread
page count of this little paper- about something like scientific notation. The problem will not go away if
back is about five times less than you ignore it. The same applies to essential mathematical skills that you are
that of the typical “kitchen sink” learning in this course for the first time, such as vector addition.
textbook, but the intellectual Sometimes students tell me they keep trying to understand a certain
level is actually pretty challeng- topic in the book, and it just doesn’t make sense. The worst thing you can
ing. possibly do in that situation is to keep on staring at the same page. Every
Conceptual Physics, Paul Hewitt. textbook explains certain things badly — even mine! — so the best thing to
Scott Foresman, Glenview, Ill. do in this situation is to look at a different book. Instead of college text-
This is the excellent book used books aimed at the same mathematical level as the course you’re taking, you
for Physics 130 here at Fullerton may in some cases find that high school books or books at a lower math
College. Only simple algebra is level give clearer explanations. The three books listed on the left are, in my
used. opinion, the best introductory physics books available, although they would
not be appropriate as the primary textbook for a college-level course for
science majors.
Finally, when reviewing for an exam, don’t simply read back over the
text and your lecture notes. Instead, try to use an active method of review-
ing, for instance by discussing some of the discussion questions with
another student, or doing homework problems you hadn’t done the first
time.

Section 0.3 How to Learn Physics 21

0.4 Self-Evaluation
The introductory part of a book like this is hard to write, because every
student arrives at this starting point with a different preparation. One
student may have grown up in another country and so may be completely
comfortable with the metric system, but may have had an algebra course in
which the instructor passed too quickly over scientific notation. Another
student may have already taken calculus, but may have never learned the
metric system. The following self-evaluation is a checklist to help you figure
out what you need to study to be prepared for the rest of the course.

If you disagree with this statement... you should study this section:

I am familiar with the basic metric units of meters,
kilograms, and seconds, and the most common metric 0.5 Basics of the Metric System
prefixes: milli- (m), kilo- (k), and centi- (c).
I know about the Newton, a unit of force 0.6 The Newton, the Metric Unit of Force

I am familiar with these less common metric prefixes:
0.7 Less Common Metric Prefixes
mega- (M), micro- (µ), and nano- (n).

I am comfortable with scientific notation. 0.8 Scientific Notation

I can confidently do metric conversions. 0.9 Conversions

I understand the purpose and use of significant figures. 0.10 Significant Figures

It wouldn’t hurt you to skim the sections you think you already know
about, and to do the self-checks in those sections.

0.5 Basics of the Metric System
The metric system
Units were not standardized until fairly recently in history, so when the
physicist Isaac Newton gave the result of an experiment with a pendulum,
he had to specify not just that the string was 37 7/8 inches long but that it
was “37 7/8 London inches long.” The inch as defined in Yorkshire would
have been different. Even after the British Empire standardized its units, it
was still very inconvenient to do calculations involving money, volume,
distance, time, or weight, because of all the odd conversion factors, like 16
ounces in a pound, and 5280 feet in a mile. Through the nineteenth
century, schoolchildren squandered most of their mathematical education
in preparing to do calculations such as making change when a customer in a
shop offered a one-crown note for a book costing two pounds, thirteen
shillings and tuppence. The dollar has always been decimal, and British
money went decimal decades ago, but the United States is still saddled with
the antiquated system of feet, inches, pounds, ounces and so on.
Every country in the world besides the U.S. has adopted a system of
units known in English as the “metric system.” This system is entirely


decimal, thanks to the same eminently logical people who brought about
the French Revolution. In deference to France, the system’s official name is
the Système International, or SI, meaning International System. (The
phrase “SI system” is therefore redundant.)
The wonderful thing about the SI is that people who live in countries
more modern than ours do not need to memorize how many ounces there
are in a pound, how many cups in a pint, how many feet in a mile, etc. The
whole system works with a single, consistent set of prefixes (derived from
Greek) that modify the basic units. Each prefix stands for a power of ten,
and has an abbreviation that can be combined with the symbol for the unit.
For instance, the meter is a unit of distance. The prefix kilo- stands for 103,
so a kilometer, 1 km, is a thousand meters.
The basic units of the metric system are the meter for distance, the
second for time, and the gram for mass.
The following are the most common metric prefixes. You should
memorize them.
prefix meaning example
3
kilo- k 10 60 kg = a person’s mass
centi- c 10-2 28 cm = height of a piece of paper
milli- m 10-3 1 ms = time for one vibration of a
guitar string playing the
note D
The prefix centi-, meaning 10-2, is only used in the centimeter; a
hundredth of a gram would not be written as 1 cg but as 10 mg. The centi-
prefix can be easily remembered because a cent is 10-2 dollars. The official SI
abbreviation for seconds is “s” (not “sec”) and grams are “g” (not “gm”).
The second
The sun stood still and the moon halted until the nation had taken ven-
geance on its enemies...
Joshua 10:12-14
Absolute, true, and mathematical time, of itself, and from its own nature,
flows equably without relation to anything external...
Isaac Newton
When I stated briefly above that the second was a unit of time, it may
not have occurred to you that this was not really much of a definition. The
two quotes above are meant to demonstrate how much room for confusion
exists among people who seem to mean the same thing by a word such as
“time.” The first quote has been interpreted by some biblical scholars as
indicating an ancient belief that the motion of the sun across the sky was
not just something that occurred with the passage of time but that the sun
actually caused time to pass by its motion, so that freezing it in the sky

Section 0.5 Basic of the Metric System 23

The Time Without
Underwear Pope Gregory created our modern
“Gregorian” calendar, with its system
Unfortunately, the French of leap years, to make the length of
Revolutionary calendar never the calendar year match the length of
caught on. Each of its twelve the cycle of seasons. Not until1752 did
months was 30 days long, with Protestant England switched to the
names like Thermidor (the month new calendar. Some less educated
citizens believed that the shortening
of heat) and Germinal (the month of the month by eleven days would
of budding). To round out the year shorten their lives by the same interval.
to 365 days, a five-day period was In this illustration by William Hogarth,
added on the end of the calendar, the leaflet lying on the ground reads,
“Give us our eleven days.”
and named the sans culottides. In
modern French, sans culottides
means “time without underwear,”
would have some kind of a supernatural decelerating effect on everyone
but in the 18th century, it was a way
except the Hebrew soldiers. Many ancient cultures also conceived of time as
to honor the workers and peasants,
cyclical, rather than proceeding along a straight line as in 1998, 1999,
who wore simple clothing instead
2000, 2001,... The second quote, from a relatively modern physicist, may
of the fancy pants (culottes) of the
sound a lot more scientific, but most physicists today would consider it
aristocracy.
useless as a definition of time. Today, the physical sciences are based on
operational definitions, which means definitions that spell out the actual
steps (operations) required to measure something numerically.
Now in an era when our toasters, pens, and coffee pots tell us the time,
it is far from obvious to most people what is the fundamental operational
definition of time. Until recently, the hour, minute, and second were
defined operationally in terms of the time required for the earth to rotate
about its axis. Unfortunately, the Earth’s rotation is slowing down slightly,
and by 1967 this was becoming an issue in scientific experiments requiring
precise time measurements. The second was therefore redefined as the time
required for a certain number of vibrations of the light waves emitted by a
cesium atoms in a lamp constructed like a familiar neon sign but with the
neon replaced by cesium. The new definition not only promises to stay
constant indefinitely, but for scientists is a more convenient way of calibrat-
ing a clock than having to carry out astronomical measurements.
Self-Check
What is a possible operational definition of how strong a person is?

The meter
107 m The French originally defined the meter as 10-7 times the distance from
the equator to the north pole, as measured through Paris (of course). Even if
the definition was operational, the operation of traveling to the north pole
and laying a surveying chain behind you was not one that most working
scientists wanted to carry out. Fairly soon, a standard was created in the
form of a metal bar with two scratches on it. This definition persisted until
1960, when the meter was redefined as the distance traveled by light in a
vacuum over a period of (1/299792458) seconds.

A dictionary might define “strong” as “posessing powerful muscles,” but that’s not an operational definition, because
it doesn’t say how to measure strength numerically. One possible operational definition would be the number of
pounds a person can bench press.


The kilogram
The third base unit of the SI is the kilogram, a unit of mass. Mass is
intended to be a measure of the amount of a substance, but that is not an
operational definition. Bathroom scales work by measuring our planet’s
gravitational attraction for the object being weighed, but using that type of
scale to define mass operationally would be undesirable because gravity
varies in strength from place to place on the earth.
There’s a surprising amount of disagreement among physics textbooks
about how mass should be defined, but here’s how it’s actually handled by
the few working physicists who specialize in ultra-high-precision measure-
ments. They maintain a physical object in Paris, which is the standard
kilogram, a cylinder made of platinum-iridium alloy. Duplicates are
checked against this mother of all kilograms by putting the original and the
copy on the two opposite pans of a balance. Although this method of
comparison depends on gravity, the problems associated with differences in
gravity in different geographical locations are bypassed, because the two
objects are being compared in the same place. The duplicates can then be
removed from the Parisian kilogram shrine and transported elsewhere in the
world.
Combinations of metric units
Just about anything you want to measure can be measured with some
combination of meters, kilograms, and seconds. Speed can be measured in
m/s, volume in m3, and density in kg/m3. Part of what makes the SI great is
this basic simplicity. No more funny units like a cord of wood, a bolt of
cloth, or a jigger of whiskey. No more liquid and dry measure. Just a simple,
consistent set of units. The SI measures put together from meters, kilo-
grams, and seconds make up the mks system. For example, the mks unit of
speed is m/s, not km/hr.

Discussion question
Isaac Newton wrote, “...the natural days are truly unequal, though they are
commonly considered as equal, and used for a measure of time... It may be
that there is no such thing as an equable motion, whereby time may be
accurately measured. All motions may be accelerated or retarded...” Newton
was right. Even the modern definition of the second in terms of light emitted by
cesium atoms is subject to variation. For instance, magnetic fields could cause
the cesium atoms to emit light with a slightly different rate of vibration. What
makes us think, though, that a pendulum clock is more accurate than a
sundial, or that a cesium atom is a more accurate timekeeper than a pendulum
clock? That is, how can one test experimentally how the accuracies of different
time standards compare?

0.6 The Newton, the Metric Unit of Force
A force is a push or a pull, or more generally anything that can change
an object’s speed or direction of motion. A force is required to start a car
moving, to slow down a baseball player sliding in to home base, or to make
an airplane turn. (Forces may fail to change an object’s motion if they are
canceled by other forces, e.g. the force of gravity pulling you down right
now is being canceled by the force of the chair pushing up on you.) The
metric unit of force is the Newton, defined as the force which, if applied for
one second, will cause a 1-kilogram object starting from rest to reach a

Section 0.6 The Newton, the Metric Unit of Force 25

speed of 1 m/s. Later chapters will discuss the force concept in more detail.
In fact, this entire book is about the relationship between force and motion.
In the previous section, I gave a gravitational definition of mass, but by
defining a numerical scale of force, we can also turn around and define a
scale of mass without reference to gravity. For instance, if a force of two
Newtons is required to accelerate a certain object from rest to 1 m/s in 1 s,
then that object must have a mass of 2 kg. From this point of view, mass
characterizes an object’s resistance to a change in its motion, which we call
inertia or inertial mass. Although there is no fundamental reason why an
object’s resistance to a change in its motion must be related to how strongly
gravity affects it, careful and precise experiments have shown that the
inertial definition and the gravitational definition of mass are highly
consistent for a variety of objects. It therefore doesn’t really matter for any
practical purpose which definition one adopts.
Discussion Question
Spending a long time in weightlessness is unhealthy. One of the most
important negative effects experienced by astronauts is a loss of muscle and
bone mass. Since an ordinary scale won’t work for an astronaut in orbit, what
is a possible way of monitoring this change in mass? (Measuring the
astronaut’s waist or biceps with a measuring tape is not good enough, because
it doesn’t tell anything about bone mass, or about the replacement of muscle
with fat.)

0.7 Less Common Metric Prefixes
The following are three metric prefixes which, while less common than
Nine little the ones discussed previously, are well worth memorizing.
prefix meaning example
10-9 nano nuns
mega- M 106 6.4 Mm = radius of the earth

mix micro- µ 10 -6
1 µm = diameter of a human hair
10-6 micro
nano- n 10-9 0.154 nm = distance between carbon
milky nuclei in an ethane molecule
10-3 milli
Note that the abbreviation for micro is the Greek letter mu, µ — a
common mistake is to confuse it with m (milli) or M (mega).
103 kilo
There are other prefixes even less common, used for extremely large and
mugs. small quantities. For instance, 1 femtometer=10-15 m is a convenient unit
106 mega
of distance in nuclear physics, and 1 gigabyte=109 bytes is used for comput-
This is a mnemonic to help you ers’ hard disks. The international committee that makes decisions about the
remember the most important SI has recently even added some new prefixes that sound like jokes, e.g. 1
metric prefixes. The word "little" yoctogram = 10-24 g is about half the mass of a proton. In the immediate
is to remind you that the list starts
future, however, you’re unlikely to see prefixes like “yocto-” and “zepto-”
with the prefixes used for small
quantities and builds upward. The used except perhaps in trivia contests at science-fiction conventions or other
exponent changes by 3 with each geekfests.
step, except that of course we do
not need a special prefix for 100,
which equals one.


Self-Check
Suppose you could slow down time so that according to your perception, a
beam of light would move across a room at the speed of a slow walk. If you
perceived a nanosecond as if it was a second, how would you perceive a
microsecond?

0.8 Scientific Notation
Most of the interesting phenomena our universe has to offer are not on
the human scale. It would take about 1,000,000,000,000,000,000,000
bacteria to equal the mass of a human body. When the physicist Thomas
Young discovered that light was a wave, it was back in the bad old days
before scientific notation, and he was obliged to write that the time required
for one vibration of the wave was 1/500 of a millionth of a millionth of a
second. Scientific notation is a less awkward way to write very large and
very small numbers such as these. Here’s a quick review.
Scientific notation means writing a number in terms of a product of
something from 1 to 10 and something else that is a power of ten. For
instance,
32 = 3.2 x 101
320 = 3.2 x 102
3200 = 3.2 x 103 ...
Each number is ten times bigger than the previous one.
Since 101 is ten times smaller than 102, it makes sense to use the
notation 100 to stand for one, the number that is in turn ten times smaller
than 101. Continuing on, we can write 10-1 to stand for 0.1, the number
ten times smaller than 100. Negative exponents are used for small numbers:
3.2 = 3.2 x 100
0.32 = 3.2 x 10-1
0.032 = 3.2 x 10-2 ...
A common source of confusion is the notation used on the displays of
many calculators. Examples:
3.2 x 106 (written notation)
3.2E+6 (notation on some calculators)
3.26 (notation on some other calculators)
The last example is particularly unfortunate, because 3.26 really stands
for the number 3.2x3.2x3.2x3.2x3.2x3.2 = 1074, a totally different number
from 3.2 x 106 = 3200000. The calculator notation should never be used in
writing. It’s just a way for the manufacturer to save money by making a
simpler display.

A microsecond is 1000 times longer than a nanosecond, so it would seem like 1000 seconds, or about 20 minutes.

Section 0.8 Scientific Notation 27

Self-Check
A student learns that 104 bacteria, standing in line to register for classes at
Paramecium Community College, would form a queue of this size:

The student concludes that 102 bacteria would form a line of this length:

Why is the student incorrect?

0.9 Conversions
I suggest you avoid memorizing lots of conversion factors between SI
units and U.S. units. Suppose the United Nations sends its black helicopters
to invade California (after all who wouldn’t rather live here than in New
York City?), and institutes water fluoridation and the SI, making the use of
inches and pounds into a crime punishable by death. I think you could get
by with only two mental conversion factors:
1 inch = 2.54 cm
An object with a weight on Earth of 2.2 lb has a mass of 1 kg.
The first one is the present definition of the inch, so it’s exact. The
second one is not exact, but is good enough for most purposes. The pound
is a unit of gravitational force, while the kg is a unit of mass, which mea-
sures how hard it is to accelerate an object, not how hard gravity pulls on it.
Therefore it would be incorrect to say that 2.2 lb literally equaled 1 kg, even
approximately.
More important than memorizing conversion factors is understanding
the right method for doing conversions. Even within the SI, you may need
to convert, say, from grams to kilograms. Different people have different
ways of thinking about conversions, but the method I’ll describe here is
systematic and easy to understand. The idea is that if 1 kg and 1000 g
represent the same mass, then we can consider a fraction like
3
10 g
1 kg
to be a way of expressing the number one. This may bother you. For
instance, if you type 1000/1 into your calculator, you will get 1000, not
one. Again, different people have different ways of thinking about it, but
the justification is that it helps us to do conversions, and it works! Now if
we want to convert 0.7 kg to units of grams, we can multiply 0.7 kg by the
number one:
3
10 g
0.7 kg ×
1 kg
If you’re willing to treat symbols such as “kg” as if they were variables as
used in algebra (which they’re really not), you can then cancel the kg on top
with the kg on the bottom, resulting in

Exponents have to do with multiplication, not addition. The first line should be 100 times longer than the second,
not just twice as long.


10 3 g
/
0.7 kg × = 700 g .
1 kg /
To convert grams to kilograms, you would simply flip the fraction upside
down.
One advantage of this method is that it can easily be applied to a series
of conversions. For instance, to convert one year to units of seconds,

/
1 year × /
365 days
× /
24 hours
× / × 60 s
60 min

/
1 year /
1 day / /
1 hour 1 min
= 3.15 x 107 s .
Should that exponent be positive or negative?
A common mistake is to write the conversion fraction incorrectly. For
instance the fraction
3
10 kg
(incorrect)
1g
checking conversions does not equal one, because 103 kg is the mass of a car, and 1 g is the mass
using common sense
of a raisin. One correct way of setting up the conversion factor would be
–3
10 kg
. (correct)
1g

You can usually detect such a mistake if you take the time to check your
answer and see if it is reasonable.
If common sense doesn’t rule out either a positive or a negative expo-
nent, here’s another way to make sure you get it right. There are big prefixes
checking conversions using and small prefixes:
the idea of “compensating” big prefixes: k M
small prefixes: m µ n
(It’s not hard to keep straight which are which, since “mega” and “micro” are
evocative, and it’s easy to remember that a kilometer is bigger than a meter
and a millimeter is smaller.) In the example above, we want the top of the
fraction to be the same as the bottom. Since k is a big prefix, we need to
compensate by putting a small number like 10-3 in front of it, not a big
number like 103.
Discussion Question
Each of the following conversions contains an error. In each case, explain
what the error is.
1 kg 1 cm
(a) 1000 kg x 1000 g = 1 g (b) 50 m x 100 m = 0.5 cm

(c) "Nano" is 10-9, so there are 10-9 nm in a meter.
(d) "Micro" is 10-6, so 1 kg is 106 µg.

Section 0.9 Conversions 29

0.10 Significant Figures
An engineer is designing a car engine, and has been told that the
diameter of the pistons (which are being designed by someone else) is 5 cm.
He knows that 0.02 cm of clearance is required for a piston of this size, so
he designs the cylinder to have an inside diameter of 5.04 cm. Luckily, his
supervisor catches his mistake before the car goes into production. She
explains his error to him, and mentally puts him in the “do not promote”
category.
What was his mistake? The person who told him the pistons were 5 cm
in diameter was wise to the ways of significant figures, as was his boss, who
explained to him that he needed to go back and get a more accurate num-
ber for the diameter of the pistons. That person said “5 cm” rather than
“5.00 cm” specifically to avoid creating the impression that the number was
extremely accurate. In reality, the pistons’ diameter was 5.13 cm. They
would never have fit in the 5.04-cm cylinders.
The number of digits of accuracy in a number is referred to as the
Significant figures
communicate the number of significant figures, or “sig figs” for short. As in the example
accuracy of a number. above, sig figs provide a way of showing the accuracy of a number. In most
cases, the result of a calculation involving several pieces of data can be no
more accurate than the least accurate piece of data. In other words, “garbage
in, garbage out.” Since the 5 cm diameter of the pistons was not very
accurate, the result of the engineer’s calculation, 5.04 cm, was really not as
accurate as he thought. In general, your result should not have more than
the number of sig figs in the least accurate piece of data you started with.
The calculation above should have been done as follows:
5 cm (1 sig fig)
+ 0.04 cm (1 sig fig)
= 5 cm (rounded off to 1 sig fig)
The fact that the final result only has one significant figure then alerts you
to the fact that the result is not very accurate, and would not be appropriate
for use in designing the engine.
Note that the leading zeroes in the number 0.04 do not count as
significant figures, because they are only placeholders. On the other hand, a
number such as 50 cm is ambiguous — the zero could be intended as a
significant figure, or it might just be there as a placeholder. The ambiguity
involving trailing zeroes can be avoided by using scientific notation, in
which 5 x 101 cm would imply one sig fig of accuracy, while 5.0 x 101 cm
would imply two sig figs.
Dealing correctly with significant figures can save you time! Often,
students copy down numbers from their calculators with eight significant
figures of precision, then type them back in for a later calculation. That’s a
waste of time, unless your original data had that kind of incredible preci-
sion.


The rules about significant figures are only rules of thumb, and are not
a substitute for careful thinking. For instance, $20.00 + $0.05 is $20.05. It
need not and should not be rounded off to $20. In general, the sig fig rules
work best for multiplication and division, and we also apply them when
doing a complicated calculation that involves many types of operations. For
simple addition and subtraction, it makes more sense to maintain a fixed
number of digits after the decimal point. When in doubt, don’t use the sig
fig rules at all: just observe the effect on your final result when you change
one piece of your initial data by the maximum amount by which you think
it could have been inaccurate.
Self-Check
How many significant figures are there in each of the following measurements?
(a) 9.937 m
(b) 4.0 s
(c) 0.0000037 kg

(a) 4; (b) 2; (c) 2

Section 0.10 Significant Figures 31

Summary
Selected Vocabulary
matter ............................... Anything that is affected by gravity.
light................................... Anything that can travel from one place to another through empty space
and can influence matter, but is not affected by gravity.
operational definition ........ A definition that states what operations should be carried out to measure
the thing being defined.
Système International ........ A fancy name for the metric system.
mks system ........................ The use of metric units based on the meter, kilogram, and second. Ex-
ample: meters per second is the mks unit of speed, not cm/s or km/hr.
mass .................................. A numerical measure of how difficult it is to change an object’s motion.
significant figures .............. Digits that contribute to the accuracy of a measurement.
Notation
m ...................................... symbol for mass, or the meter, the metric distance unit
kg ...................................... kilogram, the metric unit of mass
s ........................................ second, the metric unit of time
M- ..................................... the metric prefix mega-, 106
k- ...................................... the metric prefix kilo-, 103
m- ..................................... the metric prefix milli-, 10-3
µ- ...................................... the metric prefix micro-, 10-6
n- ...................................... the metric prefix nano-, 10-9
Summary
Physics is the use of the scientific method to study the behavior of light and matter. The scientific method
requires a cycle of theory and experiment, theories with both predictive and explanatory value, and
reproducible experiments.
The metric system is a simple, consistent framework for measurement built out of the meter, the kilogram,
and the second plus a set of prefixes denoting powers of ten. The most systematic method for doing
conversions is shown in the following example:
–3
370 ms × 10 s = 0.37 s
1 ms
Mass is a measure of the amount of a substance. Mass can be defined gravitationally, by comparing an
object to a standard mass on a double-pan balance, or in terms of inertia, by comparing the effect of a force
on an object to the effect of the same force on a standard mass. The two definitions are found experimentally
to be proportional to each other to a high degree of precision, so we usually refer simply to “mass,” without
bothering to specify which type.
A force is that which can change the motion of an object. The metric unit of force is the Newton, defined as
the force required to accelerate a standard 1-kg mass from rest to a speed of 1 m/s in 1 s.
Scientific notation means, for example, writing 3.2x105 rather than 320000.
Writing numbers with the correct number of significant figures correctly communicates how accurate they
are. As a rule of thumb, the final result of a calculation is no more accurate than, and should have no more
significant figures than, the least accurate piece of data.


Homework Problems
1. Correct use of a calculator: (a) Calculate 74658 on a calcula-
53222 + 97554
tor.
[Self-check: The most common mistake results in 97555.40.]
(b) Which would be more like the price of a TV, and which would be more
like the price of a house, $ 3.5x105 or $ 3.55?
2. Compute the following things. If they don't make sense because of units,
say so.
(a) 3 cm + 5 cm (b) 1.11 m + 22 cm
(c) 120 miles + 2.0 hours (d) 120 miles / 2.0 hours
3. Your backyard has brick walls on both ends. You measure a distance of
23.4 m from the inside of one wall to the inside of the other. Each wall is
29.4 cm thick. How far is it from the outside of one wall to the outside of
the other? Pay attention to significant figures.
4 . The speed of light is 3.0x108 m/s. Convert this to furlongs per fort-
night. A furlong is 220 yards, and a fortnight is 14 days. An inch is 2.54
cm.
5 . Express each of the following quantities in micrograms: (a) 10 mg, (b)
104 g, (c) 10 kg, (d) 100x103 g, (e) 1000 ng.
6 S. Convert 134 mg to units of kg, writing your answer in scientific
notation.
7. In the last century, the average age of the onset of puberty for girls has
decreased by several years. Urban folklore has it that this is because of
hormones fed to beef cattle, but it is more likely to be because modern girls
have more body fat on the average and possibly because of estrogen-
mimicking chemicals in the environment from the breakdown of pesticides.
A hamburger from a hormone-implanted steer has about 0.2 ng of estrogen
(about double the amount of natural beef ). A serving of peas contains about
300 ng of estrogen. An adult woman produces about 0.5 mg of estrogen per
day (note the different unit!). (a) How many hamburgers would a girl have
to eat in one day to consume as much estrogen as an adult woman’s daily
production? (b) How many servings of peas?
8 S. The usual definition of the mean (average) of two numbers a and b is
(a+b)/2. This is called the arithmetic mean. The geometric mean, however,
is defined as (ab)1/2. For the sake of definiteness, let’s say both numbers have
units of mass. (a) Compute the arithmetic mean of two numbers that have
units of grams. Then convert the numbers to units of kilograms and
recompute their mean. Is the answer consistent? (b) Do the same for the
geometric mean. (c) If a and b both have units of grams, what should we
call the units of ab? Does your answer make sense when you take the square
root? (d) Suppose someone proposes to you a third kind of mean, called the
superduper mean, defined as (ab)1/3. Is this reasonable?

S A solution is given in the back of the book. A difficult problem.
A computerized answer check is available. ∫ A problem that requires calculus.

Homework Problems 33

Life would be very different if
you were the size of an insect.

1 Scaling and Order-of-
Magnitude Estimates
1.1 Introduction
Why can’t an insect be the size of a dog? Some skinny stretched-out cells
in your spinal cord are a meter tall — why does nature display no single
cells that are not just a meter tall, but a meter wide, and a meter thick as
well? Believe it or not, these are questions that can be answered fairly easily
without knowing much more about physics than you already do. The only
mathematical technique you really need is the humble conversion, applied
to area and volume.
Amoebas this size are seldom Area and volume
encountered. Area can be defined by saying that we can copy the shape of interest
onto graph paper with 1 cm x 1 cm squares and count the number of
squares inside. Fractions of squares can be estimated by eye. We then say the
area equals the number of squares, in units of square cm. Although this
might seem less “pure” than computing areas using formulae like A=πr2 for
a circle or A=wh/2 for a triangle, those formulae are not useful as definitions
of area because they cannot be applied to irregularly shaped areas.
Units of square cm are more commonly written as cm2 in science. Of
course, the unit of measurement symbolized by “cm” is not an algebra
symbol standing for a number that can be literally multiplied by itself. But
it is advantageous to write the units of area that way and treat the units as if
they were algebra symbols. For instance, if you have a rectangle with an area
of 6 m2 and a width of 2 m, then calculating its length as (6 m2)/(2 m)=3 m
gives a result that makes sense both numerically and in terms of units. This
algebra-style treatment of the units also ensures that our methods of
35

converting units work out correctly. For instance, if we accept the fraction
100 cm
1m
as a valid way of writing the number one, then one times one equals one, so
we should also say that one can be represented by
100 cm × 100 cm
1m 1m
which is the same as
10000 cm 2
.
1 m2
That means the conversion factor from square meters to square centi-
meters is a factor of 104, i.e. a square meter has 104 square centimeters in
it.
All of the above can be easily applied to volume as well, using one-
cubic-centimeter blocks instead of squares on graph paper.
To many people, it seems hard to believe that a square meter equals
10000 square centimeters, or that a cubic meter equals a million cubic
centimeters — they think it would make more sense if there were 100 cm2
in 1 m2, and 100 cm3 in 1 m3, but that would be incorrect. The examples
shown in the figure below aim to make the correct answer more believable,
using the traditional U.S. units of feet and yards. (One foot is 12 inches,
and one yard is three feet.)

1 ft 1 yd = 3 ft

1 ft2 1 yd2 = 9 ft2 1 ft3
1 yd3 = 27 ft3

Self-Check
Based on the figure, convince yourself that there are 9 ft2 in a square yard ,
and 27 ft3 in a cubic yard, then demonstrate the same thing symbolically (i.e.
with the method using fractions that equal one).

Discussion question
A. How many square centimeters are there in a square inch? (1 inch=2.54 cm)
First find an approximate answer by making a drawing, then derive the
conversion factor more accurately using the symbolic method.

1 yd2x(3 ft/1 yd)2=9 ft2. 1 yd3x(3 ft/1 yd)3=27 ft3.

36 Chapter 1 Scaling and Order-of-Magnitude Estimates

Galileo Galilei (1564-1642) was a Renaissance Italian who brought the scientific
method to bear on physics, creating the modern version of the science. Coming
from a noble but very poor family, Galileo had to drop out of medical school at
the University of Pisa when he ran out of money. Eventually becoming a lecturer
in mathematics at the same school, he began a career as a notorious
troublemaker by writing a burlesque ridiculing the university’s regulations — he
was forced to resign, but found a new teaching position at Padua. He invented
the pendulum clock, investigated the motion of falling bodies, and discovered
the moons of Jupiter. The thrust of his life’s work was to discredit Aristotle’s
physics by confronting it with contradictory experiments, a program which paved
the way for Newton’s discovery of the relationship between force and motion. In
Chapter 3 we’ll come to the story of Galileo’s ultimate fate at the hands of the
Church.

1.2 Scaling of Area and Volume
Great fleas have lesser fleas
Upon their backs to bite ‘em.
And lesser fleas have lesser still,
And so ad infinitum.
Jonathan Swift
Now how do these conversions of area and volume relate to the ques-
tions I posed about sizes of living things? Well, imagine that you are shrunk
like Alice in Wonderland to the size of an insect. One way of thinking
about the change of scale is that what used to look like a centimeter now
The small boat holds up just fine.
looks like perhaps a meter to you, because you’re so much smaller. If area
and volume scaled according to most people’s intuitive, incorrect expecta-
tions, with 1 m2 being the same as 100 cm2, then there would be no
particular reason why nature should behave any differently on your new,
reduced scale. But nature does behave differently now that you’re small. For
instance, you will find that you can walk on water, and jump to many times
your own height. The physicist Galileo Galilei had the basic insight that the
A larger boat built with the same scaling of area and volume determines how natural phenomena behave
proportions as the small one will differently on different scales. He first reasoned about mechanical struc-
collapse under its own weight. tures, but later extended his insights to living things, taking the then-radical
point of view that at the fundamental level, a living organism should follow
the same laws of nature as a machine. We will follow his lead by first
discussing machines and then living things.
Galileo on the behavior of nature on large and small scales
A boat this large needs to have timbers
that are thicker compared to its size. One of the world’s most famous pieces of scientific writing is Galileo’s
Dialogues Concerning the Two New Sciences. Galileo was an entertaining
writer who wanted to explain things clearly to laypeople, and he livened up
his work by casting it in the form of a dialogue among three people. Salviati
is really Galileo’s alter ego. Simplicio is the stupid character, and one of the
reasons Galileo got in trouble with the Church was that there were rumors
that Simplicio represented the Pope. Sagredo is the earnest and intelligent
student, with whom the reader is supposed to identify. (The following
excerpts are from the 1914 translation by Crew and de Salvio.)

Section 1.2 Scaling of Area and Volume 37

This plank is the longest it This plank is made out of the
can be without collapsing same kind of wood. It is twice
under its own weight. If it as thick, twice as long, and
was a hundredth of an inch twice as wide. It will collapse
longer, it would collapse. under its own weight.
(After Galileo's original drawing.)

SALVIATI: ...we asked the reason why [shipbuilders] employed stocks,
scaffolding, and bracing of larger dimensions for launching a big vessel than
they do for a small one; and [an old man] answered that they did this in order
to avoid the danger of the ship parting under its own heavy weight, a danger to
which small boats are not subject?
SAGREDO: Yes, that is what I mean; and I refer especially to his last assertion
which I have always regarded as false...; namely, that in speaking of these and
other similar machines one cannot argue from the small to the large, because
many devices which succeed on a small scale do not work on a large scale.
Now, since mechanics has its foundations in geometry, where mere size [ is
unimportant], I do not see that the properties of circles, triangles, cylinders,
cones and other solid figures will change with their size. If, therefore, a large
machine be constructed in such a way that its parts bear to one another the
same ratio as in a smaller one, and if the smaller is sufficiently strong for the
purpose for which it is designed, I do not see why the larger should not be able
to withstand any severe and destructive tests to which it may be subjected.
Salviati contradicts Sagredo:
SALVIATI: ...Please observe, gentlemen, how facts which at first seem
improbable will, even on scant explanation, drop the cloak which has hidden
them and stand forth in naked and simple beauty. Who does not know that a
horse falling from a height of three or four cubits will break his bones, while a
dog falling from the same height or a cat from a height of eight or ten cubits will
suffer no injury? Equally harmless would be the fall of a grasshopper from a
tower or the fall of an ant from the distance of the moon.

The point Galileo is making here is that small things are sturdier in
proportion to their size. There are a lot of objections that could be raised,
however. After all, what does it really mean for something to be “strong”, to
be “strong in proportion to its size,” or to be strong “out of proportion to its
size?” Galileo hasn’t spelled out operational definitions of things like
“strength,” i.e. definitions that spell out how to measure them numerically.
Also, a cat is shaped differently from a horse — an enlarged photograph
of a cat would not be mistaken for a horse, even if the photo-doctoring
experts at the National Inquirer made it look like a person was riding on its
back. A grasshopper is not even a mammal, and it has an exoskeleton
instead of an internal skeleton. The whole argument would be a lot more
convincing if we could do some isolation of variables, a scientific term that
means to change only one thing at a time, isolating it from the other
variables that might have an effect. If size is the variable whose effect we’re


interested in seeing, then we don’t really want to compare things that are
different in size but also different in other ways.
Also, Galileo is doing something that would be frowned on in modern
science: he is mixing experiments whose results he has actually observed
(building boats of different sizes), with experiments that he could not
possibly have done (dropping an ant from the height of the moon).
After this entertaining but not scientifically rigorous beginning, Galileo
starts to do something worthwhile by modern standards. He simplifies
everything by considering the strength of a wooden plank. The variables
involved can then be narrowed down to the type of wood, the width, the
thickness, and the length. He also gives an operational definition of what it
means for the plank to have a certain strength “in proportion to its size,” by
introducing the concept of a plank that is the longest one that would not
snap under its own weight if supported at one end. If you increased its
length by the slightest amount, without increasing its width or thickness, it
would break. He says that if one plank is the same shape as another but a
different size, appearing like a reduced or enlarged photograph of the other,
Galileo discusses planks made of then the planks would be strong “in proportion to their sizes” if both were
wood, but the concept may be easier just barely able to support their own weight.
to imagine with clay. All three clay rods
in the figure were originally the same He now relates how he has done actual experiments with such planks,
shape. The medium-sized one was and found that, according to this operational definition, they are not strong
twice the height, twice the length, and in proportion to their sizes. The larger one breaks. He makes sure to tell the
twice the width of the small one, and
similarly the large one was twice as
reader how important the result is, via Sagredo’s astonished response:
big as the medium one in all its linear SAGREDO: My brain already reels. My mind, like a cloud momentarily illuminated
dimensions. The big one has four by a lightning flash, is for an instant filled with an unusual light, which now
times the linear dimensions of the beckons to me and which now suddenly mingles and obscures strange, crude
small one, 16 times the cross-sectional ideas. From what you have said it appears to me impossible to build two
area when cut perpendicular to the similar structures of the same material, but of different sizes and have them
page, and 64 times the volume. That proportionately strong.
means that the big one has 64 times
the weight to support, but only 16 times In other words, this specific experiment, using things like wooden
the strength compared to the smallest planks that have no intrinsic scientific interest, has very wide implications
one. because it points out a general principle, that nature acts differently on
different scales.
To finish the discussion, Galileo gives an explanation. He says that the
strength of a plank (defined as, say, the weight of the heaviest boulder you
could put on the end without breaking it) is proportional to its cross-
sectional area, that is, the surface area of the fresh wood that would be
exposed if you sawed through it in the middle. Its weight, however, is
proportional to its volume.
How do the volume and cross-sectional area of the longer plank
compare with those of the shorter plank? We have already seen, while
discussing conversions of the units of area and volume, that these quantities
don’t act the way most people naively expect. You might think that the
volume and area of the longer plank would both be doubled compared to
the shorter plank, so they would increase in proportion to each other, and
the longer plank would be equally able to support its weight. You would be
wrong, but Galileo knows that this is a common misconception, so he has


Salviati address the point specifically:
SALVIATI: ...Take, for example, a cube two inches on a side so that each face
has an area of four square inches and the total area, i.e., the sum of the six
faces, amounts to twenty-four square inches; now imagine this cube to be
sawed through three times [with cuts in three perpendicular planes] so as to
divide it into eight smaller cubes, each one inch on the side, each face one
inch square, and the total surface of each cube six square inches instead of
full size twenty-four in the case of the larger cube. It is evident therefore, that the
surface of the little cube is only one-fourth that of the larger, namely, the ratio
of six to twenty-four; but the volume of the solid cube itself is only one-eighth;
the volume, and hence also the weight, diminishes therefore much more
rapidly than the surface... You see, therefore, Simplicio, that I was not mistaken
when ... I said that the surface of a small solid is comparatively greater than
that of a large one.
The same reasoning applies to the planks. Even though they are not
cubes, the large one could be sawed into eight small ones, each with half the
length, half the thickness, and half the width. The small plank, therefore,
has more surface area in proportion to its weight, and is therefore able to
support its own weight while the large one breaks.
Scaling of area and volume for irregularly shaped objects
You probably are not going to believe Galileo’s claim that this has deep
implications for all of nature unless you can be convinced that the same is
3/4 size true for any shape. Every drawing you’ve seen so far has been of squares,
rectangles, and rectangular solids. Clearly the reasoning about sawing things
up into smaller pieces would not prove anything about, say, an egg, which
cannot be cut up into eight smaller egg-shaped objects with half the length.
Is it always true that something half the size has one quarter the surface
area and one eighth the volume, even if it has an irregular shape? Take the
example of a child’s violin. Violins are made for small children in lengths
that are either half or 3/4 of the normal length, accommodating their small
hands. Let’s study the surface area of the front panels of the three violins.
Consider the square in the interior of the panel of the full-size violin. In
the 3/4-size violin, its height and width are both smaller by a factor of 3/4,
half size so the area of the corresponding, smaller square becomes 3/4x3/4=9/16 of
the original area, not 3/4 of the original area. Similarly, the corresponding
square on the smallest violin has half the height and half the width of the
original one, so its area is 1/4 the original area, not half.
The same reasoning works for parts of the panel near the edge, such as
the part that only partially fills in the other square. The entire square scales
down the same as a square in the interior, and in each violin the same
fraction (about 70%) of the square is full, so the contribution of this part to
the total area scales down just the same.
Since any small square region or any small region covering part of a
square scales down like a square object, the entire surface area of an irregu-
larly shaped object changes in the same manner as the surface area of a
square: scaling it down by 3/4 reduces the area by a factor of 9/16, and so
on.
In general, we can see that any time there are two objects with the same
shape, but different linear dimensions (i.e. one looks like a reduced photo of
the other), the ratio of their areas equals the ratio of the squares of their
linear dimensions:


2
A1 L1
= .
A2 L2
Note that it doesn’t matter where we choose to measure the linear size, L, of
an object. In the case of the violins, for instance, it could have been mea-
sured vertically, horizontally, diagonally, or even from the bottom of the left
f-hole to the middle of the right f-hole. We just have to measure it in a
consistent way on each violin. Since all the parts are assumed to shrink or
expand in the same manner, the ratio L1/L2 is independent of the choice of
measurement.
It is also important to realize that it is completely unnecessary to have a
formula for the area of a violin. It is only possible to derive simple formulas
for the areas of certain shapes like circles, rectangles, triangles and so on, but
that is no impediment to the type of reasoning we are using.
Sometimes it is inconvenient to write all the equations in terms of
ratios, especially when more than two objects are being compared. A more
compact way of rewriting the previous equation is
A∝L 2 .
The symbol “∝” means “is proportional to.” Scientists and engineers often
speak about such relationships verbally using the phrases “scales like” or
“goes like,” for instance “area goes like length squared.”
All of the above reasoning works just as well in the case of volume.
Volume goes like length cubed:
V ∝ L3 .
If different objects are made of the same material with the same density,
ρ=m/V, then their masses, m=ρV, are proportional to L3, and so are their
weights. (The symbol for density is ρ, the lower-case Greek letter “rho”.)
An important point is that all of the above reasoning about scaling only
applies to objects that are the same shape. For instance, a piece of paper is
larger than a pencil, but has a much greater surface-to-volume ratio.
One of the first things I learned as a teacher was that students were not
very original about their mistakes. Every group of students tends to come
up with the same goofs as the previous class. The following are some
examples of correct and incorrect reasoning about proportionality.


Example: scaling of the area of a triangle
(a) Question: In fig. (a), the larger triangle has sides twice as long.
How many times greater is its area?
Correct solution #1: Area scales in proportion to the square of the
linear dimensions, so the larger triangle has four times more area
(22=4).
Correct solution #2: You could cut the larger triangle into four of
the smaller size, as shown in fig. (b), so its area is four times greater.
(This solution is correct, but it would not work for a shape like a
circle, which can’t be cut up into smaller circles.)
(b)
Correct solution #3: The area of a triangle is given by
1
A= 2 bh, where b is the base and h is the height. The areas of the
triangles are
1
A1 = 2 b1h1
1
A2 = 2 b2h2
The big triangle has four times more
area than the little one. 1
= 2 (2b1)(2h1)
= 2b1h1
A2/A1 = (2b1h1)/( 1 b1h1)
2
=4
(Although this solution is correct, it is a lot more work than
solution #1, and it can only be used in this case because a triangle is
a simple geometric shape, and we happen to know a formula for its
area.)
1
Correct solution #4: The area of a triangle is A = 2 bh. The
comparison of the areas will come out the same as long as the ratios
of the linear sizes of the triangles is as specified, so let’s just say
b1=1.00 m and b2=2.00 m. The heights are then also h1=1.00 m
and h2=2.00 m, giving areas A1=0.50 m2 and A2=2.00 m2, so A2/
A1=4.00.
(The solution is correct, but it wouldn’t work with a shape for
whose area we don’t have a formula. Also, the numerical calculation
might make the answer of 4.00 appear inexact, whereas solution #1
makes it clear that it is exactly 4.)
1
Incorrect solution: The area of a triangle is A = 2 bh, and if you
plug in b=2.00 m and h=2.00 m, you get A=2.00 m2, so the bigger
triangle has 2.00 times more area. (This solution is incorrect
because no comparison has been made with the smaller triangle.)


(c) Example: scaling of the volume of a sphere
Question: In figure (c), the larger sphere has a radius that is five
times greater. How many times greater is its volume?
Correct solution #1: Volume scales like the third power of the
linear size, so the larger sphere has a volume that is 125 times
greater (53=125).
Correct solution #2: The volume of a sphere is V= 4 πr3, so
3
The big sphere has 125 times more V1 = 4 πr 3
volume than the little one. 3 1

V2 = 4 πr 3
3 2

= 4 π(5r ) 3
1
3

= 500πr 3
1
3

V2/V1 = 500 πr 3 / 4 πr 3
1
3 3 1
= 125
Incorrect solution: The volume of a sphere is V= 4 πr3, so
3

V1 = 4 πr 3
3 1

V2 = 4 πr 3
3 2

= 4 π ⋅ 5r 3
1
3

= 20 πr 3
3 1

V2/V1=( 20 πr 3 )/( 4 πr 3 )
3 1 3 1
=5
(The solution is incorrect because (5r1)3 is not the same as 5r 3 .)
1

S S
(d) The 48-point “S” has 1.78 times
Example: scaling of a more complex shape
Question: The first letter “S” in fig. (d) is in a 36-point font, the
second in 48-point. How many times more ink is required to make
the larger “S”?
more area than the 36-point “S.” Correct solution: The amount of ink depends on the area to be
covered with ink, and area is proportional to the square of the linear
dimensions, so the amount of ink required for the second “S” is
greater by a factor of (48/36)2=1.78.
Incorrect solution: The length of the curve of the second “S” is
longer by a factor of 48/36=1.33, so 1.33 times more ink is
required.
(The solution is wrong because it assumes incorrectly that the width
of the curve is the same in both cases. Actually both the width and
the length of the curve are greater by a factor of 48/36, so the area is
greater by a factor of (48/36)2=1.78.)


Discussion questions
A. A toy fire engine is 1/30 the size of the real one, but is constructed from the
same metal with the same proportions. How many times smaller is its weight?
How many times less red paint would be needed to paint it?
B. Galileo spends a lot of time in his dialog discussing what really happens
when things break. He discusses everything in terms of Aristotle’s now-
discredited explanation that things are hard to break, because if something
breaks, there has to be a gap between the two halves with nothing in between,
at least initially. Nature, according to Aristotle, “abhors a vacuum,” i.e. nature
doesn’t “like” empty space to exist. Of course, air will rush into the gap
immediately, but at the very moment of breaking, Aristotle imagined a vacuum
in the gap. Is Aristotle’s explanation of why it is hard to break things an
experimentally testable statement? If so, how could it be tested
experimentally?

1.3 Scaling Applied to Biology
Organisms of different sizes with the same shape
The first of the following graphs shows the approximate validity of the
proportionality m∝L3 for cockroaches (redrawn from McMahon and
Bonner). The scatter of the points around the curve indicates that some
cockroaches are proportioned slightly differently from others, but in general
the data seem well described by m∝L3. That means that the largest cock-
roaches the experimenter could raise (is there a 4-H prize?) had roughly the
same shape as the smallest ones.
Another relationship that should exist for animals of different sizes
shaped in the same way is that between surface area and body mass. If all
the animals have the same average density, then body mass should be
proportional to the cube of the animal’s linear size, m∝L3, while surface
area should vary proportionately to L2. Therefore, the animals’ surface areas
should be proportional to m2/3. As shown in the second graph, this relation-
ship appears to hold quite well for the dwarf siren, a type of salamander.
Notice how the curve bends over, meaning that the surface area does not
increase as quickly as body mass, e.g. a salamander with eight times more
body mass will have only four times more surface area.
This behavior of the ratio of surface area to mass (or, equivalently, the
ratio of surface area to volume) has important consequences for mammals,
which must maintain a constant body temperature. It would make sense for
the rate of heat loss through the animal’s skin to be proportional to its
surface area, so we should expect small animals, having large ratios of
surface area to volume, to need to produce a great deal of heat in compari-
son to their size to avoid dying from low body temperature. This expecta-
tion is borne out by the data of the third graph, showing the rate of oxygen
consumption of guinea pigs as a function of their body mass. Neither an
animal’s heat production nor its surface area is convenient to measure, but
in order to produce heat, the animal must metabolize oxygen, so oxygen
consumption is a good indicator of the rate of heat production. Since
surface area is proportional to m2/3, the proportionality of the rate of oxygen
consumption to m2/3 is consistent with the idea that the animal needs to
produce heat at a rate in proportion to its surface area. Although the smaller
animals metabolize less oxygen and produce less heat in absolute terms, the
amount of food and oxygen they must consume is greater in proportion to
their own mass. The Etruscan pigmy shrew, weighing in at 2 grams as an


Body mass, m, versus leg
1000 length, L, for the cockroach 1000
Periplaneta americana.
The data points rep-
resent individual
specimens, and the 800
750 curve is a fit to the

surface area (cm2 )
data of the form
body mass (mg)

m=kL 3 , where k is
a constant. 600
Surface
area versus
500 body mass for
dwarf sirens, a
400 species of sala-
mander (Pseudo-
branchus striatus ).
250 The data points
200 represent individual
specimens, and the curve is
a fit of the form A=km2/3 .

0 0
0 1 2 3 0 500 1000
body mass (g)
length of leg segment (mm)

Diameter versus length
8 of the third lumbar
5 vertebrae of adult
African Bovidae
7
(antelopes and oxen).
The smallest animal
oxygen consumption (mL/min)

6 4 represented is the
cat-sized Gunther's
dik-dik, and the
diameter (cm)

5 largest is the
3 850-kg giant
eland. The
4
solid curve is
a fit of the form d=kL 3/2 ,
3 2
and the dashed
line is a linear
Rate of oxygen
fit. (After
2 consumption versus
McMahon and
body mass for guinea 1 Bonner, 1983.)
pigs at rest. The
1
curve is a fit of the
form (rate)=km 2/3 .
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6
body mass (kg) length (cm)

45

adult, is at about the lower size limit for mammals. It must eat continually,
consuming many times its body weight each day to survive.
Changes in shape to accommodate changes in size
Large mammals, such as elephants, have a small ratio of surface area to
volume, and have problems getting rid of their heat fast enough. An
elephant cannot simply eat small enough amounts to keep from producing
excessive heat, because cells need to have a certain minimum metabolic rate
to run their internal machinery. Hence the elephant’s large ears, which add
to its surface area and help it to cool itself. Previously, we have seen several
examples of data within a given species that were consistent with a fixed
shape, scaled up and down in the cases of individual specimens. The
elephant’s ears are an example of a change in shape necessitated by a change
in scale.
Large animals also must be able to support their own weight. Returning
to the example of the strengths of planks of different sizes, we can see that if
the strength of the plank depends on area while its weight depends on
volume, then the ratio of strength to weight goes as follows:
strength/weight ∝ A/V ∝ 1/L .
Galileo’s original drawing, showing Thus, the ability of objects to support their own weights decreases
how larger animals’ bones must be inversely in proportion to their linear dimensions. If an object is to be just
greater in diameter compared to their barely able to support its own weight, then a larger version will have to be
lengths.
proportioned differently, with a different shape.
Since the data on the cockroaches seemed to be consistent with roughly
similar shapes within the species, it appears that the ability to support its
own weight was not the tightest design constraint that Nature was working
under when she designed them. For large animals, structural strength is
important. Galileo was the first to quantify this reasoning and to explain
why, for instance, a large animal must have bones that are thicker in
proportion to their length. Consider a roughly cylindrical bone such as a leg
bone or a vertebra. The length of the bone, L, is dictated by the overall
linear size of the animal, since the animal’s skeleton must reach the animal’s
whole length. We expect the animal’s mass to scale as L3, so the strength of
the bone must also scale as L3. Strength is proportional to cross-sectional
area, as with the wooden planks, so if the diameter of the bone is d, then
d2∝ L3
or
d ∝ L 3/2 .
If the shape stayed the same regardless of size, then all linear dimensions,
including d and L, would be proportional to one another. If our reasoning
holds, then the fact that d is proportional to L3/2, not L, implies a change in
proportions of the bone. As shown in the graph on the previous page, the
vertebrae of African Bovidae follow the rule d ∝ L3/2 fairly well. The
vertebrae of the giant eland are as chunky as a coffee mug, while those of a
Gunther’s dik-dik are as slender as the cap of a pen.


A. Single-celled animals must passively absorb nutrients and oxygen from their
surroundings, unlike humans who have lungs to pump air in and out and a
heart to distribute the oxygenated blood throughout their bodies. Even the cells
composing the bodies of multicellular animals must absorb oxygen from a
nearby capillary through their surfaces. Based on these facts, explain why cells
are always microscopic in size.
B. The reasoning of the previous question would seem to be contradicted by
the fact that human nerve cells in the spinal cord can be as much as a meter
long, although their widths are still very small. Why is this possible?

1.4 Order-of-Magnitude Estimates
It is the mark of an instructed mind to rest satisfied with the degree of preci-
sion that the nature of the subject permits and not to seek an exactness
where only an approximation of the truth is possible.
Aristotle
It is a common misconception that science must be exact. For instance,
in the Star Trek TV series, it would often happen that Captain Kirk would
ask Mr. Spock, “Spock, we’re in a pretty bad situation. What do you think
are our chances of getting out of here?” The scientific Mr. Spock would
answer with something like, “Captain, I estimate the odds as 237.345 to
one.” In reality, he could not have estimated the odds with six significant
figures of accuracy, but nevertheless one of the hallmarks of a person with a
good education in science is the ability to make estimates that are likely to
be at least somewhere in the right ballpark. In many such situations, it is
often only necessary to get an answer that is off by no more than a factor of
ten in either direction. Since things that differ by a factor of ten are said to
differ by one order of magnitude, such an estimate is called an order-of-
magnitude estimate. The tilde, ~, is used to indicate that things are only of
the same order of magnitude, but not exactly equal, as in
odds of survival ~ 100 to one .
The tilde can also be used in front of an individual number to emphasize
that the number is only of the right order of magnitude.
Although making order-of-magnitude estimates seems simple and
natural to experienced scientists, it’s a mode of reasoning that is completely
unfamiliar to most college students. Some of the typical mental steps can be
illustrated in the following example.

Section 1.4 Order-of-Magnitude Estimates 47

Example: Cost of transporting tomatoes
Question: Roughly what percentage of the price of a tomato comes
from the cost of transporting it in a truck?
The following incorrect solution illustrates one of the main ways you can go
wrong in order-of-magnitude estimates.
Incorrect solution: Let’s say the trucker needs to make a $400
profit on the trip. Taking into account her benefits, the cost of gas,
and maintenance and payments on the truck, let’s say the total cost
is more like $2000. I’d guess about 5000 tomatoes would fit in the
back of the truck, so the extra cost per tomato is 40 cents. That
means the cost of transporting one tomato is comparable to the cost
of the tomato itself. Transportation really adds a lot to the cost of
produce, I guess.
The problem is that the human brain is not very good at estimating
area or volume, so it turns out the estimate of 5000 tomatoes fitting in the
truck is way off. That’s why people have a hard time at those contests where
you are supposed to estimate the number of jellybeans in a big jar. Another
example is that most people think their families use about 10 gallons of
water per day, but in reality the average is about 300 gallons per day. When
estimating area or volume, you are much better off estimating linear
dimensions, and computing volume from the linear dimensions. Here’s a
better solution:
Better solution: As in the previous solution, say the cost of the trip
is $2000. The dimensions of the bin are probably 4 m x 2 m x 1 m,
for a volume of 8 m3. Since the whole thing is just an order-of-
magnitude estimate, let’s round that off to the nearest power of ten,
10 m3. The shape of a tomato is complicated, and I don’t know any
formula for the volume of a tomato shape, but since this is just an
estimate, let’s pretend that a tomato is a cube, 0.05 m x 0.05 m x
0.05, for a volume of 1.25x10-4 m3. Since this is just a rough
estimate, let’s round that to 10-4 m3. We can find the total number
of tomatoes by dividing the volume of the bin by the volume of one
tomato: 10 m3 / 10-4 m3 = 105 tomatoes. The transportation cost
per tomato is $2000/105 tomatoes=$0.02/tomato. That means that
transportation really doesn’t contribute very much to the cost of a
tomato.
Approximating the shape of a tomato as a cube is an example of another
1m general strategy for making order-of-magnitude estimates. A similar situa-
tion would occur if you were trying to estimate how many m2 of leather
could be produced from a herd of ten thousand cattle. There is no point in
trying to take into account the shape of the cows’ bodies. A reasonable plan
of attack might be to consider a spherical cow. Probably a cow has roughly
the same surface area as a sphere with a radius of about 1 m, which would
be 4π(1 m)2. Using the well-known facts that pi equals three, and four
times three equals about ten, we can guess that a cow has a surface area of
about 10 m2, so the herd as a whole might yield 105 m2 of leather.


The following list summarizes the strategies for getting a good order-of-
magnitude estimate.
(1) Don’t even attempt more than one significant figure of precision.
(2) Don’t guess area or volume directly. Guess linear dimensions and
get area or volume from them.
(3) When dealing with areas or volumes of objects with complex
shapes, idealize them as if they were some simpler shape, a cube or a
sphere, for example.
(4) Check your final answer to see if it is reasonable. If you estimate
that a herd of ten thousand cattle would yield 0.01 m2 of leather,
then you have probably made a mistake with conversion factors
somewhere.

Section 1.4 Order-of-Magnitude Estimates 49

Summary
Notation
∝ ....................................... is proportional to
~ ........................................ on the order of, is on the order of
Summary
Nature behaves differently on large and small scales. Galileo showed that this results fundamentally from
the way area and volume scale. Area scales as the second power of length, A∝L2, while volume scales as
length to the third power, V∝L3.
An order of magnitude estimate is one in which we do not attempt or expect an exact answer. The main
reason why the uninitiated have trouble with order-of-magnitude estimates is that the human brain does not
intuitively make accurate estimates of area and volume. Estimates of area and volume should be approached
by first estimating linear dimensions, which one’s brain has a feel for.

Homework Problems
1 . How many cubic inches are there in a cubic foot? The answer is not
12.
2. Assume a dog's brain is twice is great in diameter as a cat's, but each
animal's brain cells are the same size and their brains are the same shape. In
addition to being a far better companion and much nicer to come home to,
how many times more brain cells does a dog have than a cat? The answer is
not 2.
3 . The population density of Los Angeles is about 4000 people/km2.
That of San Francisco is about 6000 people/km2. How many times farther
away is the average person's nearest neighbor in LA than in San Francisco?
The answer is not 1.5.
4. A hunting dog's nose has about 10 square inches of active surface. How
is this possible, since the dog's nose is only about 1 in x 1 in x 1 in = 1 in3?
After all, 10 is greater than 1, so how can it fit?
5. Estimate the number of blades of grass on a football field.
6. In a computer memory chip, each bit of information (a 0 or a 1) is stored
in a single tiny circuit etched onto the surface of a silicon chip. A typical
chip stores 64 Mb (megabytes) of data, where a byte is 8 bits. Estimate (a)
the area of each circuit, and (b) its linear size.
7. Suppose someone built a gigantic apartment building, measuring 10 km
x 10 km at the base. Estimate how tall the building would have to be to
have space in it for the entire world's population to live.
8. A hamburger chain advertises that it has sold 10 billion Bongo Burgers.
Estimate the total mass of feed required to raise the cows used to make the
burgers.
9. Estimate the volume of a human body, in cm3.
10 S. How many cm2 is 1 mm2?
11 S. Compare the light-gathering powers of a 3-cm-diameter telescope and
a 30-cm telescope.



12. S. One step on the Richter scale corresponds to a factor of 100 in terms
of the energy absorbed by something on the surface of the Earth, e.g. a
house. For instance, a 9.3-magnitude quake would release 100 times more
energy than an 8.3. The energy spreads out from the epicenter as a wave,
and for the sake of this problem we’ll assume we’re dealing with seismic
waves that spread out in three dimensions, so that we can visualize them as
hemispheres spreading out under the surface of the earth. If a certain 7.6-
magnitude earthquake and a certain 5.6-magnitude earthquake produce the
same amount of vibration where I live, compare the distances from my
house to the two epicenters.
13. In Europe, a piece of paper of the standard size, called A4, is a little
narrower and taller than its American counterpart. The ratio of the height
to the width is the square root of 2, and this has some useful properties. For
instance, if you cut an A4 sheet from left to right, you get two smaller sheets
that have the same proportions. You can even buy sheets of this smaller size,
and they’re called A5. There is a whole series of sizes related in this way, all
with the same proportions. (a) Compare an A5 sheet to an A4 in terms of
area and linear size. (b) The series of paper sizes starts from an A0 sheet,
which has an area of one square meter. Suppose we had a series of boxes
defined in a similar way: the B0 box has a volume of one cubic meter, two
B1 boxes fit exactly inside an B0 box, and so on. What would be the
dimensions of a B0 box?

51

Motion in One
Dimension
I didn’t learn until I was nearly through with college that I could understand a book much better if I
mentally outlined it for myself before I actually began reading. It’s a technique that warns my
brain to get little cerebral file folders ready for the different topics I’m going to learn, and as I’m
reading it allows me to say to myself, “Oh, the reason they’re talking about this now is because
they’re preparing for this other thing that comes later,” or “I don’t need to sweat the details of this
idea now, because they’re going to explain it in more detail later on.”
At this point, you’re about to dive in to the main subjects of this book, which are force and motion.
The concepts you’re going to learn break down into the following three areas:
kinematics — how to describe motion numerically
dynamics — how force affects motion
vectors — a mathematical way of handling the three-dimensional nature of force and motion
Roughly speaking, that’s the order in which we’ll cover these three areas, but the earlier chapters
do contain quite a bit of preparation for the later topics. For instance, even before the present
point in the book you’ve learned about the Newton, a unit of force. The discussion of force
properly belongs to dynamics, which we aren’t tackling head-on for a few more chapters, but I’ve
found that when I teach kinematics it helps to be able to refer to forces now and then to show
why it makes sense to define certain kinematical concepts. And although I don’t explicitly
introduce vectors until ch. 8, the groundwork is being laid for them in earlier chapters.
Here’s a roadmap to the rest of the book:

kinematics dynamics vectors
preliminaries
chapters 0-1

motion in one
dimension
chapters 2-6

motion in three
dimensions
chapters 7-9

gravity:
chapter 10

53

2 Velocity and Relative
Motion
2.1 Types of Motion
If you had to think consciously in order to move your body, you would
be severely disabled. Even walking, which we consider to be no great feat,
requires an intricate series of motions that your cerebrum would be utterly
incapable of coordinating. The task of putting one foot in front of the
other is controlled by the more primitive parts of your brain, the ones that
have not changed much since the mammals and reptiles went their separate
evolutionary ways. The thinking part of your brain limits itself to general
directives such as “walk faster,” or “don’t step on her toes,” rather than
micromanaging every contraction and relaxation of the hundred or so
muscles of your hips, legs, and feet.
Physics is all about the conscious understanding of motion, but we’re
obviously not immediately prepared to understand the most complicated
types of motion. Instead, we’ll use the divide-and-conquer technique.
Rotation.
We’ll first classify the various types of motion, and then begin our campaign
with an attack on the simplest cases. To make it clear what we are and are
not ready to consider, we need to examine and define carefully what types
of motion can exist.
Rigid-body motion distinguished from motion that changes
an object’s shape
Nobody, with the possible exception of Fred Astaire, can simply glide
forward without bending their joints. Walking is thus an example in which
there is both a general motion of the whole object and a change in the shape
of the object. Another example is the motion of a jiggling water balloon as
it flies through the air. We are not presently attempting a mathematical
description of the way in which the shape of an object changes. Motion
Simultaneous rotation and
without a change in shape is called rigid-body motion. (The word “body”
motion through space.
is often used in physics as a synonym for “object.”)
Center-of-mass motion as opposed to rotation
A ballerina leaps into the air and spins around once before landing. We
feel intuitively that her rigid-body motion while her feet are off the ground
consists of two kinds of motion going on simultaneously: a rotation and a
motion of her body as a whole through space, along an arc. It is not
immediately obvious, however, what is the most useful way to define the
distinction between rotation and motion through space. Imagine that you
attempt to balance a chair and it falls over. One person might say that the
One person might say that the
tipping chair was only rotating only motion was a rotation about the chair’s point of contact with the floor,
in a circle about its point of but another might say that there was both rotation and motion down and
contact with the floor, but to the side.
another could describe it as
having both rotation and
motion through space.

54 Chapter 2 Velocity and Relative Motion

The leaping dancer’s motion is
complicated, but the motion of her
center of mass
center of mass is simple.

No matter what point you hang the It turns out that there is one particularly natural and useful way to make
pear from, the string lines up with the a clear definition, but it requires a brief digression. Every object has a
pear’s center of mass. The center of balance point, referred to in physics as the center of mass. For a two-
mass can therefore be defined as the
intersection of all the lines made by dimensional object such as a cardboard cutout, the center of mass is the
hanging the pear in this way. Note that point at which you could hang the object from a string and make it balance.
the X in the figure should not be In the case of the ballerina (who is likely to be three-dimensional unless her
interpreted as implying that the center diet is particularly severe), it might be a point either inside or outside her
of mass is on the surface — it is
actually inside the pear.
body, depending on how she holds her arms. Even if it is not practical to
attach a string to the balance point itself, the center of mass can be defined
as shown in the figure on the left.
Why is the center of mass concept relevant to the question of classifying
rotational motion as opposed to motion through space? As illustrated in
the figure above, it turns out that the motion of an object’s center of mass is
nearly always far simpler than the motion of any other part of the object.
The motion of an object’s center of The ballerina’s body is a large object with a complex shape. We might
mass is usually much simpler than the expect that her motion would be much more complicated that the motion
motion of any other point on it.
of a small, simply-shaped object, say a marble, thrown up at the same angle
as the angle at which she leapt. But it turns out that the motion of the
ballerina’s center of mass is exactly the same as the motion of the marble.
That is, the motion of the center of mass is the same as the motion the
ballerina would have if all her mass was concentrated at a point. By restrict-
ing our attention to the motion of the center of mass, we can therefore
simplify things greatly.
The same leaping dancer, viewed from
above. Her center of mass traces a
straight line, but a point away from her
center of mass, such as her elbow,
traces the much more complicated
path shown by the dots.
We can now replace the ambiguous idea of “motion as a whole through
space” with the more useful and better defined concept of “center-of-mass
motion.” The motion of any rigid body can be cleanly split into rotation
and center-of-mass motion. By this definition, the tipping chair does have
both rotational and center-of-mass motion. Concentrating on the center of

Section 2.1 Types of Motion 55

mass motion allows us to make a simplified model of the motion, as if a
complicated object like a human body was just a marble or a point-like
particle. Science really never deals with reality; it deals with models of
geometrical reality.
center
Note that the word “center” in “center of mass” is not meant to imply
that the center of mass must lie at the geometrical center of an object. A car
center of mass wheel that has not been balanced properly has a center of mass that does
not coincide with its geometrical center. An object such as the human body
does not even have an obvious geometrical center.
An improperly balanced wheel has a It can be helpful to think of the center of mass as the average location of
center of mass that is not at its
all the mass in the object. With this interpretation, we can see for example
geometric center. When you get a new
tire, the mechanic clamps little weights that raising your arms above your head raises your center of mass, since the
to the rim to balance the wheel.

A fixed point on the dancer’s body
follows a trajectory that is flatter than
what we expect, creating an illusion
of flight.

center of mass
fixed point on dancer's body

higher position of the arms’ mass raises the average.
Ballerinas and professional basketball players can create an illusion of
flying horizontally through the air because our brains intuitively expect
them to have rigid-body motion, but the body does not stay rigid while
executing a grand jete or a slam dunk. The legs are low at the beginning
and end of the jump, but come up higher at the middle. Regardless of what
the limbs do, the center of mass will follow the same arc, but the low
position of the legs at the beginning and end means that the torso is higher
compared to the center of mass, while in the middle of the jump it is lower
compared to the center of mass. Our eye follows the motion of the torso
and tries to interpret it as the center-of-mass motion of a rigid body. But
since the torso follows a path that is flatter than we expect, this attempted
center
interpretation fails, and we experience an illusion that the person is flying
of mass horizontally. Another interesting example from the sports world is the high
The high-jumper’s body passes over
jump, in which the jumper’s curved body passes over the bar, but the center
the bar, but his center of mass passes of mass passes under the bar! Here the jumper lowers his legs and upper
under it. body at the peak of the jump in order to bring his waist higher compared to
Photo by Dunia Young.
the center of mass.
Later in this course, we’ll find that there are more fundamental reasons
(based on Newton’s laws of motion) why the center of mass behaves in such
a simple way compared to the other parts of an object. We’re also postpon-
ing any discussion of numerical methods for finding an object’s center of
mass. Until later in the course, we will only deal with the motion of objects’


centers of mass.
Center-of-mass motion in one dimension
In addition to restricting our study of motion to center-of-mass motion,
we will begin by considering only cases in which the center of mass moves
along a straight line. This will include cases such as objects falling straight
down, or a car that speeds up and slows down but does not turn.
Note that even though we are not explicitly studying the more complex
aspects of motion, we can still analyze the center-of-mass motion while
ignoring other types of motion that might be occurring simultaneously .
For instance, if a cat is falling out of a tree and is initially upside-down, it
goes through a series of contortions that bring its feet under it. This is
definitely not an example of rigid-body motion, but we can still analyze the
motion of the cat’s center of mass just as we would for a dropping rock.
Self-Check
Consider a person running, a person pedaling on a bicycle, a person coasting
on a bicycle, and a person coasting on ice skates. In which cases is the
center-of-mass motion one-dimensional? Which cases are examples of rigid-
body motion?

2.2 Describing Distance and Time
Center-of-mass motion in one dimension is particularly easy to deal
with because all the information about it can be encapsulated in two
variables: x, the position of the center of mass relative to the origin, and t,
which measures a point in time. For instance, if someone supplied you with
a sufficiently detailed table of x and t values, you would know pretty much
all there was to know about the motion of the object’s center of mass.
A point in time as opposed to duration
In ordinary speech, we use the word “time” in two different senses,
which are to be distinguished in physics. It can be used, as in “a short time”
or “our time here on earth,” to mean a length or duration of time, or it can
be used to indicate a clock reading, as in “I didn’t know what time it was,”
or “now’s the time.” In symbols, t is ordinarily used to mean a point in
time, while ∆t signifies an interval or duration in time. The capital Greek
letter delta, ∆, means “the change in...,” i.e. a duration in time is the change
or difference between one clock reading and another. The notation ∆t does
not signify the product of two numbers, ∆ and t, but rather one single
number, ∆t. If a matinee begins at a point in time t=1 o’clock and ends at
t=3 o’clock, the duration of the movie was the change in t,
∆t = 3 hours - 1 hour = 2 hours .
To avoid the use of negative numbers for ∆t, we write the clock reading
“after” to the left of the minus sign, and the clock reading “before” to the
right of the minus sign. A more specific definition of the delta notation is
therefore that delta stands for “after minus before.”
Even though our definition of the delta notation guarantees that ∆t is
positive, there is no reason why t can’t be negative. If t could not be nega-
tive, what would have happened one second before t=0? That doesn’t mean
Coasting on a bike and coasting on skates give one-dimensional center-of-mass motion, but running and pedaling
require moving body parts up and down, which makes the center of mass move up and down. The only example of
rigid-body motion is coasting on skates. (Coasting on a bike is not rigid-body motion, because the wheels twist.)

Section 2.2 Describing Distance and Time 57

that time “goes backward” in the sense that adults can shrink into infants
and retreat into the womb. It just means that we have to pick a reference
point and call it t=0, and then times before that are represented by negative
values of t.
Although a point in time can be thought of as a clock reading, it is
usually a good idea to avoid doing computations with expressions such as
“2:35” that are combinations of hours and minutes. Times can instead be
expressed entirely in terms of a single unit, such as hours. Fractions of an
hour can be represented by decimals rather than minutes, and similarly if a
problem is being worked in terms of minutes, decimals can be used instead
of seconds.
Self-Check
Of the following phrases, which refer to points in time, which refer to time
intervals, and which refer to time in the abstract rather than as a measurable
number?
(a) “The time has come.”
(b) “Time waits for no man.”
(c) “The whole time, he had spit on his chin.”

Position as opposed to change in position
As with time, a distinction should be made between a point in space,
symbolized as a coordinate x, and a change in position, symbolized as ∆x.
As with t, x can be negative. If a train is moving down the tracks, not
only do you have the freedom to choose any point along the tracks and call
it x=0, but it’s also up to you to decide which side of the x=0 point is
positive x and which side is negative x.
Since we’ve defined the delta notation to mean “after minus before,” it
is possible that ∆x will be negative, unlike ∆t which is guaranteed to be
positive. Suppose we are describing the motion of a train on tracks linking
Tucson and Chicago. As shown in the figure, it is entirely up to you to
decide which way is positive.

Chicago Chicago

Joplin Joplin

Enid ∆x0 Enid ∆x0

x0 x0
x=0 x=0
x0 x0
Tucson Tucson

Two equally valid ways of describing the motion of a train from Tucson to
Chicago. In the first example, the train has a positive ∆x as it goes from Enid to
Joplin. In the second example, the same train going forward in the same
direction has a negative ∆x.

(a) a point in time; (b) time in the abstract sense; (c) a time interval


Note that in addition to x and ∆x, there is a third quantity we could
define, which would be like an odometer reading, or actual distance
traveled. If you drive 10 miles, make a U-turn, and drive back 10 miles,
then your ∆x is zero, but your car’s odometer reading has increased by 20
miles. However important the odometer reading is to car owners and used
car dealers, it is not very important in physics, and there is not even a
standard name or notation for it. The change in position, ∆x, is more useful
because it is so much easier to calculate: to compute ∆x, we only need to
know the beginning and ending positions of the object, not all the informa-
tion about how it got from one position to the other.
Self-Check
A ball hits the floor, bounces to a height of one meter, falls, and hits the floor
again. Is the ∆x between the two impacts equal to zero, one, or two meters?

Frames of reference
The example above shows that there are two arbitrary choices you have
to make in order to define a position variable, x. You have to decide where
to put x=0, and also which direction will be positive. This is referred to as
choosing a coordinate system or choosing a frame of reference. (The two terms
are nearly synonymous, but the first focuses more on the actual x variable,
while the second is more of a general way of referring to one’s point of
view.) As long as you are consistent, any frame is equally valid. You just
don’t want to change coordinate systems in the middle of a calculation.
Have you ever been sitting in a train in a station when suddenly you
notice that the station is moving backward? Most people would describe the
situation by saying that you just failed to notice that the train was moving
— it only seemed like the station was moving. But this shows that there is
yet a third arbitrary choice that goes into choosing a coordinate system:
valid frames of reference can differ from each other by moving relative to
one another. It might seem strange that anyone would bother with a
coordinate system that was moving relative to the earth, but for instance the
frame of reference moving along with a train might be far more convenient
for describing things happening inside the train.

Zero, because the “after” and “before” values of x are the same.

Section 2.2 Describing Distance and Time 59

2.3 Graphs of Motion; Velocity.
Motion with constant velocity
In example (a), an object is moving at constant speed in one direction.
30
We can tell this because every two seconds, its position changes by five
25 ∆t meters.
20 In algebra notation, we’d say that the graph of x vs. t shows the same
∆x
x change in position, ∆x=5.0 m, over each interval of ∆t=2.0 s. The object’s
(m) 15
velocity or speed is obtained by calculating v=∆x/∆t=(5.0 m)/(2.0 s)=2.5 m/
10
s. In graphical terms, the velocity can be interpreted as the slope of the line.
5 Since the graph is a straight line, it wouldn’t have mattered if we’d taken a
0 longer time interval and calculated v=∆x/∆t=(10.0 m)/(4.0 s). The answer
0 2 4 6 8 10 would still have been the same, 2.5 m/s.
t (s)
Note that when we divide a number that has units of meters by another
(a) Motion with constant velocity.
number that has units of seconds, we get units of meters per second, which
can be written m/s. This is another case where we treat units as if they were
algebra symbols, even though they’re not.
In example (b), the object is moving in the opposite direction: as time
progresses, its x coordinate decreases. Recalling the definition of the ∆
30 notation as “after minus before,” we find that ∆t is still positive, but ∆x
must be negative. The slope of the line is therefore negative, and we say
25 ∆t
that the object has a negative velocity, v=∆x/∆t=(-5.0 m)/(2.0 s)=-2.5 m/s.
20
∆x We’ve already seen that the plus and minus signs of ∆x values have the
x
interpretation of telling us which direction the object moved. Since ∆t is
(m) 15
10
always positive, dividing by ∆t doesn’t change the plus or minus sign, and
the plus and minus signs of velocities are to be interpreted in the same way.
5 In graphical terms, a positive slope characterizes a line that goes up as we go
0 to the right, and a negative slope tells us that the line went down as we went
0 2 4 6 8 10 to the right.
t (s)
Motion with changing velocity
(b) Motion that decreases x is
represented with negative values of ∆x
Now what about a graph like example (c)? This might be a graph of a
and v. car’s motion as the driver cruises down the freeway, then slows down to look
at a car crash by the side of the road, and then speeds up again, disap-
pointed that there is nothing dramatic going on such as flames or babies
trapped in their car seats. (Note that we are still talking about one-dimen-
sional motion. Just because the graph is curvy doesn’t mean that the car’s
path is curvy. The graph is not like a map, and the horizontal direction of
the graph represents the passing of time, not distance.)
30
25
Example (c) is similar to example (a) in that the object moves a total of
25.0 m in a period of 10.0 s, but it is no longer true that it makes the same
20 amount of progress every second. There is no way to characterize the entire
x
(m)
15 graph by a certain velocity or slope, because the velocity is different at every
10 moment. It would be incorrect to say that because the car covered 25.0 m
in 10.0 s, its velocity was 2.5 m/s . It moved faster than that at the begin-
5
ning and end, but slower in the middle. There may have been certain
0 instants at which the car was indeed going 2.5 m/s, but the speedometer
0 2 4 6 8 10 swept past that value without “sticking,” just as it swung through various
t (s) other values of speed. (I definitely want my next car to have a speedometer
(c) Motion with changing velocity. calibrated in m/s and showing both negative and positive values.)


30 We assume that our speedometer tells us what is happening to the speed
∆t of our car at every instant, but how can we define speed mathematically in a
25
∆x case like this? We can’t just define it as the slope of the curvy graph, because
20
a curve doesn’t have a single well-defined slope as does a line. A mathemati-
x
(m)
15 cal definition that corresponded to the speedometer reading would have to
10 be one that attached a different velocity value to a single point on the curve,
i.e. a single instant in time, rather than to the entire graph. If we wish to
5
define the speed at one instant such as the one marked with a dot, the best
0 way to proceed is illustrated in (d), where we have drawn the line through
0 2 4 6 8 10 that point called the tangent line, the line that “hugs the curve.” We can
t (s) then adopt the following definition of velocity:
(d) The velocity at any given moment
is defined as the slope of the tangent definition of velocity
line through the relevant point on the The velocity of an object at any given moment is the slope of
graph.
the tangent line through the relevant point on its x-t graph.

One interpretation of this definition is that the velocity tells us how many
meters the object would have traveled in one second, if it had continued
moving at the same speed for at least one second. To some people the
graphical nature of this definition seems “inaccurate” or “not mathemati-
cal.” The equation v=∆x/∆t by itself, however, is only valid if the velocity is
30
constant, and so cannot serve as a general definition.
25
∆x Example
20 Question: What is the velocity at the point shown with a dot on
x
15 the graph?
(m)
Solution: First we draw the tangent line through that point. To
10
find the slope of the tangent line, we need to pick two points on
5 ∆t=4.0 s it. Theoretically, the slope should come out the same regardless
of which two points we picked, but in practical terms we’ll be able
0
to measure more accurately if we pick two points fairly far apart,
0 2 4 6 8 10 such as the two white diamonds. To save work, we pick points
t (s)
that are directly above labeled points on the t axis, so that ∆t=4.0
Example: finding the velocity at the s is easy to read off. One diamond lines up with x≈17.5 m, the
point indicated with the dot. other with x≈26.5 m, so ∆x=9.0 m. The velocity is ∆x/∆t=2.2 m/s.
Conventions about graphing
The placement of t on the horizontal axis and x on the upright axis may
seem like an arbitrary convention, or may even have disturbed you, since
30
your algebra teacher always told you that x goes on the horizontal axis and y
25 goes on the upright axis. There is a reason for doing it this way, however.
20 In example (e), we have an object that reverses its direction of motion twice.
x It can only be in one place at any given time, but there can be more than
(m) 15 one time when it is at a given place. For instance, this object passed
10 through x=17 m on three separate occasions, but there is no way it could
5 have been in more than one place at t=5.0 s. Resurrecting some terminol-
ogy you learned in your trigonometry course, we say that x is a function of
0
t, but t is not a function of x. In situations such as this, there is a useful
0 2 4 6 8 10
convention that the graph should be oriented so that any vertical line passes
t (s)
through the curve at only one point. Putting the x axis across the page and
(e) Reversing the direction of t upright would have violated this convention. To people who are used to
motion. interpreting graphs, a graph that violates this convention is as annoying as

Section 2.3 Graphs of Motion; Velocity. 61

fingernails scratching on a chalkboard. We say that this is a graph of “x
versus t.” If the axes were the other way around, it would be a graph of “t
versus x.” I remember the “versus” terminology by visualizing the labels on
the x and t axes and remembering that when you read, you go from left to
right and from top to bottom.
A. An ant walks forward, pauses, then runs quickly ahead. It then suddenly
reverses direction and walks slowly back the way it came. Which graph could
represent its motion?

1 2 3
x x x

t t t
4 5 6
x x x

t t t

B. The figure shows a sequence of positions for two racing tractors. Compare
the tractors’ velocities as the race progresses. When do they have the same
velocity?

t=0 s t=1 s t=2 s t=3 s t=4 s t=5 s t=6 s t=7 s

t=0 s t=1 s t=2 s t=3 s t=4 s t=5 s t=6 s t=7 s

C. If an object had a straight-line motion graph with ∆x=0 and ∆t≠0, what would
be true about its velocity? What would this look like on a graph? What about
∆t=0 and ∆x≠0?
D. If an object has a wavy motion graph like the one in example (e) on the
previous page, which are the points at which the object reverses its direction?
What is true about the object’s velocity at these points?
E. Discuss anything unusual about the following three graphs.


1 2 3
x x x

t t t
F. I have been using the term “velocity” and avoiding the more common
English word “speed,” because some introductory physics texts define them to
x mean different things. They use the word “speed,” and the symbol “s” to mean
the absolute value of the velocity, s=|v|. Although I have thrown in my lot with
the minority of books that don’t emphasize this distinction in technical
vocabulary, there are clearly two different concepts here. Can you think of an
t example of a graph of x vs. t in which the object has constant speed, but not
constant velocity?
Discussion question G. G. In the graph on the left, describe how the object’s velocity changes.
H. Two physicists duck out of a boring scientific conference to go get beer. On
the way to the bar, they witness an accident in which a pedestrian is injured by
a hit-and-run driver. A criminal trial results, and they must testify. In her
testimony, Dr. Transverz Waive says, “The car was moving along pretty fast, I’d
say the velocity was +40 mi/hr. They saw the old lady too late, and even
though they slammed on the brakes they still hit her before they stopped.
Then they made a U turn and headed off at a velocity of about -20 mi/hr, I’d
say.” Dr. Longitud N.L. Vibrasheun says, “He was really going too fast, maybe
his velocity was -35 or -40 mi/hr. After he hit Mrs. Hapless, he turned around
and left at a velocity of, oh, I’d guess maybe +20 or +25 mi/hr.” Is their
testimony contradictory? Explain.

Section 2.3 Graphs of Motion; Velocity. 63

2.4 The Principle of Inertia
Physical effects relate only to a change in velocity
Consider two statements that were at one time made with the utmost
seriousness:
People like Galileo and Copernicus who say the earth is rotating must be
crazy. We know the earth can’t be moving. Why, if the earth was really
turning once every day, then our whole city would have to be moving
hundreds of leagues in an hour. That’s impossible! Buildings would shake
on their foundations. Gale-force winds would knock us over. Trees would fall
down. The Mediterranean would come sweeping across the east coasts of
Spain and Italy. And furthermore, what force would be making the world
turn?
All this talk of passenger trains moving at forty miles an hour is sheer
hogwash! At that speed, the air in a passenger compartment would all be
forced against the back wall. People in the front of the car would suffocate,
and people at the back would die because in such concentrated air, they
wouldn’t be able to expel a breath.
Some of the effects predicted in the first quote are clearly just based on
a lack of experience with rapid motion that is smooth and free of vibration.
But there is a deeper principle involved. In each case, the speaker is assum-
ing that the mere fact of motion must have dramatic physical effects. More
subtly, they also believe that a force is needed to keep an object in motion:
the first person thinks a force would be needed to maintain the earth’s
rotation, and the second apparently thinks of the rear wall as pushing on
the air to keep it moving.
Common modern knowledge and experience tell us that these people’s
predictions must have somehow been based on incorrect reasoning, but it is
not immediately obvious where the fundamental flaw lies. It’s one of those
things a four-year-old could infuriate you by demanding a clear explanation
of. One way of getting at the fundamental principle involved is to consider
how the modern concept of the universe differs from the popular concep-
tion at the time of the Italian Renaissance. To us, the word “earth” implies a
planet, one of the nine planets of our solar system, a small ball of rock and
dirt that is of no significance to anyone in the universe except for members
of our species, who happen to live on it. To Galileo’s contemporaries,
however, the earth was the biggest, most solid, most important thing in all
of creation, not to be compared with the wandering lights in the sky known
as planets. To us, the earth is just another object, and when we talk loosely
about “how fast” an object such as a car “is going,” we really mean the car-
object’s velocity relative to the earth-object.
Motion is relative
There is nothing special about motion
or lack of motion relative to the planet According to our modern world-view, it really isn’t that reasonable to
earth. expect that a special force should be required to make the air in the train
have a certain velocity relative to our planet. After all, the “moving” air in
the “moving” train might just happen to have zero velocity relative to some
other planet we don’t even know about. Aristotle claimed that things
“naturally” wanted to be at rest, lying on the surface of the earth. But
experiment after experiment has shown that there is really nothing so


(a) (b) (c)

(d) (e) (f)

This Air Force doctor volunteered to ride a rocket sled as a medical experiment. The obvious effects on
his head and face are not because of the sled's speed but because of its rapid changes in speed: increasing
in (b) and (c), and decreasing in (e) and (f).In (d) his speed is greatest, but because his speed is not
increasing or decreasing very much at this moment, there is little effect on him.

special about being at rest relative to the earth. For instance, if a mattress
falls out of the back of a truck on the freeway, the reason it rapidly comes to
rest with respect to the planet is simply because of friction forces exerted by
the asphalt, which happens to be attached to the planet.
Galileo’s insights are summarized as follows:

The Principle of Inertia
No force is required to maintain motion with constant velocity
in a straight line, and absolute motion does not cause any
observable physical effects.

There are many examples of situations that seem to disprove the
principle of inertia, but these all result from forgetting that friction is a
force. For instance, it seems that a force is needed to keep a sailboat in
motion. If the wind stops, the sailboat stops too. But the wind’s force is not
the only force on the boat; there is also a frictional force from the water. If
the sailboat is cruising and the wind suddenly disappears, the backward
frictional force still exists, and since it is no longer being counteracted by
the wind’s forward force, the boat stops. To disprove the principle of inertia,
we would have to find an example where a moving object slowed down
even though no forces whatsoever were acting on it.

Section 2.4 The Principle of Inertia 65

Self-Check
What is incorrect about the following supposed counterexamples to the
principle of inertia?
(1) When astronauts blast off in a rocket, their huge velocity does cause
a physical effect on their bodies — they get pressed back into their
seats, the flesh on their faces gets distorted, and they have a hard time
lifting their arms.
(2) When you’re driving in a convertible with the top down, the wind in
your face is an observable physical effect of your absolute motion.
A. A passenger on a cruise ship finds, while the ship is docked, that he can
leap off of the upper deck and just barely make it into the pool on the lower
deck. If the ship leaves dock and is cruising rapidly, will this adrenaline junkie
still be able to make it?
B. You are a passenger in the open basket hanging under a helium balloon.
The balloon is being carried along by the wind at a constant velocity. If you are
holding a flag in your hand, will the flag wave? If so, which way? [Based on a
question from PSSC Physics.]
C. Aristotle stated that all objects naturally wanted to come to rest, with the
unspoken implication that “rest” would be interpreted relative to the surface of
the earth. Suppose we could transport Aristotle to the moon, put him in a
space suit, and kick him out the door of the spaceship and into the lunar
landscape. What would he expect his fate to be in this situation? If intelligent

ship's direction
of motion

pool

Discussion question A. Discussion question B.

creatures inhabited the moon, and one of them independently came up with
the equivalent of Aristotelian physics, what would they think about objects
coming to rest?
D. The bottle is sitting on a level table in a train’s dining car, but the surface of
the beer is tilted. What can you infer about the motion of the train?

Discussion question D.

(1) The effect only occurs during blastoff, when their velocity is changing. Once the rocket engines stop firing, their
velocity stops changing, and they no longer feel any effect. (2) It is only an observable effect of your motion relative
to the air.


2.5 Addition of Velocities
Addition of velocities to describe relative motion
Since absolute motion cannot be unambiguously measured, the only
way to describe motion unambiguously is to describe the motion of one
object relative to another. Symbolically, we can write vPQ for the velocity of
object P relative to object Q.
Relative velocities Velocities measured with respect to different reference points can be
add together. compared by addition. In the figure below, the ball’s velocity relative to the
couch equals the ball’s velocity relative to the truck plus the truck’s velocity
relative to the couch:
v BC = v BT + v TC
= 5 cm/s + 10 cm/s
The same equation can be used for any combination of three objects, just
by substituting the relevant subscripts for B, T, and C. Just remember to
write the equation so that the velocities being added have the same sub-
script twice in a row. In this example, if you read off the subscripts going
from left to right, you get BC...=...BTTC. The fact that the two “inside”
subscripts on the right are the same means that the equation has been set up
correctly. Notice how subscripts on the left look just like the subscripts on
the right, but with the two T’s eliminated.

In one second, Green Dino and the Purple Dino and the couch both
truck both moved forward 10 cm, so their moved backward 10 cm in 1 s, so they
velocity was 10 cm/s. The ball moved had a velocity of -10 cm/s. During the same
forward 15 cm, so it had v=15 cm/s. period of time, the ball got 5 cm closer to
me, so it was going +5 cm/s.

These two highly competent physicists disagree on absolute velocities, but they would agree on relative
velocities. Purple Dino considers the couch to be at rest, while Green Dino thinks of the truck as being at rest.
They agree, however, that the truck’s velocity relative to the couch is vTC=10 cm/s, the ball’s velocity relative
to the truck is vBT=5 cm/s, and the ball’s velocity relative to the couch is vBC=vBT+vTC=15 cm/s.

Section 2.5 Addition of Velocities 67

Negative velocities in relative motion
My discussion of how to interpret positive and negative signs of velocity
may have left you wondering why we should bother. Why not just make
velocity positive by definition? The original reason why negative numbers
were invented was that bookkeepers decided it would be convenient to use
the negative number concept for payments to distinguish them from
receipts. It was just plain easier than writing receipts in black and payments
in red ink. After adding up your month’s positive receipts and negative
payments, you either got a positive number, indicating profit, or a negative
number, showing a loss. You could then show the that total with a high-
tech “+” or “-” sign, instead of looking around for the appropriate bottle of
ink.
Nowadays we use positive and negative numbers for all kinds of things,
but in every case the point is that it makes sense to add and subtract those
things according to the rules you learned in grade school, such as “minus a
If you consistently label velocities as positive minus makes a plus, why this is true we need not discuss.” Adding velocities
or negative depending on their directions,
then adding velocities will also give signs has the significance of comparing relative motion, and with this interpreta-
that consistently relate to direction. tion negative and positive velocities can used within a consistent framework.
For example, the truck’s velocity relative to the couch equals the truck’s
velocity relative to the ball plus the ball’s velocity relative to the couch:
v TC = v TB + v BC
= –5 cm/s + 15 cm/s
= 10 cm/s
If we didn’t have the technology of negative numbers, we would have
had to remember a complicated set of rules for adding velocities: (1) if the
two objects are both moving forward, you add, (2) if one is moving forward
and one is moving backward, you subtract, but (3) if they’re both moving
backward, you add. What a pain that would have been.

A. Interpret the general rule v AB =–v BA in words.

B. If we have a specific situation where v AB+ v BC = 0 , what is going on?


2.6 Graphs of Velocity Versus Time
Since changes in velocity play such a prominent role in physics, we need
a better way to look at changes in velocity than by laboriously drawing
tangent lines on x-versus-t graphs. A good method is to draw a graph of
velocity versus time. The examples on the left show the x-t and v-t graphs
that might be produced by a car starting from a traffic light, speeding up,
20
cruising for a while at constant speed, and finally slowing down for a stop
sign. If you have an air freshener hanging from your rear-view mirror, then
x you will see an effect on the air freshener during the beginning and ending
(m) periods when the velocity is changing, but it will not be tilted during the
period of constant velocity represented by the flat plateau in the middle of
the v-t graph.
0 Students often mix up the things being represented on these two types
of graphs. For instance, many students looking at the top graph say that
the car is speeding up the whole time, since “the graph is becoming greater.”
v 2 What is getting greater throughout the graph is x, not v.
(m/s) Similarly, many students would look at the bottom graph and think it
showed the car backing up, because “it’s going backwards at the end.” But
what is decreasing at the end is v, not x. Having both the x-t and v-t graphs
0 in front of you like this is often convenient, because one graph may be
0 4 8 easier to interpret than the other for a particular purpose. Stacking them
t (s) like this means that corresponding points on the two graphs’ time axes are
lined up with each other vertically. However, one thing that is a little
counterintuitive about the arrangement is that in a situation like this one
involving a car, one is tempted to visualize the landscape stretching along
the horizontal axis of one of the graphs. The horizontal axes, however,
represent time, not position. The correct way to visualize the landscape is
by mentally rotating the horizon 90 degrees counterclockwise and imagin-
ing it stretching along the upright axis of the x-t graph, which is the only
axis that represents different positions in space.

2.7 ∫ Applications of Calculus
The integral symbol, ∫, in the heading for this section indicates that it is
meant to be read by students in calculus-based physics. Students in an
algebra-based physics course should skip these sections. The calculus-related
sections in this book are meant to be usable by students who are taking
calculus concurrently, so at this early point in the physics course I do not
assume you know any calculus yet. This section is therefore not much more
than a quick preview of calculus, to help you relate what you’re learning in
the two courses.
Newton was the first person to figure out the tangent-line definition of
velocity for cases where the x-t graph is nonlinear. Before Newton, nobody
had conceptualized the description of motion in terms of x-t and v-t graphs.
In addition to the graphical techniques discussed in this chapter, Newton
also invented a set of symbolic techniques called calculus. If you have an
equation for x in terms of t, calculus allows you, for instance, to find an
equation for v in terms of t. In calculus terms, we say that the function v(t)

Section 2.6 Graphs of Velocity Versus Time 69

is the derivative of the function x(t). In other words, the derivative of a
function is a new function that tells how rapidly the original function was
changing. We now use neither Newton’s name for his technique (he called it
“the method of fluxions”) nor his notation. The more commonly used
notation is due to Newton’s German contemporary Leibnitz, whom the
English accused of plagiarizing the calculus from Newton. In the Leibnitz
notation, we write

v = dx
dt
to indicate that the function v(t) equals the slope of the tangent line of the
graph of x(t) at every time t. The Leibnitz notation is meant to evoke the
delta notation, but with a very small time interval. Because the dx and dt are
thought of as very small ∆x’s and ∆t’s, i.e. very small differences, the part of
calculus that has to do with derivatives is called differential calculus.
Differential calculus consists of three things:
• The concept and definition of the derivative, which is covered in
this book, but which will be discussed more formally in your math
course.
• The Leibnitz notation described above, which you’ll need to get
more comfortable with in your math course.
• A set of rules for that allows you to find an equation for the
derivative of a given function. For instance, if you happened to
have a situation where the position of an object was given by the
equation x=2t7, you would be able to use those rules to find dx/
dt=14t6. This bag of tricks is covered in your math course.


Summary
Selected Vocabulary
center of mass .................... the balance point of an object
velocity .............................. the rate of change of position; the slope of the tangent line on an x-t
graph.
Notation
x ........................................ a point in space
t ........................................ a point in time, a clock reading
∆ ....................................... “change in;” the value of a variable afterwards minus its value before
∆x ..................................... a distance, or more precisely a change in x, which may be less than the
distance traveled; its plus or minus sign indicates direction
∆t ...................................... a duration of time
v ........................................ velocity
vAB .................................................... the velocity of object A relative to object B
Standard Terminology Avoided in This Book
displacement ..................... a name for the symbol ∆x.
speed ................................. the absolute value of the velocity, i.e. the velocity stripped of any informa-
tion about its direction
Summary
An object’s center of mass is the point at which it can be balanced. For the time being, we are studying the
mathematical description only of the motion of an object’s center of mass in cases restricted to one dimension.
The motion of an object’s center of mass is usually far simpler than the motion of any of its other parts.
It is important to distinguish location, x, from distance, ∆x, and clock reading, t, from time interval ∆t. When
an object’s x-t graph is linear, we define its velocity as the slope of the line, ∆x/∆t. When the graph is curved,
we generalize the definition so that the velocity is the slope of the tangent line at a given point on the graph.
Galileo’s principle of inertia states that no force is required to maintain motion with constant velocity in a
straight line, and absolute motion does not cause any observable physical effects. Things typically tend to
reduce their velocity relative to the surface of our planet only because they are physically rubbing against the
planet (or something attached to the planet), not because there is anything special about being at rest with
respect to the earth’s surface. When it seems, for instance, that a force is required to keep a book sliding
across a table, in fact the force is only serving to cancel the contrary force of friction.
Absolute motion is not a well-defined concept, and if two observers are not at rest relative to one another
they will disagree about the absolute velocities of objects. They will, however, agree about relative velocities. If
object A is in motion relative to object B, and B is in motion relative to C, then A’s velocity relative to C is given
by vAC=vAB+vBC. Positive and negative signs are used to indicate the direction of an object’s motion.

Summary 71

Homework Problems
1 . The graph shows the motion of a car stuck in stop-and-go freeway
90 traffic. (a) If you only knew how far the car had gone during this entire
time period, what would you think its velocity was? (b) What is the car’s
80
maximum velocity?
70
60 2. (a) Let θ be the latitude of a point on the Earth's surface. Derive an
distance 50
algebra equation for the distance, L, traveled by that point during one
(m) 40 rotation of the Earth about its axis, i.e. over one day, expressed in terms of
L, θ, and R, the radius of the earth. Check: Your equation should give L=0
30
for the North Pole.
20
10 (b) At what speed is Fullerton, at latitude θ=34°, moving with the
0
rotation of the Earth about its axis? Give your answer in units of mi/h. [See
0 4 8 12 the table in the back of the book for the relevant data.]
time (s)
Problem 1. 3 . A person is parachute jumping. During the time between when she
leaps out of the plane and when she opens her chute, her altitude is given by
the equation
y=(10000 m) - (50 m/s)[t+(5.0 s)e-t/5.0 s] .
Problem 7.
Find her velocity at t=7.0 s. (This can be done on a calculator, without
knowing calculus.) Because of air resistance, her velocity does not increase
at a steady rate as it would for an object falling in vacuum.
4 S. A light-year is a unit of distance used in astronomy, and defined as the
distance light travels in one year. The speed of light is 3.0x108 m/s. Find
how many meters there are in one light-year, expressing your answer in
scientific notation.
5 S. You’re standing in a freight train, and have no way to see out. If you
have to lean to stay on your feet, what, if anything, does that tell you about
the train’s velocity? Its acceleration? Explain.
6 ∫. A honeybee’s position as a function of time is given by x=10t-t3, where t
is in seconds and x in meters. What is its velocity at t=3.0 s?
7 S. The figure shows the motion of a point on the rim of a rolling wheel.
(The shape is called a cycloid.) Suppose bug A is riding on the rim of the
wheel on a bicycle that is rolling, while bug B is on the spinning wheel of a
bike that is sitting upside down on the floor. Bug A is moving along a
cycloid, while bug B is moving in a circle. Both wheels are doing the same
number of revolutions per minute. Which bug has a harder time holding
on, or do they find it equally difficult?
8 . Peanut plants fold up their leaves at night. Estimate the top speed of
the tip of one of the leaves shown in the figure, expressing your result in
scientific notation in SI units..

Problem 8.



3 Acceleration and
Free Fall
3.1 The Motion of Falling Objects
The motion of falling objects is the simplest and most common ex-
ample of motion with changing velocity. The early pioneers of physics had a
correct intuition that the way things drop was a message directly from
Nature herself about how the universe worked. Other examples seem less
likely to have deep significance. A walking person who speeds up is making
a conscious choice. If one stretch of a river flows more rapidly than another,
it may be only because the channel is narrower there, which is just an
accident of the local geography. But there is something impressively consis-
tent, universal, and inexorable about the way things fall.
Stand up now and simultaneously drop a coin and a bit of paper side by
side. The paper takes much longer to hit the ground. That’s why Aristotle
wrote that heavy objects fell more rapidly. Europeans believed him for two
thousand years.
Now repeat the experiment, but make it into a race between the coin
and your shoe. My own shoe is about 50 times heavier than the nickel I had
handy, but it looks to me like they hit the ground at exactly the same
moment. So much for Aristotle! Galileo, who had a flair for the theatrical,
did the experiment by dropping a bullet and a heavy cannonball from a tall
tower. Aristotle’s observations had been incomplete, his interpretation a vast
oversimplification.
Galileo dropped a cannonball and a
musketball simultaneously from a It is inconceivable that Galileo was the first person to observe a discrep-
tower, and observed that they hit the ancy with Aristotle’s predictions. Galileo was the one who changed the
ground at nearly the same time. This
course of history because he was able to assemble the observations into a
contradicted Aristotle’s long-accepted
idea that heavier objects fell faster. coherent pattern, and also because he carried out systematic quantitative
(numerical) measurements rather than just describing things qualitatively.
Why is it that some objects, like the coin and the shoe, have similar
motion, but others, like a feather or a bit of paper, are different? Galileo
Galileo and the Church
Galileo’s contradiction of Aristotle had serious consequences. He was interrogated by the Church authorities and
convicted of teaching that the earth went around the sun as a matter of fact and not, as he had promised previously,
as a mere mathematical hypothesis. He was placed under permanent house arrest, and forbidden to write about or
teach his theories. Immediately after being forced to recant his claim that the earth revolved around the sun, the old
man is said to have muttered defiantly “and yet it does move.”
The story is dramatic, but there are some omissions in the commonly taught heroic version. There was a rumor
that the Simplicio character represented the Pope. Also, some of the ideas Galileo advocated had controversial religious
overtones. He believed in the existence of atoms, and atomism was thought by some people to contradict the Church’s
doctrine of transubstantiation, which said that in the Catholic mass, the blessing of the bread and wine literally
transformed them into the flesh and blood of Christ. His support for a cosmology in which the earth circled the sun
was also disreputable because one of its supporters, Giordano Bruno, had also proposed a bizarre synthesis of Christianity
with the ancient Egyptian religion.
73

speculated that in addition to the force that always pulls objects down, there
was an upward force exerted by the air. Anyone can speculate, but Galileo
went beyond speculation and came up with two clever experiments to probe
the issue. First, he experimented with objects falling in water, which probed
the same issues but made the motion slow enough that he could take time
measurements with a primitive pendulum clock. With this technique, he
established the following facts:
• All heavy, streamlined objects (for example a steel rod dropped
point-down) reach the bottom of the tank in about the same
amount of time, only slightly longer than the time they would take
to fall the same distance in air.
• Objects that are lighter or less streamlined take a longer time to
reach the bottom.
This supported his hypothesis about two contrary forces. He imagined
an idealized situation in which the falling object did not have to push its
way through any substance at all. Falling in air would be more like this ideal
case than falling in water, but even a thin, sparse medium like air would be
sufficient to cause obvious effects on feathers and other light objects that
were not streamlined. Today, we have vacuum pumps that allow us to suck
nearly all the air out of a chamber, and if we drop a feather and a rock side
by side in a vacuum, the feather does not lag behind the rock at all.
How the speed of a falling object increases with time
Galileo’s second stroke of genius was to find a way to make quantitative
measurements of how the speed of a falling object increased as it went
along. Again it was problematic to make sufficiently accurate time measure-
ments with primitive clocks, and again he found a tricky way to slow things
down while preserving the essential physical phenomena: he let a ball roll
down a slope instead of dropping it vertically. The steeper the incline, the
more rapidly the ball would gain speed. Without a modern video camera,
Galileo had invented a way to make a slow-motion version of falling.

Velocity increases more gradually on
the gentle slope, but the motion is
otherwise the same as the motion of
a falling object.

Although Galileo’s clocks were only good enough to do accurate
experiments at the smaller angles, he was confident after making a system-
atic study at a variety of small angles that his basic conclusions were gener-
ally valid. Stated in modern language, what he found was that the velocity-
v versus-time graph was a line. In the language of algebra, we know that a line
has an equation of the form y=ax+b, but our variables are v and t, so it
would be v=at+b. (The constant b can be interpreted simply as the initial
t velocity of the object, i.e. its velocity at the time when we started our clock,
The v-t graph of a falling object is a which we conventionally write as v o .)
line.

74 Chapter 3 Acceleration and Free Fall

Self-Check
An object is rolling down an incline. After it has been rolling for a short time, it
is found to travel 13 cm during a certain one-second interval. During the
second after that, if goes 16 cm. How many cm will it travel in the second after
that?

A contradiction in Aristotle’s reasoning
Galileo’s inclined-plane experiment disproved the long-accepted claim
by Aristotle that a falling object had a definite “natural falling speed”
proportional to its weight. Galileo had found that the speed just kept on
increasing, and weight was irrelevant as long as air friction was negligible.
Not only did Galileo prove experimentally that Aristotle had been wrong,
but he also pointed out a logical contradiction in Aristotle’s own reasoning.
(a) Galileo’s experiments show that all Simplicio, the stupid character, mouths the accepted Aristotelian wisdom:
falling objects have the same motion
if air resistance is negligible. SIMPLICIO: There can be no doubt but that a particular body ... has a
fixed velocity which is determined by nature...
SALVIATI: If then we take two bodies whose natural speeds are
different, it is clear that, [according to Aristotle], on uniting the two, the
more rapid one will be partly held back by the slower, and the slower
will be somewhat hastened by the swifter. Do you not agree with me
in this opinion?
SIMPLICIO: You are unquestionably right.
SALVIATI: But if this is true, and if a large stone moves with a speed of,
say, eight [unspecified units] while a smaller moves with a speed of
four, then when they are united, the system will move with a speed
less than eight; but the two stones when tied together make a stone
(b) (c)
larger than that which before moved with a speed of eight. Hence the
Aristotle said that heavier objects fell heavier body moves with less speed than the lighter; an effect which
faster than lighter ones. If two rocks is contrary to your supposition. Thus you see how, from your
are tied together, that makes an extra-
assumption that the heavier body moves more rapidly than the lighter
heavy rock, (b), which should fall
faster. But Aristotle’s theory would also one, I infer that the heavier body moves more slowly.
predict that the light rock would hold [tr. Crew and De Salvio]
back the heavy rock, resulting in a
slower fall, (c). What is gravity?
The physicist Richard Feynman liked to tell a story about how when he
was a little kid, he asked his father, “Why do things fall?” As an adult, he
praised his father for answering, “Nobody knows why things fall. It’s a deep
mystery, and the smartest people in the world don’t know the basic reason
for it.” Contrast that with the average person’s off-the-cuff answer, “Oh, it’s
because of gravity.” Feynman liked his father’s answer, because his father
realized that simply giving a name to something didn’t mean that you
understood it. The radical thing about Galileo’s and Newton’s approach to
science was that they concentrated first on describing mathematically what
really did happen, rather than spending a lot of time on untestable specula-
tion such as Aristotle’s statement that “Things fall because they are trying to
reach their natural place in contact with the earth.” That doesn’t mean that
science can never answer the “why” questions. Over the next month or two
as you delve deeper into physics, you will learn that there are more funda-
mental reasons why all falling objects have v-t graphs with the same slope,
regardless of their mass. Nevertheless, the methods of science always impose
limits on how deep our explanation can go.
Its speed increases at a steady rate, so in the next second it will travel 19 cm.

Section 3.1 The Motion of Falling Objects 75

3.2 Acceleration
Definition of acceleration for linear v-t graphs
Galileo’s experiment with dropping heavy and light objects from a
tower showed that all falling objects have the same motion, and his in-
clined-plane experiments showed that the motion was described by v=ax+vo.
The initial velocity vo depends on whether you drop the object from rest or
throw it down, but even if you throw it down, you cannot change the slope,
a, of the v-t graph.
Since these experiments show that all falling objects have linear v-t
graphs with the same slope, the slope of such a graph is apparently an
important and useful quantity. We use the word acceleration, and the
symbol a, for the slope of such a graph. In symbols, a=∆v/∆t. The accelera-
tion can be interpreted as the amount of speed gained in every second, and
it has units of velocity divided by time, i.e. “meters per second per second,”
or m/s/s. Continuing to treat units as if they were algebra symbols, we
simplify “m/s/s” to read “m/s2.” Acceleration can be a useful quantity for
describing other types of motion besides falling, and the word and the
symbol “a” can be used in a more general context. We reserve the more
specialized symbol “g” for the acceleration of falling objects, which on the
surface of our planet equals 9.8 m/s2. Often when doing approximate
calculations or merely illustrative numerical examples it is good enough to
use g=10 m/s2, which is off by only 2%.
Example
Question: A despondent physics student jumps off a bridge, and
falls for three seconds before hitting the water. How fast is he
going when he hits the water?
30 Solution: Approximating g as 10 m/s2, he will gain 10 m/s of
20 speed each second. After one second, his velocity is 10 m/s,
v
after two seconds it is 20 m/s, and on impact, after falling for
(m/s)10
three seconds, he is moving at 30 m/s.
0
0 1 2 3
t (s) Example: extracting acceleration from a graph
Question: The x-t and v-t graphs show the motion of a car
starting from a stop sign. What is the car’s acceleration?
Solution: Acceleration is defined as the slope of the v-t graph.
The graph rises by 3 m/s during a time interval of 3 s, so the
4
acceleration is (3 m/s)/(3 s)=1 m/s2.
x (m)

Incorrect solution #1: The final velocity is 3 m/s, and
2 acceleration is velocity divided by time, so the acceleration is (3
m/s)/(10 s)=0.3 m/s2.
# The solution is incorrect because you can’t find the slope of a
graph from one point. This person was just using the point at the
3 right end of the v-t graph to try to find the slope of the curve.
Incorrect solution #2: Velocity is distance divided by time so
v (m/s)

2 v=(4.5 m)/(3 s)=1.5 m/s. Acceleration is velocity divided by time,
so a=(1.5 m/s)/(3 s)=0.5 m/s2.
1 # The solution is incorrect because velocity is the slope of the
tangent line. In a case like this where the velocity is changing,
0 you can’t just pick two points on the x-t graph and use them to
7 8 9 10 find the velocity.
t (s)


Example: converting g to different units
Question: What is g in units of cm/s2?
Solution: The answer is going to be how many cm/s of speed a
falling object gains in one second. If it gains 9.8 m/s in one
second, then it gains 980 cm/s in one second, so g=980 cm/s2.
Alternatively, we can use the method of fractions that equal one:

/×
9.8 m 100 cm 980 cm
=
2 s2
s /
1m
Question: What is g in units of miles/hour2?
Solution:
2
9.8 m × 1 mile × 3600 s = 7.9×10 4 mile / hour 2
s2 1600 m 1 hour

This large number can be interpreted as the speed, in miles per
hour, you would gain by falling for one hour. Note that we had to
square the conversion factor of 3600 s/hour in order to cancel
out the units of seconds squared in the denominator.
Question: What is g in units of miles/hour/s?
Solution:
9.8 m × 1 mile × 3600 s = 22 mile/hour/s
s2 1600 m 1 hour
This is a figure that Americans will have an intuitive feel for. If
your car has a forward acceleration equal to the acceleration of a
falling object, then you will gain 22 miles per hour of speed every
second. However, using mixed time units of hours and seconds
like this is usually inconvenient for problem-solving. It would be
like using units of foot-inches for area instead of ft2 or in2.
The acceleration of gravity is different in different locations.
Everyone knows that gravity is weaker on the moon, but actually it is
not even the same everywhere on Earth, as shown by the sampling of
numerical data in the following table.

elevation g
location latitude (m) (m/s2)
north pole 90° N 0 9.8322
Reykjavik, Iceland 64° N 0 9.8225
Fullerton, California 34° N 0 9.7957
Guayaquil, Ecuador 2° S 0 9.7806
Mt. Cotopaxi, Ecuador 1° S 5896 9.7624
Mt. Everest 28° N 8848 9.7643

The main variables that relate to the value of g on Earth are latitude and
elevation. Although you have not yet learned how g would be calculated
based on any deeper theory of gravity, it is not too hard to guess why g
depends on elevation. Gravity is an attraction between things that have
Section 3.1 The Motion of Falling Objects 77

mass, and the attraction gets weaker with increasing distance. As you ascend
from the seaport of Guayaquil to the nearby top of Mt. Cotopaxi, you are
distancing yourself from the mass of the planet. The dependence on latitude
occurs because we are measuring the acceleration of gravity relative to the
earth’s surface, but the earth’s rotation causes the earth’s surface to fall out
from under you. (We will discuss both gravity and rotation in more detail
later in the course.)
Much more spectacular differences in the strength of gravity can be
observed away from the Earth’s surface:

location g (m/s2)
asteroid Vesta (surface) 0.3
Earth's moon (surface) 1.6
Mars (surface) 3.7
Earth (surface) 9.8
Jupiter (cloud-tops) 26
Sun (visible surface) 270
typical neutron star (surface) 1012
infinite according to
black hole (center) some theories, on the
order of 1052
according to others

A typical neutron star is not so different in size from a large asteroid, but is
orders of magnitude more massive, so the mass of a body definitely corre-
This false-color map shows variations lates with the g it creates. On the other hand, a neutron star has about the
in the strength of the earth’s gravity.
Purple areas have the strongest
same mass as our Sun, so why is its g billions of times greater? If you had the
gravity, yellow the weakest. The over- misfortune of being on the surface of a neutron star, you’d be within a few
all trend toward weaker gravity at the thousand miles of all its mass, whereas on the surface of the Sun, you’d still
equator and stronger gravity at the be millions of miles from most if its mass.
poles has been artificially removed to
allow the weaker local variations to
show up. The map covers only the
oceans because of the technique used
to make it: satellites look for bulges
and depressions in the surface of the
ocean. A very slight bulge will occur
over an undersea mountain, for
instance, because the mountain’s
gravitational attraction pulls water
toward it. The US government
originally began collecting data like
these for military use, to correct for the
deviations in the paths of missiles. The
data have recently been released for
scientific and commercial use (e.g.
searching for sites for off-shore oil
wells).


A. What is wrong with the following definitions of g?
(a) “g is gravity.”
(b) “g is the speed of a falling object.”
(c) “g is how hard gravity pulls on things.”
B. When advertisers specify how much acceleration a car is capable of, they
do not give an acceleration as defined in physics. Instead, they usually specify
how many seconds are required for the car to go from rest to 60 miles/hour.
Suppose we use the notation “a” for the acceleration as defined in physics,
and “acar ad” for the quantity used in advertisements for cars. In the US’s non-
metric system of units, what would be the units of a and acar ad? How would the
use and interpretation of large and small, positive and negative values be
different for a as opposed to acar ad?
C. Two people stand on the edge of a cliff. As they lean over the edge, one
person throws a rock down, while the other throws one straight up with an
exactly opposite initial velocity. Compare the speeds of the rocks on impact at
the bottom of the cliff.

3.3 Positive and Negative Acceleration
Gravity always pulls down, but that does not mean it always speeds
things up. If you throw a ball straight up, gravity will first slow it down to
a= a= a= v=0 and then begin increasing its speed. When I took physics in high
−10 m/s2 −10 m/s2 −10 m/s2 school, I got the impression that positive signs of acceleration indicated
speeding up, while negative accelerations represented slowing down, i.e.
deceleration. Such a definition would be inconvenient, however, because we
would then have to say that the same downward tug of gravity could
produce either a positive or a negative acceleration. As we will see in the
following example, such a definition also would not be the same as the slope
of the v-t graph
Let’s study the example of the rising and falling ball. In the example of
2 the person falling from a bridge, I assumed positive velocity values without
calling attention to it, which meant I was assuming a coordinate system
x (m)

whose x axis pointed down. In this example, where the ball is reversing
1 direction, it is not possible to avoid negative velocities by a tricky choice of
axis, so let’s make the more natural choice of an axis pointing up. The ball’s
velocity will initially be a positive number, because it is heading up, in the
same direction as the x axis, but on the way back down, it will be a negative
number. As shown in the figure, the v-t graph does not do anything special
5 at the top of the ball’s flight, where v equals 0. Its slope is always negative.
In the left half of the graph, the negative slope indicates a positive velocity
v (m/s)

0 that is getting closer to zero. On the right side, the negative slope is due to a
negative velocity that is getting farther from zero, so we say that the ball is
-5
speeding up, but its velocity is decreasing!
To summarize, what makes the most sense is to stick with the original
0.5 1 1.5 definition of acceleration as the slope of the v-t graph, ∆v/∆t. By this
t (s) definition, it just isn’t necessarily true that things speeding up have positive
acceleration while things slowing down have negative acceleration. The
word “deceleration” is not used much by physicists, and the word “accelera-
tion” is used unblushingly to refer to slowing down as well as speeding up:
“There was a red light, and we accelerated to a stop.”
Example
Question: In the example above, what happens if you calculate

Section 3.3 Positive and Negative Acceleration 79

the acceleration between t=1.0 and 1.5 s?
Answer: Reading from the graph, it looks like the velocity is
about -1 m/s at t=1.0 s, and around -6 m/s at t=1.5 s. The
acceleration, figured between these two points, is

a = ∆v =
( – 6 m / s) – ( – 1 m / s)
= – 10 m / s 2 .
∆t (1.5 s) – (1.0 s)
Even though the ball is speeding up, it has a negative
acceleration.
Another way of convincing you that this way of handling the plus and
minus signs makes sense is to think of a device that measures acceleration.
After all, physics is supposed to use operational definitions, ones that relate
to the results you get with actual measuring devices. Consider an air
freshener hanging from the rear-view mirror of your car. When you speed
up, the air freshener swings backward. Suppose we define this as a positive
reading. When you slow down, the air freshener swings forward, so we’ll
call this a negative reading on our accelerometer. But what if you put the car
in reverse and start speeding up backwards? Even though you’re speeding
up, the accelerometer responds in the same way as it did when you were
going forward and slowing down. There are four possible cases:

accelerom- slope of direction of
eter v-t force acting
motion of car swings graph on car

forward, speeding up backward + forward

forward, slowing down forward - backward

backward, speeding up forward - backward

backward, slowing down backward + forward

Note the consistency of the three right-hand columns — nature is
trying to tell us that this is the right system of classification, not the left-
hand column.


Because the positive and negative signs of acceleration depend on the
choice of a coordinate system, the acceleration of an object under the
influence of gravity can be either positive or negative. Rather than having to
write things like “g=9.8 m/s2 or -9.8 m/s2” every time we want to discuss g’s
numerical value, we simply define g as the absolute value of the acceleration
of objects moving under the influence of gravity. We consistently let g=9.8
m/s2, but we may have either a=g or a=-g, depending on our choice of a
coordinate system.
Example
Question: A person kicks a ball, which rolls up a sloping street,
comes to a halt, and rolls back down again. The ball has
constant acceleration. The ball is initially moving at a velocity of
4.0 m/s, and after 10.0 s it has returned to where it started. At the
end, it has sped back up to the same speed it had initially, but in
the opposite direction. What was its acceleration?
Solution: By giving a positive number for the initial velocity, the
statement of the question implies a coordinate axis that points up
the slope of the hill. The “same” speed in the opposite direction
should therefore be represented by a negative number, -4.0 m/s.
The acceleration is a=∆v/∆t=(vafter-vbefore)/10.0 s=[(-4.0 m/s)-(4.0
m/s)]/10.0 s=-0.80 m/s2. The acceleration was no different during
the upward part of the roll than on the downward part of the roll.
Incorrect solution: Acceleration is ∆v/∆t, and at the end it’s not
moving any faster or slower than when it started, so ∆v=0 and
a=0.
# The velocity does change, from a positive number to a
negative number.
A. A child repeatedly jumps up and down on a trampoline. Discuss the sign
and magnitude of his acceleration, including both the time when he is in the air
and the time when his feet are in contact with the trampoline.
B. Sally is on an amusement park ride which begins with her chair being
hoisted straight up a tower at a constant speed of 60 miles/hour. Despite stern
warnings from her father that he’ll take her home the next time she
misbehaves, she decides that as a scientific experiment she really needs to
release her corndog over the side as she’s on the way up. She does not throw
it. She simply sticks it out of the car, lets it go, and watches it against the
background of the sky, with no trees or buildings as reference points. What
does the corndog’s motion look like as observed by Sally? Does its speed ever
appear to her to be zero? What acceleration does she observe it to have: is it
ever positive? negative? zero? What would her enraged father answer if asked
for a similar description of its motion as it appears to him, standing on the
ground?
C. Can an object maintain a constant acceleration, but meanwhile reverse the
direction of its velocity?
D. Can an object have a velocity that is positive and increasing at the same
time that its acceleration is decreasing?
E. The four figures show a refugee from a Picasso painting blowing on a rolling
water bottle. In some cases the person’s blowing is speeding the bottle up, but
Discussion question B. in others it is slowing it down. The arrow inside the bottle shows which


direction it is going, and a coordinate system is shown at the bottom of each
figure. In each case, figure out the plus or minus signs of the velocity and
acceleration. It may be helpful to draw a v-t graph in each case.

(a) (b)

x x

(c) (d)

x x

3.4 Varying Acceleration
So far we have only been discussing examples of motion for which the
v-t graph is linear. If we wish to generalize our definition to v-t graphs that
are more complex curves, the best way to proceed is similar to how we
defined velocity for curved x-t graphs:
definition of acceleration
The acceleration of an object at any instant is the slope of the tangent
line passing through its v-versus-t graph at the relevant point.
Example: a skydiver
Question: The graphs show the results of a fairly realistic
computer simulation of the motion of a skydiver, including the
600 effects of air friction. The x axis has been chosen pointing down,
so x is increasing as she falls. Find (a) the skydiver’s
acceleration at t=3.0 s, and also (b) at t=7.0 s.
400 Solution: I’ve added tangent lines at the two points in question.
x (m)

(a) To find the slope of the tangent line, I pick two points on the
200

0
50
40
v (m/s)

30
20
10
0
0 2 4 6 8 10 12 14
t (s)


(9.0 s, 52 m/s)
(7.0 s, 47 m/s)

50

40

30

v (m/s)
20
(5.0 s, 42 m/s)

10
(3.0 s, 28 m/s)
0
0 2 4 6 8 10 12 14
t (s)
line (not necessarily on the actual curve): (3.0 s, 28 m/s) and (5.0
s, 42 m/s). The slope of the tangent line is (42 m/s-28 m/s)/(5.0 s
- 3.0 s)=7.0 m/s2.
(b) Two points on this tangent line are (7.0 s, 47 m/s) and (9.0 s,
52 m/s). The slope of the tangent line is (52 m/s-47 m/s)/(9.0 s -
7.0 s)=2.5 m/s2.
Physically, what’s happening is that at t=3.0 s, the skydiver is not
yet going very fast, so air friction is not yet very strong. She
therefore has an acceleration almost as great as g. At t=7.0 s,
she is moving almost twice as fast (about 100 miles per hour),
and air friction is extremely strong, resulting in a significant
departure from the idealized case of no air friction.
In the above example, the x-t graph was not even used in the solution of
the problem, since the definition of acceleration refers to the slope of the v-t
graph. It is possible, however, to interpret an x-t graph to find out some-
thing about the acceleration. An object with zero acceleration, i.e. constant
velocity, has an x-t graph that is a straight line. A straight line has no
curvature. A change in velocity requires a change in the slope of the x-t
graph, which means that it is a curve rather than a line. Thus acceleration
a=0 a=0 relates to the curvature of the x-t graph. Figure (c) shows some examples.
x x
In the skydiver example, the x-t graph was more strongly curved at the
beginning, and became nearly straight at the end. If the x-t graph is nearly
t t
straight, then its slope, the velocity, is nearly constant, and the acceleration
small, large, is therefore small. We can thus interpret the acceleration as representing the
x positive x positive curvature of the x-t graph. If the “cup” of the curve points up, the accelera-
a a tion is positive, and if it points down, the acceleration is negative.
t t
Since the relationship between a and v is analogous to the relationship
a0
x large, x
negative a0
a
t t


600
10

x (m)
x (m)
400
5
200 position
0 0 slope of
50 tangent line

v (m/s)
v (m/s) 4 40
3
2
30
20
velocity curvature
=rate of change of position
1 10
0 0 slope of
tangent line

a (m/s2)
a (m/s2)

1
5
acceleration
=rate of change of velocity
0 0
0 1 2 3 4 5 0 2 4 6 8 101214
t (s) t (s)
(a) (b) (c)
between v and x, we can also make graphs of acceleration as a function of
time, as shown in figures (a) and (b) above.
Figure (c) summarizes the relationships among the three types of
graphs.

A. Describe in words how the changes in the a-t graph for the skydiver relate
to the behavior of the v-t graph.
B. Explain how each set of graphs contains inconsistencies.

1 2 3
x x x

v v v

a a a

t t t
Discussion question B.


3.5 The Area Under the Velocity-Time Graph
A natural question to ask about falling objects is how fast they fall, but
Galileo showed that the question has no answer. The physical law that he
discovered connects a cause (the attraction of the planet Earth’s mass) to an
effect, but the effect is predicted in terms of an acceleration rather than a
20 (a) velocity. In fact, no physical law predicts a definite velocity as a result of a
specific phenomenon, because velocity cannot be measured in absolute
terms, and only changes in velocity relate directly to physical phenomena.
v (m/s)

10 The unfortunate thing about this situation is that the definitions of
velocity and acceleration are stated in terms of the tangent-line technique,
which lets you go from x to v to a, but not the other way around. Without a
technique to go backwards from a to v to x, we cannot say anything quanti-
0
tative, for instance, about the x-t graph of a falling object. Such a technique
0 2 4 6 8 does exist, and I used it to make the x-t graphs in all the examples above.
t (s)
First let’s concentrate on how to get x information out of a v-t graph. In
example (a), an object moves at a speed of 20 m/s for a period of 4.0 s. The
20 (b) distance covered is ∆x=v∆t=(20 m/s)x(4.0 s)=80 m. Notice that the quanti-
ties being multiplied are the width and the height of the shaded rectangle
— or, strictly speaking, the time represented by its width and the velocity
v (m/s)

represented by its height. The distance of ∆x=80 m thus corresponds to the
10
area of the shaded part of the graph.
The next step in sophistication is an example like (b), where the object
moves at a constant speed of 10 m/s for two seconds, then for two seconds
0
at a different constant speed of 20 m/s. The shaded region can be split into
0 2 4 6 8 a small rectangle on the left, with an area representing ∆x=20 m, and a taller
t (s)
one on the right, corresponding to another 40 m of motion. The total
distance is thus 60 m, which corresponds to the total area under the graph.

20 An example like (c) is now just a trivial generalization; there is simply a
(c)
large number of skinny rectangular areas to add up. But notice that graph
(c) is quite a good approximation to the smooth curve (d). Even though we
have no formula for the area of a funny shape like (d), we can approximate
v (m/s)

10 its area by dividing it up into smaller areas like rectangles, whose area is
easier to calculate. If someone hands you a graph like (d) and asks you to
find the area under it, the simplest approach is just to count up the little
0
rectangles on the underlying graph paper, making rough estimates of
fractional rectangles as you go along.
0 2 4 6 8
t (s)

20 (d)
v (m/s)

10

0
0 2 4 6 8
t (s)

Section 3.5 The Area Under the Velocity-Time Graph 85

0.5 m
20
1m 1.5 m

1m 1.5 m

1.5 m 1.5 m

2m 1.5 m

2m 1.5 m
10

v (m/s)
2m 1.5 m

0.5 m 2m 1.5 m

2m 2m 1.5 m

1m 2m 2m 1.5 m

2m 2m 2m 1.5 m
0

0 2 4 6 8
t (s)

That’s what I’ve done above. Each rectangle on the graph paper is 1.0 s
wide and 2 m/s tall, so it represents 2 m. Adding up all the numbers gives
∆x=41 m. If you needed better accuracy, you could use graph paper with
smaller rectangles.
It’s important to realize that this technique gives you ∆x, not x. The v-t
graph has no information about where the object was when it started.
The following are important points to keep in mind when applying this
technique:
• If the range of v values on your graph does not extend down to
zero, then you will get the wrong answer unless you compensate by
adding in the area that is not shown.
• As in the example, one rectangle on the graph paper does not
necessarily correspond to one meter of distance.
• Negative velocity values represent motion in the opposite direction,
so area under the t axis should be subtracted, i.e. counted as
“negative area.”
• Since the result is a ∆x value, it only tells you xafter-xbefore, which
may be less than the actual distance traveled. For instance, the
object could come back to its original position at the end, which
would correspond to ∆x=0, even though it had actually moved a
nonzero distance.
Finally, note that one can find ∆v from an a-t graph using an entirely
analogous method. Each rectangle on the a-t graph represents a certain
amount of velocity change.
Discussion question
Roughly what would a pendulum’s v-t graph look like? What would happen
when you applied the area-under-the-curve technique to find the pendulum’s
∆x for a time period covering many swings?


3.6 Algebraic Results for Constant Acceleration
Although the area-under-the-curve technique can be applied to any
graph, no matter how complicated, it may be laborious to carry out, and if
fractions of rectangles must be estimated the result will only be approxi-
mate. In the special case of motion with constant acceleration, it is possible
v to find a convenient shortcut which produces exact results. When the
∆t acceleration is constant, the v-t graph is a straight line, as shown in the
figure. The area under the curve can be divided into a triangle plus a
rectangle, both of whose areas can be calculated exactly: A=bh for a rect-
angle and A=1 2 bh for a triangle. The height of the rectangle is the initial
∆v
velocity, vo, and the height of the triangle is the change in velocity from
beginning to end, ∆v. The object’s ∆x is therefore given by the equation
∆x = v o∆t + 1 ∆v∆t . This can be simplified a little by using the definition of
vo 2
t acceleration, a=∆v/∆t to eliminate ∆v, giving

∆x = v o∆t + 1 a∆t 2 [motion with constant acceleration] .
2

Since this is a second-order polynomial in ∆t, the graph of ∆x versus ∆t
is a parabola, and the same is true of a graph of x versus t — the two graphs
differ only by shifting along the two axes. Although I have derived the
equation using a figure that shows a positive vo, positive a, and so on, it still
turns out to be true regardless of what plus and minus signs are involved.
Another useful equation can be derived if one wants to relate the change
in velocity to the distance traveled. This is useful, for instance, for finding
the distance needed by a car to come to a stop. For simplicity, we start by
deriving the equation for the special case of vo=0, in which the final velocity
vf is a synonym for ∆v. Since velocity and distance are the variables of
interest, not time, we take the equation ∆x = 1 2 a∆t2 and use ∆t=∆v/a to
eliminate ∆t. This gives ∆x = 1 2 (∆v)2/a, which can be rewritten as

v 2 = 2a∆x [motion with constant acceleration, v o = 0] .
f

For the more general case where v o ≠ 0 , we skip the tedious algebra
leading to the more general equation,

v 2 = v 2+2a∆x [motion with constant acceleration] .
f o

To help get this all organized in your head, first let’s categorize the
variables as follows:
Variables that change during motion with constant acceleration:
x, v, t
Variable that doesn’t change:
a

Section 3.6 Algebraic Results for Constant Acceleration 87

If you know one of the changing variables and want to find another,
there is always an equation that relates those two:

vf2 = vo2 + 2a∆x v
x
∆v
a=
∆t
∆x = vo∆t + 1 a∆t2
2

t
The symmetry among the three variables is imperfect only because the
equation relating x and t includes the initial velocity.
There are two main difficulties encountered by students in applying
these equations:
• The equations apply only to motion with constant acceleration. You
can’t apply them if the acceleration is changing.
• Students are often unsure of which equation to use, or may cause
themselves unnecessary work by taking the longer path around the
triangle in the chart above. Organize your thoughts by listing the
variables you are given, the ones you want to find, and the ones you
aren’t given and don’t care about.
Example
Question: You are trying to pull an old lady out of the way of an
oncoming truck. You are able to give her an acceleration of 20 m/
s2. Starting from rest, how much time is required in order to move
her 2 m?
Solution: First we organize our thoughts:
Variables given: ∆x, a, vo
Variables desired: ∆t
Irrelevant variables: vf
Consulting the triangular chart above, the equation we need is
clearly ∆x = v o∆t + 1 a∆t , since it has the four variables of
2
2
interest and omits the irrelevant one. Eliminating the vo term and
∆
solving for ∆t gives ∆t = 2 a x =0.4 s.
A Check that the units make sense in the three equations derived in this
section.
B. In chapter 1, I gave examples of correct and incorrect reasoning about
proportionality, using questions about the scaling of area and volume. Try to
translate the incorrect modes of reasoning shown there into mistakes about the
following question: If the acceleration of gravity on Mars is 1/3 that on Earth,
how many times longer does it take for a rock to drop the same distance on
Mars?


3.7* Biological Effects of Weightlessness
The usefulness of outer space was brought home to North Americans in
1998 by the unexpected failure of the communications satellite that had
been handling almost all of the continent’s cellular phone traffic. Compared
to the massive economic and scientific payoffs of satellites and space probes,
human space travel has little to boast about after four decades. Sending
people into orbit has just been too expensive to be an effective scientific or
commercial activity. The 1986 Challenger disaster dealt a blow to NASA’s
confidence, and with the end of the cold war, U.S. prestige as a superpower
was no longer a compelling reason to send Americans into space. All that
may change soon, with a new generation of much cheaper reusable space-
ships. (The space shuttle is not truly reusable. Retrieving the boosters out of
the ocean is no cheaper than building new ones, but NASA brings them
back and uses them over for public relations, to show how frugal they are.)
Space tourism is even beginning to make economic sense! No fewer than
three private companies are now willing to take your money for a reserva-
tion on a two-to-four minute trip into space, although none of them has a
firm date on which to begin service. Within a decade, a space cruise may be
the new status symbol among those sufficiently rich and brave.
Space sickness
Well, rich, brave, and possessed of an iron stomach. Travel agents will
probably not emphasize the certainty of constant space-sickness. For us
animals evolved to function in g=9.8 m/s2, living in g=0 is extremely
unpleasant. The early space program focused obsessively on keeping the
astronaut-trainees in perfect physical shape, but it soon became clear that a
body like a Greek demigod’s was no defense against that horrible feeling
Artist’s conceptions of the X-33
spaceship, a half-scale uncrewed that your stomach was falling out from under you and you were never going
version of the planned VentureStar to catch up. Our inner ear, which normally tells us which way is down,
vehicle, which was supposed to cut the tortures us when down is nowhere to be found. There is contradictory
cost of sending people into space by information about whether anyone ever gets over it; the “right stuff ” culture
an order of magnitude. The X-33
program was canceled in March 2001
creates a strong incentive for astronauts to deny that they are sick.
due to technical failures and budget Effects of long space missions
overruns, so the Space Shuttle will
remain the U.S.’s only method of
Worse than nausea are the health-threatening effects of prolonged
sending people into space for the weightlessness. The Russians are the specialists in long-term missions, in
forseeable future. which cosmonauts suffer harm to their blood, muscles, and, most impor-
Courtesy of NASA.
tantly, their bones.
The effects on the muscles and skeleton appear to be similar to those
experienced by old people and people confined to bed for a long time.
Everyone knows that our muscles get stronger or weaker depending on the
amount of exercise we get, but the bones are likewise adaptable. Normally
old bone mass is continually being broken down and replaced with new
material, but the balance between its loss and replacement is upset when
people do not get enough weight-bearing exercise. The main effect is on the
bones of the lower body. More research is required to find out whether
astronauts’ loss of bone mass is due to faster breaking down of bone, slower
replacement, or both. It is also not known whether the effect can be sup-
pressed via diet or drugs.
The other set of harmful physiological effects appears to derive from the
redistribution of fluids. Normally, the veins and arteries of the legs are

Section 3.7* Biological Effects of Weightlessness 89

tightly constricted to keep gravity from making blood collect there. It is
uncomfortable for adults to stand on their heads for very long, because the
head’s blood vessels are not able to constrict as effectively. Weightless
astronauts’ blood tends to be expelled by the constricted blood vessels of the
lower body, and pools around their hearts, in their thoraxes, and in their
heads. The only immediate result is an uncomfortable feeling of bloatedness
in the upper body, but in the long term, a harmful chain of events is set in
motion. The body’s attempts to maintain the correct blood volume are most
sensitive to the level of fluid in the head. Since astronauts have extra fluid in
their heads, the body thinks that the over-all blood volume has become too
great. It responds by decreasing blood volume below normal levels. This
U.S. and Russian astronauts
aboard the International Space
increases the concentration of red blood cells, so the body then decides that
Station, October 2000. the blood has become too thick, and reduces the number of blood cells. In
missions lasting up to a year or so, this is not as harmful as the musculo-
skeletal effects, but it is not known whether longer period in space would
bring the red blood cell count down to harmful levels.
Reproduction in space
For those enthralled by the romance of actual human colonization of
space, human reproduction in weightlessness becomes an issue. An already-
pregnant Russian cosmonaut did spend some time in orbit in the 1960’s,
and later gave birth to a normal child on the ground. Recently, one of
NASA’s public relations concerns about the space shuttle program has been
to discourage speculation about space sex, for fear of a potential taxpayers’
backlash against the space program as an expensive form of exotic pleasure.
Scientific work has been concentrated on studying plant and animal
reproduction in space. Green plants, fungi, insects, fish, and amphibians
have all gone through at least one generation in zero-gravity experiments
without any serious problems. In many cases, animal embryos conceived in
orbit begin by developing abnormally, but later in development they seem
The International Space Station, to correct themselves. However, chicken embryos fertilized on earth less
September 2000. The space station than 24 hours before going into orbit have failed to survive. Since chickens
will not rotate to provide simulated
gravity. The completed station will are the organisms most similar to humans among the species investigated so
be much bigger than it is in this far, it is not at all certain that humans could reproduce successfully in a
picture. zero-gravity space colony.
Simulated gravity
If humans are ever to live and work in space for more than a year or so,
More on Apparent Weightlessness the only solution is probably to build spinning space stations to provide the
Astronauts in orbit are not really illusion of weight, as discussed in section 9.2. Normal gravity could be
weightless; they are only a few simulated, but tourists would probably enjoy g=2 m/s2 or 5 m/s2. Space
hundred miles up, so they are still
enthusiasts have proposed entire orbiting cities built on the rotating cylin-
affected strongly by the Earth’s
gravity. Section 10.3 of this book der plan. Although science fiction has focused on human colonization of
discusses why they experience relatively earthlike bodies such as our moon, Mars, and Jupiter’s icy moon
apparent weightlessness. Europa, there would probably be no practical way to build large spinning
More on Simulated Gravity structures on their surfaces. If the biological effects of their 2-3 m/s2
For more information on simulating gravitational accelerations are as harmful as the effect of g=0, then we may
gravity by spinning a spacecraft, see be left with the surprising result that interplanetary space is more hospitable
section 9.2 of this book.
to our species than the moons and planets.


3.8 ∫ Applications of Calculus
In the Applications of Calculus section at the end of the previous
chapter, I discussed how the slope-of-the-tangent-line idea related to the
calculus concept of a derivative, and the branch of calculus known as
differential calculus. The other main branch of calculus, integral calculus,
has to do with the area-under-the-curve concept discussed in section 3.5 of
this chapter. Again there is a concept, a notation, and a bag of tricks for
doing things symbolically rather than graphically. In calculus, the area
under the v-t graph between t=t1 and t=t2 is notated like this:
t2
area under the curve = ∆x = v dt
t1

The expression on the right is called an integral, and the s-shaped symbol,
the integral sign, is read as “integral of....”
Integral calculus and differential calculus are closely related. For in-
stance, if you take the derivative of the function x(t), you get the function
v(t), and if you integrate the function v(t), you get x(t) back again. In other
words, integration and differentiation are inverse operations. This is known
as the fundamental theorem of calculus.
On an unrelated topic, there is a special notation for taking the deriva-
tive of a function twice. The acceleration, for instance, is the second (i.e.
double) derivative of the position, because differentiating x once gives v, and
then differentiating v gives a. This is written as
2
a=d x2
.
dt
The seemingly inconsistent placement of the twos on the top and bottom
confuses all beginning calculus students. The motivation for this funny
notation is that acceleration has units of m/s2, and the notation correctly
suggests that: the top looks like it has units of meters, the bottom seconds2.
The notation is not meant, however, to suggest that t is really squared.

Section 3.8 ∫ Applications of Calculus 91

Summary
Selected Vocabulary
gravity ............................... A general term for the phenomenon of attraction between things having
mass. The attraction between our planet and a human-sized object causes
the object to fall.
acceleration ....................... The rate of change of velocity; the slope of the tangent line on a v-t graph.
Notation
a ........................................ acceleration
g ........................................ the acceleration of objects in free fall
Summary
Galileo showed that when air resistance is negligible all falling bodies have the same motion regardless of
mass. Moreover, their v-t graphs are straight lines. We therefore define a quantity called acceleration as the
slope, ∆v/∆t, of an object’s v-t graph. In cases other than free fall, the v-t graph may be curved, in which case
the definition is generalized as the slope of a tangent line on the v-t graph. The acceleration of objects in free
fall varies slightly across the surface of the earth, and greatly on other planets.
Positive and negative signs of acceleration are defined according to whether the v-t graph slopes up or
down. This definition has the advantage that a force in a given direction always produces the same sign of
acceleration.
The area under the v-t graph gives ∆x, and analogously the area under the a-t graph gives ∆v.
For motion with constant acceleration, the following three equations hold:

∆x = v o∆t + 1 a∆t 2
2

v f2 = v o + 2a∆x
2

a = ∆v
∆t
They are not valid if the acceleration is changing.


Homework Problems
1 . The graph represents the velocity of a bee along a straight line. At t=0,
the bee is at the hive. (a) When is the bee farthest from the hive? (b) How
far is the bee at its farthest point from the hive? (c) At t=13 s, how far is the
bee from the hive? [Hint: Try problem 19 first.]

5
4
3
2
1
velocity
(m/s) 0
-1
-2
-3
-4
0 1 2 3 4 5 6 7 8 9 10 11 12 13
time (s)

2. A rock is dropped in a pond. Draw plots of its position versus time,
velocity versus time, and acceleration versus time. Include its whole motion,
from the moment it is dropped to the moment it comes to rest on the
bottom of the pond.
3. In an 18th-century naval battle, a cannon ball is shot horizontally, passes
through the side of an enemy ship's hull, flies across the galley, and lodges
in a bulkhead. Draw plots of its horizontal position, velocity, and accelera-
tion as functions of time, starting while it is inside the cannon and has not
yet been fired, and ending when it comes to rest.
4. Draw graphs of position, velocity, and acceleration as functions of time
for a person bunjee jumping. (In bunjee jumping, a person has a stretchy
elastic cord tied to his/her ankles, and jumps off of a high platform. At the
bottom of the fall, the cord brings the person up short. Presumably the
Problem 3.
person bounces up a little.)
5. A ball rolls down the ramp shown in the figure below, consisting of a
circular knee, a straight slope, and a circular bottom. For each part of the
ramp, tell whether the ball’s velocity is increasing, decreasing, or constant,
and also whether the ball’s acceleration is increasing, decreasing, or con-
stant. Explain your answers. Assume there is no air friction or rolling
resistance. Hint: Try problem 20 first. [Based on a problem by Hewitt.]
Problem 5.



6. At the end of its arc, the velocity of a pendulum is zero. Is its acceleration
also zero at this point? Explain using a v-t graph.
7. What is the acceleration of a car that moves at a steady velocity of 100
km/h for 100 seconds? Explain your answer.
8. A physics homework question asks, If you start from rest and accelerate
at 1.54 m/s2 for 3.29 s, how far do you travel by the end of that time? A
student answers as follows:
1.54 x 3.29 = 5.07 m
His Aunt Wanda is good with numbers, but has never taken physics. She
doesn't know the formula for the distance traveled under constant accelera-
tion over a given amount of time, but she tells her nephew his answer
cannot be right. How does she know?
9 . You are looking into a deep well. It is dark, and you cannot see the
bottom. You want to find out how deep it is, so you drop a rock in, and you
hear a splash 3 seconds later. Approximately how deep is the well?
10 . You take a trip in your spaceship to another star. Setting off, you
increase your speed at a constant acceleration. Once you get half-way there,
you start decelerating, at the same rate, so that by the time you get there,
you have slowed down to zero speed. You see the tourist attractions, and
then head home by the same method.
(a) Find a formula for the time, T, required for the round trip, in terms of d,
the distance from our sun to the star, and a, the magnitude of the accelera-
tion. Note that the acceleration is not constant over the whole trip, but the
trip can be broken up into constant-acceleration parts.
(b) The nearest star to the Earth (other than our own sun) is Proxima
Centauri, at a distance of d=4x1016 m. Suppose you use an acceleration of
a=10 m/s2, just enough to compensate for the lack of true gravity and make
you feel comfortable. How long does the round trip take, in years?
(c) Using the same numbers for d and a, find your maximum speed.
Compare this to the speed of light, which is 3.0x108 m/s. (Later in this
course, you will learn that there are some new things going on in physics
when one gets close to the speed of light, and that it is impossible to exceed
the speed of light. For now, though, just use the simpler ideas you've
learned so far.)
11. You climb half-way up a tree, and drop a rock. Then you climb to the
top, and drop another rock. How many times greater is the velocity of the
second rock on impact? Explain. (The answer is not two times greater.)
12. Two children stand atop a tall building. One drops a rock over the edge,


while simultaneously the second throws a rock downward so that it has an
initial speed of 10 m/s. Compare the accelerations of the two objects while
in flight.
13 ∫. A person is parachute jumping. During the time between when she
leaps out of the plane and when she opens her chute, her altitude is given by
an equation of the form

y = b – c t + ke – t / k ,

where e is the base of natural logarithms, and b, c, and k are constants.
Because of air resistance, her velocity does not increase at a steady rate as it
would for an object falling in vacuum.
(a) What units would b, c, and k have to have for the equation to make
sense?
(b) Find the person's velocity, v, as a function of time. [You will need to use
the chain rule, and the fact that d(ex)/dx=ex.]
(c) Use your answer from part (b) to get an interpretation of the constant c.
[Hint: e –x approaches zero for large values of x.]
x (d) Find the person's acceleration, a, as a function of time.
t
(e) Use your answer from part (b) to show that if she waits long enough to
open her chute, her acceleration will become very small.
14 S. The top part of the figure shows the position-versus-time graph for an
v t object moving in one dimension. On the bottom part of the figure, sketch
the corresponding v-versus-t graph.
15 S. On New Year's Eve, a stupid person fires a pistol straight up. The
Problem 14.
bullet leaves the gun at a speed of 100 m/s. How long does it take before
the bullet hits the ground?
16 S. If the acceleration of gravity on Mars is 1/3 that on Earth, how many
times longer does it take for a rock to drop the same distance on Mars?
Ignore air resistance.
17 S∫. A honeybee’s position as a function of time is given by x=10t-t3,
where t is in seconds and x in meters. What is its acceleration at t=3.0 s?
18 S. In July 1999, Popular Mechanics carried out tests to find which car
sold by a major auto maker could cover a quarter mile (402 meters) in the
shortest time, starting from rest. Because the distance is so short, this type
of test is designed mainly to favor the car with the greatest acceleration, not
the greatest maximum speed (which is irrelevant to the average person). The
winner was the Dodge Viper, with a time of 12.08 s. The car’s top (and
presumably final) speed was 118.51 miles per hour (52.98 m/s). (a) If a car,
starting from rest and moving with constant acceleration, covers a quarter
mile in this time interval, what is its acceleration? (b) What would be the
final speed of a car that covered a quarter mile with the constant accelera-
tion you found in part a? (c) Based on the discrepancy between your answer
in part b and the actual final speed of the Viper, what do you conclude
about how its acceleration changed over time?


5 19 S. The graph represents the motion of a rolling ball that bounces off of a
wall. When does the ball return to the location it had at t=0?
20 S. (a) The ball is released at the top of the ramp shown in the figure.
v
(m/s) 0 Friction is negligible. Use physical reasoning to draw v-t and a-t graphs.
Assume that the ball doesn’t bounce at the point where the ramp changes
slope. (b) Do the same for the case where the ball is rolled up the slope from
–5 the right side, but doesn’t quite have enough speed to make it over the top.
0 5 10 21 S. You drop a rubber ball, and it repeatedly bounces vertically. Draw
t (s) graphs of position, velocity, and acceleration as functions of time.
Problem 19. 22 S. Starting from rest, a ball rolls down a ramp, traveling a distance L and
picking up a final speed v. How much of the distance did the ball have to
cover before achieving a speed of v/2? [Based on a problem by Arnold
Arons.]

Problem 20.


Even as great and skeptical a genius as Galileo was unable to
make much progress on the causes of motion. It was not until a
generation later that Isaac Newton (1642-1727) was able to attack
the problem successfully. In many ways, Newton’s personality was
the opposite of Galileo’s. Where Galileo agressively publicized his
ideas, Newton had to be coaxed by his friends into publishing a book
on his physical discoveries. Where Galileo’s writing had been popular
and dramatic, Newton originated the stilted, impersonal style that most
people think is standard for scientific writing. (Scientific journals today
encourage a less ponderous style, and papers are often written in the
first person.) Galileo’s talent for arousing animosity among the rich
and powerful was matched by Newton’s skill at making himself a
popular visitor at court. Galileo narrowly escaped being burned at the
stake, while Newton had the good fortune of being on the winning
side of the revolution that replaced King James II with William and
Mary of Orange, leading to a lucrative post running the English royal
mint.
Newton discovered the relationship between force and motion,
and revolutionized our view of the universe by showing that the same
physical laws applied to all matter, whether living or nonliving, on or
off of our planet’s surface. His book on force and motion, the
Mathematical Principles of Natural Philosophy, was uncontradicted
by experiment for 200 years, but his other main work, Optics, was on
the wrong track due to his conviction that light was composed of
Isaac Newton particles rather than waves. Newton was also an avid alchemist and
an astrologer, an embarrassing fact that modern scientists would like
to forget.

4 Force and Motion
If I have seen farther than others, it is because I have stood on the shoul-
ders of giants.
Newton, referring to Galileo

4.1 Force
We need only explain changes in motion, not motion itself
So far you’ve studied the measurement of motion in some detail, but
not the reasons why a certain object would move in a certain way. This
chapter deals with the “why” questions. Aristotle’s ideas about the causes of
motion were completely wrong, just like all his other ideas about physical
science, but it will be instructive to start with them, because they amount to
a road map of modern students’ incorrect preconceptions.
Aristotle said motion had to be caused Aristotle thought he needed to explain both why motion occurs and
by a force. To explain why an arrow why motion might change. Newton inherited from Galileo the important
kept flying after the bowstring was no counter-Aristotelian idea that motion needs no explanation, that it is only
longer pushing on it, he said the air
rushed around behind the arrow and
changes in motion that require a physical cause.
pushed it forward. We know this is Aristotle gave three reasons for motion:
wrong, because an arrow shot in a
vacuum chamber does not instantly • Natural motion, such as falling, came from the tendency of objects
drop to the floor as it leaves the bow. to go to their “natural” place, on the ground, and come to rest.
Galileo and Newton realized that a
• Voluntary motion was the type of motion exhibited by animals,
force would only be needed to change
the arrow’s motion, not to make its which moved because they chose to.
motion continue. • Forced motion occurred when an object was acted on by some
other object that made it move.

© 1998 Benjamin Crowell 97

Motion changes due to an interaction between two objects
In the Aristotelian theory, natural motion and voluntary motion are
one-sided phenomena: the object causes its own motion. Forced motion is
supposed to be a two-sided phenomenon, because one object imposes its
“commands” on another. Where Aristotle conceived of some of the phe-
nomena of motion as one-sided and others as two-sided, Newton realized
that a change in motion was always a two-sided relationship of a force
acting between two physical objects.
The one-sided “natural motion” description of falling makes a crucial
omission. The acceleration of a falling object is not caused by its own
“natural” tendencies but by an attractive force between it and the planet
Earth. Moon rocks brought back to our planet do not “want” to fly back up
to the moon because the moon is their “natural” place. They fall to the floor
when you drop them, just like our homegrown rocks. As we’ll discuss in
more detail later in this course, gravitational forces are simply an attraction
that occurs between any two physical objects. Minute gravitational forces
can even be measured between human-scale objects in the laboratory.
The idea of natural motion also explains incorrectly why things come to
rest. A basketball rolling across a beach slows to a stop because it is interact-
ing with the sand via a frictional force, not because of its own desire to be at
rest. If it was on a frictionless surface, it would never slow down. Many of
Aristotle’s mistakes stemmed from his failure to recognize friction as a force.
The concept of voluntary motion is equally flawed. You may have been
a little uneasy about it from the start, because it assumes a clear distinction
between living and nonliving things. Today, however, we are used to having
the human body likened to a complex machine. In the modern world-view,
“Our eyes receive blue light reflected the border between the living and the inanimate is a fuzzy no-man’s land
from this painting because Monet inhabited by viruses, prions, and silicon chips. Furthermore, Aristotle’s
wanted to represent water with the statement that you can take a step forward “because you choose to” inap-
color blue.” This is a valid statement propriately mixes two levels of explanation. At the physical level of explana-
at one level of explanation, but physics
works at the physical level of tion, the reason your body steps forward is because of a frictional force
explanation, in which blue light gets acting between your foot and the floor. If the floor was covered with a
to your eyes because it is reflected by puddle of oil, no amount of “choosing to” would enable you to take a
blue pigments in the paint. graceful stride forward.
Forces can all be measured on the same numerical scale
In the Aristotelian-scholastic tradition, the description of motion as
natural, voluntary, or forced was only the broadest level of classification, like
splitting animals into birds, reptiles, mammals, and amphibians. There
might be thousands of types of motion, each of which would follow its own
rules. Newton’s realization that all changes in motion were caused by two-
sided interactions made it seem that the phenomena might have more in
common than had been apparent. In the Newtonian description, there is
only one cause for a change in motion, which we call force. Forces may be
of different types, but they all produce changes in motion according to the
same rules. Any acceleration that can be produced by a magnetic force can
equally well be produced by an appropriately controlled stream of water. We
can speak of two forces as being equal if they produce the same change in
motion when applied in the same situation, which means that they pushed
or pulled equally hard in the same direction.

98 Chapter 4 Force and Motion

The idea of a numerical scale of force and the newton unit were intro-
duced in chapter 0. To recapitulate briefly, a force is when a pair of objects
push or pull on each other, and one newton is the force required to acceler-
ate a 1-kg object from rest to a speed of 1 m/s in 1 second.
More than one force on an object
As if we hadn’t kicked poor Aristotle around sufficiently, his theory has
another important flaw, which is important to discuss because it corre-
sponds to an extremely common student misconception. Aristotle con-
ceived of forced motion as a relationship in which one object was the boss
and the other “followed orders.” It therefore would only make sense for an
object to experience one force at a time, because an object couldn’t follow
orders from two sources at once. In the Newtonian theory, forces are
numbers, not orders, and if more than one force acts on an object at once,
the result is found by adding up all the forces. It is unfortunate that the use
the English word “force” has become standard, because to many people it
suggests that you are “forcing” an object to do something. The force of the
earth’s gravity cannot “force” a boat to sink, because there are other forces
acting on the boat. Adding them up gives a total of zero, so the boat
accelerates neither up nor down.
Objects can exert forces on each other at a distance
Aristotle declared that forces could only act between objects that were
touching, probably because he wished to avoid the type of occult specula-
tion that attributed physical phenomena to the influence of a distant and
invisible pantheon of gods. He was wrong, however, as you can observe
when a magnet leaps onto your refrigerator or when the planet earth exerts
gravitational forces on objects that are in the air. Some types of forces, such
as friction, only operate between objects in contact, and are called contact
forces. Magnetism, on the other hand, is an example of a noncontact force.
Although the magnetic force gets stronger when the magnet is closer to
your refrigerator, touching is not required.
Weight
In physics, an object’s weight , FW, is defined as the earth’s gravitational
force on it. The SI unit of weight is therefore the Newton. People com-
monly refer to the kilogram as a unit of weight, but the kilogram is a unit of
mass, not weight. Note that an object’s weight is not a fixed property of that
object. Objects weigh more in some places than in others, depending on the
local strength of gravity. It is their mass that always stays the same. A
+8 N
-3 N baseball pitcher who can throw a 90-mile-per-hour fastball on earth would
+4 N not be able to throw any faster on the moon, because the ball’s inertia
would still be the same.
+2 N Positive and negative signs of force
We’ll start by considering only cases of one-dimensional center-of-mass
In this example, positive signs have
motion in which all the forces are parallel to the direction of motion, i.e.
been used consistently for forces to either directly forward or backward. In one dimension, plus and minus
the right, and negative signs for forces signs can be used to indicate directions of forces, as shown in the figure. We
to the left. The numerical value of a can then refer generically to addition of forces, rather than having to speak
force carries no information about the sometimes of addition and sometimes of subtraction. We add the forces
place on the saxophone where the
force is applied. shown in the figure and get 11 N. In general, we should choose a one-

Section 4.1 Force 99

dimensional coordinate system with its x axis parallel the direction of
motion. Forces that point along the positive x axis are positive, and forces in
the opposite direction are negative. Forces that are not directly along the x
axis cannot be immediately incorporated into this scheme, but that’s OK,
because we’re avoiding those cases for now.
In chapter 0, I defined 1 N as the force that would accelerate a 1-kg mass from
rest to 1 m/s in 1 s. Anticipating the following section, you might guess that 2
N could be defined as the force that would accelerate the same mass to twice
the speed, or twice the mass to the same speed. Is there an easier way to
define 2 N based on the definition of 1 N?

4.2 Newton’s First Law
We are now prepared to make a more powerful restatement of the
principle of inertia.

Newton's First Law
If the total force on an object is zero, its center of mass
continues in the same state of motion.

In other words, an object initially at rest is predicted to remain at rest if the
total force on it is zero, and an object in motion remains in motion with the
same velocity in the same direction. The converse of Newton’s first law is
also true: if we observe an object moving with constant velocity along a
straight line, then the total force on it must be zero.
In a future physics course or in another textbook, you may encounter
the term net force, which is simply a synonym for total force.
What happens if the total force on an object is not zero? It accelerates.
Numerical prediction of the resulting acceleration is the topic of Newton’s
second law, which we’ll discuss in the following section.
This is the first of Newton’s three laws of motion. It is not important to
memorize which of Newton’s three laws are numbers one, two, and three. If
a future physics teacher asks you something like, “Which of Newton’s laws
are you thinking of,” a perfectly acceptable answer is “The one about
constant velocity when there’s zero total force.” The concepts are more
important than any specific formulation of them. Newton wrote in Latin,
and I am not aware of any modern textbook that uses a verbatim translation
of his statement of the laws of motion. Clear writing was not in vogue in
Newton’s day, and he formulated his three laws in terms of a concept now
called momentum, only later relating it to the concept of force. Nearly all
modern texts, including this one, start with force and do momentum later.
Example: an elevator
Question: An elevator has a weight of 5000 N. Compare the
forces that the cable must exert to raise it at constant velocity,
lower it at constant velocity, and just keep it hanging.
Answer: In all three cases the cable must pull up with a force of
exactly 5000 N. Most people think you’d need at least a little
more than 5000 N to make it go up, and a little less than 5000 N
to let it down, but that’s incorrect. Extra force from the cable is


only necessary for speeding the car up when it starts going up or
slowing it down when it finishes going down. Decreased force is
needed to speed the car up when it gets going down and to slow
it down when it finishes going up. But when the elevator is
cruising at constant velocity, Newton’s first law says that you just
need to cancel the force of the earth’s gravity.
To many students, the statement in the example that the cable’s upward
force “cancels” the earth’s downward gravitational force implies that there
has been a contest, and the cable’s force has won, vanquishing the earth’s
gravitational force and making it disappear. That is incorrect. Both forces
continue to exist, but because they add up numerically to zero, the elevator
has no center-of-mass acceleration. We know that both forces continue to
exist because they both have side-effects other than their effects on the car’s
center-of-mass motion. The force acting between the cable and the car
continues to produce tension in the cable and keep the cable taut. The
earth’s gravitational force continues to keep the passengers (whom we are
considering as part of the elevator-object) stuck to the floor and to produce
internal stresses in the walls of the car, which must hold up the floor.
Example: terminal velocity for falling objects
Question: An object like a feather that is not dense or
streamlined does not fall with constant acceleration, because air
resistance is nonnegligible. In fact, its acceleration tapers off to
nearly zero within a fraction of a second, and the feather finishes
dropping at constant speed (known as its terminal velocity). Why
does this happen?
Answer: Newton’s first law tells us that the total force on the
feather must have been reduced to nearly zero after a short time.
There are two forces acting on the feather: a downward
gravitational force from the planet earth, and an upward frictional
force from the air. As the feather speeds up, the air friction
becomes stronger and stronger, and eventually it cancels out the
earth’s gravitational force, so the feather just continues with
constant velocity without speeding up any more.
The situation for a skydiver is exactly analogous. It’s just that
the skydiver experiences perhaps a million times more
gravitational force than the feather, and it is not until she is falling
very fast that the force of air friction becomes as strong as the
gravitational force. It takes her several seconds to reach terminal
velocity, which is on the order of a hundred miles per hour.

Section 4.2 Newton’s First Law 101

More general combinations of forces
It is too constraining to restrict our attention to cases where all the
forces lie along the line of the center of mass’s motion. For one thing, we
can’t analyze any case of horizontal motion, since any object on earth will be
subject to a vertical gravitational force! For instance, when you are driving
your car down a straight road, there are both horizontal forces and vertical
forces. However, the vertical forces have no effect on the center of mass
motion, because the road’s upward force simply counteracts the earth’s
downward gravitational force and keeps the car from sinking into the
ground.
Later in the book we’ll deal with the most general case of many forces
acting on an object at any angles, using the mathematical technique of
vector addition, but the following slight generalization of Newton’s first law
allows us to analyze a great many cases of interest:
Suppose that an object has two sets of forces acting on it, one set along
the line of the object’s initial motion and another set perpendicular to
the first set. If both sets of forces cancel, then the object’s center of mass
continues in the same state of motion.

Example: a car crash
Question: If you drive your car into a brick wall, what is the
mysterious force that slams your face into the steering wheel?
Answer: Your surgeon has taken physics, so she is not going to
believe your claim that a mysterious force is to blame. She
knows that your face was just following Newton’s first law.
Immediately after your car hit the wall, the only forces acting on
your head were the same canceling-out forces that had existed
previously: the earth’s downward gravitational force and the
upward force from your neck. There were no forward or
backward forces on your head, but the car did experience a
backward force from the wall, so the car slowed down and your
face caught up.

Example: a passenger riding the subway
Question: Describe the forces acting on a person standing in a
subway train that is cruising at constant velocity.
Answer: No force is necessary to keep the person moving
relative to the ground. He will not be swept to the back of the
train if the floor is slippery. There are two vertical forces on him,
the earth’s downward gravitational force and the floor’s upward
force, which cancel. There are no horizontal forces on him at all,
so of course the total horizontal force is zero.

air's force Example: forces on a sailboat
on sail Question: If a sailboat is cruising at constant velocity with the
wind coming from directly behind it, what must be true about the
water's bouyant forces acting on it?
force on boat
Answer: The forces acting on the boat must be canceling each
water's frictional
other out. The boat is not sinking or leaping into the air, so
force on boat
evidently the vertical forces are canceling out. The vertical forces
earth's gravitational are the downward gravitational force exerted by the planet earth
force on boat and an upward force from the water.
The air is making a forward force on the sail, and if the boat
is not accelerating horizontally then the water’s backward


frictional force must be canceling it out.
Contrary to Aristotle, more force is not needed in order to
maintain a higher speed. Zero total force is always needed to
maintain constant velocity. Consider the following made-up
numbers:

boat moving at a boat moving at
low, constant a high, constant
velocity velocity
forward force of the
wind on the sail...... 10,000 N 20,000 N
backward force of
the water on the
hull........................ -10,000 N -20,000 N
total force on the
boat...................... 0N 0N

The faster boat still has zero total force on it. The forward force
on it is greater, and the backward force smaller (more negative),
but that’s irrelevant because Newton’s first law has to do with the
total force, not the individual forces.
This example is quite analogous to the one about terminal
velocity of falling objects, since there is a frictional force that
increases with speed. After casting off from the dock and raising
the sail, the boat will accelerate briefly, and then reach its
terminal velocity, at which the water’s frictional force has become
as great as the wind’s force on the sail.
A. Newton said that objects continue moving if no forces are acting on them,
but his predecessor Aristotle said that a force was necessary to keep an object
moving. Why does Aristotle’s theory seem more plausible, even though we
now believe it to be wrong? What insight was Aristotle missing about the
reason why things seem to slow down naturally?
B. In the first figure, what would have to be true about the saxophone’s initial
motion if the forces shown were to result in continued one-dimensional
motion?
C. The second figure requires an ever further generalization of the preceding
discussion. After studying the forces, what does your physical intuition tell you
will happen? Can you state in words how to generalize the conditions for one-
dimensional motion to include situations like this one?

3N 8N

2N 3N

4N

4N
Discussion question B. Discussion question C.

Section 4.2 Newton’s First Law 103

4.3 Newton’s Second Law
What about cases where the total force on an object is not zero, so that
Newton’s first law doesn’t apply? The object will have an acceleration. The
way we’ve defined positive and negative signs of force and acceleration
guarantees that positive forces produce positive accelerations, and likewise
for negative values. How much acceleration will it have? It will clearly
depend on both the object’s mass and on the amount of force.
Experiments with any particular object show that its acceleration is
directly proportional to the total force applied to it. This may seem wrong,
since we know of many cases where small amounts of force fail to move an
object at all, and larger forces get it going. This apparent failure of propor-
tionality actually results from forgetting that there is a frictional force in
addition to the force we apply to move the object. The object’s acceleration
is exactly proportional to the total force on it, not to any individual force on
it. In the absence of friction, even a very tiny force can slowly change the
velocity of a very massive object.
Experiments also show that the acceleration is inversely proportional to
the object’s mass, and combining these two proportionalities gives the
following way of predicting the acceleration of any object:
Newton’s Second Law
a = Ftotal/m ,
where
m is an object’s mass
Ftotal is the sum of the forces acting on it, and
a is the acceleration of the object’s center of mass.
We are presently restricted to the case where the forces of interest are
parallel to the direction of motion.
Example: an accelerating bus
Question: A VW bus with a mass of 2000 kg accelerates from 0
to 25 m/s (freeway speed) in 34 s. Assuming the acceleration is
constant, what is the total force on the bus?
Solution: We solve Newton’s second law for Ftotal=ma, and
substitute ∆v/∆t for a, giving
Ftotal = m∆v/∆t
= (2000 kg)(25 m/s - 0 m/s)/(34 s)
= 1.5 kN .
A generalization
As with the first law, the second law can be easily generalized to include
a much larger class of interesting situations:
Suppose an object is being acted on by two sets of forces, one set
lying along the object’s initial direction of motion and another set
acting along a perpendicular line. If the forces perpendicular to the
initial direction of motion cancel out, then the object accelerates
along its original line of motion according to a=Ftotal/m.
The relationship between mass and weight
Mass is different from weight, but they’re related. An apple’s mass tells


us how hard it is to change its motion. Its weight measures the strength of
the gravitational attraction between the apple and the planet earth. The
apple’s weight is less on the moon, but its mass is the same. Astronauts
assembling the International Space Station in zero gravity cannot just pitch
massive modules back and forth with their bare hands; the modules are
weightless, but not massless.
We have already seen the experimental evidence that when weight (the
force of the earth’s gravity) is the only force acting on an object, its accelera-
tion equals the constant g, and g depends on where you are on the surface of
the earth, but not on the mass of the object. Applying Newton’s second law
A simple double-pan balance works by then allows us to calculate the magnitude of the gravitational force on any
comparing the weight forces exerted object in terms of its mass:
by the earth on the contents of the two
pans. Since the two pans are at almost |FW| = mg .
the same location on the earth’s
surface, the value of g is essentially
(The equation only gives the magnitude, i.e. the absolute value, of FW,
the same for each one, and equality because we’re defining g as a positive number, so it equals the absolute value
of weight therefore also implies of a falling object’s acceleration.)
equality of mass.
Example: calculating terminal velocity
Question: Experiments show that the force of air friction on a
falling object such as a skydiver or a feather can be
approximated fairly well with the equation |Fair|=cρAv2, where c is
a constant, ρ is the density of the air, A is the cross-sectional
area of the object as seen from below, and v is the object’s
velocity. Predict the object’s terminal velocity, i.e. the final
velocity it reaches after a long time.
Solution: As the object accelerates, its greater v causes the
upward force of the air to increase until finally the gravitational
force and the force of air friction cancel out, after which the
object continues at constant velocity. We choose a coordinate
system in which positive is up, so that the gravitational force is
negative and the force of air friction is positive. We want to find
the velocity at which
Fair + FW = 0 , i.e.
cρAv 2 – mg = 0 .
Solving for v gives
mg
vterminal = cρA
Self-Check
It is important to get into the habit of interpreting equations. These two self-
check questions may be difficult for you, but eventually you will get used to this
kind of reasoning.
(a) Interpret the equation vterminal = m g / c ρ A in the case of ρ=0.
(b) How would the terminal velocity of a 4-cm steel ball compare to that of a 1-
cm ball?

(a) The case of ρ=0 represents an object falling in a vacuum, i.e. there is no density of air. The terminal velocity
would be infinite. Physically, we know that an object falling in a vacuum would never stop speeding up, since there
would be no force of air friction to cancel the force of gravity. (b) The 4-cm ball would have a mass that was greater
by a factor of 4x4x4, but its cross-sectional area would be greater by a factor of 4x4. Its terminal velocity would be
greater by a factor of 4 3 / 4 2 =2.

Section 4.3 Newton’s Second Law 105

A. Show that the Newton can be reexpressed in terms of the three basic mks
units as the combination kg.m/s2.
B. What is wrong with the following statements?
1. “g is the force of gravity.”
x (m) t (s) 2. “Mass is a measure of how much space something takes up.”
10 1.84 C. Criticize the following incorrect statement:
20 2.86 “If an object is at rest and the total force on it is zero, it stays at rest.
30 3.80 There can also be cases where an object is moving and keeps on moving
without having any total force on it, but that can only happen when there’s
40 4.67
no friction, like in outer space.”
50 5.53 D. The table on the left gives laser timing data for Ben Johnson’s 100 m dash
60 6.38 at the 1987 World Championship in Rome. (His world record was later revoked
70 7.23 because he tested positive for steroids.) How does the total force on him
80 8.10 change over the duration of the race?
90 8.96
100 9.83
Discussion question D.

4.4 What Force Is Not
Violin teachers have to endure their beginning students’ screeching. A
frown appears on the woodwind teacher’s face as she watches her student
take a breath with an expansion of his ribcage but none in his belly. What
makes physics teachers cringe is their students’ verbal statements about
forces. Below I have listed several dicta about what force is not.
Force is not a property of one object.
A great many of students’ incorrect descriptions of forces could be cured
by keeping in mind that a force is an interaction of two objects, not a
property of one object.
Incorrect statement: “That magnet has a lot of force.”
# If the magnet is one millimeter away from a steel ball bearing,
they may exert a very strong attraction on each other, but if they
were a meter apart, the force would be virtually undetectable.
The magnet’s strength can be rated using certain electrical units
(ampere-meters2), but not in units of force.
Force is not a measure of an object’s motion.
If force is not a property of a single object, then it cannot be used as a
measure of the object’s motion.
Incorrect statement: “The freight train rumbled down the tracks
with awesome force.”
# Force is not a measure of motion. If the freight train collides
with a stalled cement truck, then some awesome forces will
occur, but if it hits a fly the force will be small.
Force is not energy.
There are two main approaches to understanding the motion of objects,
one based on force and one on a different concept, called energy. The SI
unit of energy is the Joule, but you are probably more familiar with the
calorie, used for measuring food’s energy, and the kilowatt-hour, the unit
the electric company uses for billing you. Physics students’ previous famil-
iarity with calories and kilowatt-hours is matched by their universal unfa-
miliarity with measuring forces in units of Newtons, but the precise opera-
tional definitions of the energy concepts are more complex than those of the


force concepts, and textbooks, including this one, almost universally place
the force description of physics before the energy description. During the
long period after the introduction of force and before the careful definition
of energy, students are therefore vulnerable to situations in which, without
realizing it, they are imputing the properties of energy to phenomena of
force.
Incorrect statement: “How can my chair be making an upward
force on my rear end? It has no power!”
# Power is a concept related to energy, e.g. 100-watt lightbulb
uses up 100 joules per second of energy. When you sit in a chair,
no energy is used up, so forces can exist between you and the
chair without any need for a source of power.
Force is not stored or used up.
Because energy can be stored and used up, people think force also can
be stored or used up.
Incorrect statement: “If you don’t fill up your tank with gas, you’ll
run out of force.”
# Energy is what you’ll run out of, not force.
Forces need not be exerted by living things or machines.
Transforming energy from one form into another usually requires some
kind of living or mechanical mechanism. The concept is not applicable to
forces, which are an interaction between objects, not a thing to be trans-
ferred or transformed.
Incorrect statement: “How can a wooden bench be making an
upward force on my rear end? It doesn’t have any springs or
anything inside it.”
# No springs or other internal mechanisms are required. If the
bench didn’t make any force on you, you would obey Newton’s
second law and fall through it. Evidently it does make a force on
you!
A force is the direct cause of a change in motion.
I can click a remote control to make my garage door change from being
at rest to being in motion. My finger’s force on the button, however, was
not the force that acted on the door. When we speak of a force on an object
in physics, we are talking about a force that acts directly. Similarly, when
you pull a reluctant dog along by its leash, the leash and the dog are making
forces on each other, not your hand and the dog. The dog is not even
touching your hand.
Self-Check
Which of the following things can be correctly described in terms of force?
(a) A nuclear submarine is charging ahead at full steam.
(b) A nuclear submarine’s propellers spin in the water.
(c) A nuclear submarine needs to refuel its reactor periodically.
A. Criticize the following incorrect statement: “If you shove a book across a
table, friction takes away more and more of its force, until finally it stops.”
B. You hit a tennis ball against a wall. Explain any and all incorrect ideas in the
following description of the physics involved: “The ball gets some force from
you when you hit it, and when it hits the wall, it loses part of that force, so it
doesn’t bounce back as fast. The muscles in your arm are the only things that
a force can come from.”

(a) This is motion, not force. (b) This is a description of how the sub is able to get the water to produce a forward
force on it. (c) The sub runs out of energy, not force.

Section 4.4 What Force Is Not 107

4.5 Inertial and Noninertial Frames of Reference
One day, you’re driving down the street in your pickup truck, on your
way to deliver a bowling ball. The ball is in the back of the truck, enjoying
its little jaunt and taking in the fresh air and sunshine. Then you have to
slow down because a stop sign is coming up. As you brake, you glance in
your rearview mirror, and see your trusty companion accelerating toward
you. Did some mysterious force push it forward? No, it only seems that way
because you and the car are slowing down. The ball is faithfully obeying
Newton’s first law, and as it continues at constant velocity it gets ahead
relative to the slowing truck. No forces are acting on it (other than the same
canceling-out vertical forces that were always acting on it). The ball only
appeared to violate Newton’s first law because there was something wrong
with your frame of reference, which was based on the truck.
How, then, are we to tell in which frames of reference Newton’s laws are
valid? It’s no good to say that we should avoid moving frames of reference,
because there is no such thing as absolute rest or absolute motion. All
frames can be considered as being either at rest or in motion. According to

(b) In an inertial frame of reference, which the surface of the earth
(a) In a frame of reference that moves with approximately is, the bowling ball obeys Newton's first law. It
the truck, the bowling ball appears to violate moves equal distances in equal time intervals, i.e. maintains
Newton's first law by accelerating despite constant velocity. In this frame of reference, it is the truck that
having no horizontal forces on it. appears to have a change in velocity, which makes sense, since
the road is making a horizontal force on it.


an observer in India, the strip mall that constituted the frame of reference in
panel (b) of the figure was moving along with the earth’s rotation at hun-
dreds of miles per hour.
The reason why Newton’s laws fail in the truck’s frame of reference is
not because the truck is moving but because it is accelerating. (Recall that
physicists use the word to refer either to speeding up or slowing down.)
Newton’s laws were working just fine in the moving truck’s frame of
reference as long as the truck was moving at constant velocity. It was only
when its speed changed that there was a problem. How, then, are we to tell
which frames are accelerating and which are not? What if you claim that
your truck is not accelerating, and the sidewalk, the asphalt, and the Burger
King are accelerating? The way to settle such a dispute is to examine the
motion of some object, such as the bowling ball, which we know has zero
total force on it. Any frame of reference in which the ball appears to obey
Newton’s first law is then a valid frame of reference, and to an observer in
that frame, Mr. Newton assures us that all the other objects in the universe
will obey his laws of motion, not just the ball.
Valid frames of reference, in which Newton’s laws are obeyed, are called
inertial frames of reference. Frames of reference that are not inertial are called
noninertial frames. In those frames, objects violate the principle of inertia
and Newton’s first law. While the truck was moving at constant velocity,
both it and the sidewalk were valid inertial frames. The truck became an
invalid frame of reference when it began changing its velocity.
You usually assume the ground under your feet is a perfectly inertial
frame of reference, and we made that assumption above. It isn’t perfectly
inertial, however. Its motion through space is quite complicated, being
composed of a part due to the earth’s daily rotation around its own axis, the
monthly wobble of the planet caused by the moon’s gravity, and the rota-
tion of the earth around the sun. Since the accelerations involved are
numerically small, the earth is approximately a valid inertial frame.
Noninertial frames are avoided whenever possible, and we will seldom,
if ever, have occasion to use them in this course. Sometimes, however, a
noninertial frame can be convenient. Naval gunners, for instance, get all
their data from radars, human eyeballs, and other detection systems that are
moving along with the earth’s surface. Since their guns have ranges of many
miles, the small discrepancies between their shells’ actual accelerations and
the accelerations predicted by Newton’s second law can have effects that
accumulate and become significant. In order to kill the people they want to
kill, they have to add small corrections onto the equation a=Ftotal/m. Doing
their calculations in an inertial frame would allow them to use the usual
form of Newton’s second law, but they would have to convert all their data
into a different frame of reference, which would require cumbersome
calculations.
Discussion question
If an object has a linear x-t graph in a certain inertial frame, what is the effect
on the graph if we change to a coordinate system with a different origin? What
is the effect if we keep the same origin but reverse the positive direction of the
x axis? How about an inertial frame moving alongside the object? What if we
describe the object’s motion in a noninertial frame?

Section 4.5 Inertial and Noninertial Frames of Reference 109

Summary
Selected Vocabulary
weight ............................... the force of gravity on an object, equal to mg
inertial frame ..................... a frame of reference that is not accelerating, one in which Newton’s first
law is true
noninertial frame ............... an accelerating frame of reference, in which Newton’s first law is violated
Terminology Used in Some Other Books
net force ............................ another way of saying “total force”
Notation
FW ..................................................... the weight force
Summary
Newton’s first law of motion states that if all the forces on an object cancel each other out, then the object
continues in the same state of motion. This is essentially a more refined version of Galileo’s principle of
inertia, which did not refer to a numerical scale of force.
Newton’s second law of motion allows the prediction of an object’s acceleration given its mass and the
total force on it, a=Ftotal/m. This is only the one-dimensional version of the law; the full-three dimensional
treatment will come in chapter 8, Vectors. Without the vector techniques, we can still say that the situation
remains unchanged by including an additional set of vectors that cancel among themselves, even if they are
not in the direction of motion.
Newton’s laws of motion are only true in frames of reference that are not accelerating, known as inertial
frames.


Homework Problems
1. An object is observed to be moving at constant speed along a line. Can
you conclude that no forces are acting on it? Explain. [Based on a problem
by Serway and Faughn.]
2. A car is normally capable of an acceleration of 3 m/s2. If it is towing a
trailer with half as much mass as the car itself, what acceleration can it
achieve? [Based on a problem from PSSC Physics.]
3. (a) Let T be the maximum tension that the elevator's cable can withstand
without breaking, i.e. the maximum force it can exert. If the motor is
programmed to give the car an acceleration a, what is the maximum mass
that the car can have, including passengers, if the cable is not to break?
[Numerical check, not for credit: for T=1.0x104 N and a=3.0 m/s2, your
equation should give an answer of 780 kg.] (b) Interpret the equation you
derived in the special cases of a=0 and of a downward acceleration of
magnitude g.
4 . A helicopter of mass m is taking off vertically. The only forces acting on
it are the earth's gravitational force and the force, Fair, of the air pushing up
on the propeller blades. (a) If the helicopter lifts off at t=0, what is its
vertical speed at time t? (b) Plug numbers into your equation from part a,
using m=2300 kg, Fair=27000 N, and t=4.0 s.
5 . In the 1964 Olympics in Tokyo, the best men's high jump was 2.18
m. Four years later in Mexico City, the gold medal in the same event was
for a jump of 2.24 m. Because of Mexico City's altitude (2400 m), the
acceleration of gravity there is lower than that in Tokyo by about 0.01 m/s2.
Suppose a high-jumper has a mass of 72 kg.
(a) Compare his mass and weight in the two locations.
(b) Assume that he is able to jump with the same initial vertical velocity
in both locations, and that all other conditions are the same except for
gravity. How much higher should he be able to jump in Mexico City?
(Actually, the reason for the big change between '64 and '68 was the
introduction of the Fosbury flop.)
6 ∫. A blimp is initially at rest, hovering, when at t=0 the pilot turns on the
motor of the propeller. The motor cannot instantly get the propeller going,
but the propeller speeds up steadily. The steadily increasing force between
the air and the propeller is given by the equation F=kt, where k is a con-
Problem 6. stant. If the mass of the blimp is m, find its position as a function of time.
(Assume that during the period of time you're dealing with, the blimp is
not yet moving fast enough to cause a significant backward force due to air
resistance.)
7 S. A car is accelerating forward along a straight road. If the force of the
road on the car's wheels, pushing it forward, is a constant 3.0 kN, and the
car's mass is 1000 kg, then how long will the car take to go from 20 m/s to
50 m/s?



8. Some garden shears are like a pair of scissors: one sharp blade slices past
another. In the “anvil” type, however, a sharp blade presses against a flat one
rather than going past it. A gardening book says that for people who are not
very physically strong, the anvil type can make it easier to cut tough
branches, because it concentrates the force on one side. Evaluate this claim
based on Newton’s laws. [Hint: Consider the forces acting on the branch,
and the motion of the branch.]

112

Rockets work by pushing exhaust gases out
the back. Newton’s third law says that if the
rocket exerts a backward force on the gases,
the gases must make an equal forward force
on the rocket. Rocket engines can function
above the atmosphere, unlike propellers and
jets, which work by pushing against the sur-
rounding air.

5 Analysis of Forces
5.1 Newton’s Third Law
Newton created the modern concept of force starting from his insight
that all the effects that govern motion are interactions between two objects:
unlike the Aristotelian theory, Newtonian physics has no phenomena in
which an object changes its own motion.
Is one object always the “order-giver” and the other the “order-fol-
lower”? As an example, consider a batter hitting a baseball. The bat defi-
nitely exerts a large force on the ball, because the ball accelerates drastically.
But if you have ever hit a baseball, you also know that the ball makes a force
on the bat — often with painful results if your technique is as bad as mine!
How does the ball’s force on the bat compare with the bat’s force on the
ball? The bat’s acceleration is not as spectacular as the ball’s, but maybe we
shouldn’t expect it to be, since the bat’s mass is much greater. In fact, careful
measurements of both objects’ masses and accelerations would show that
mballaball is very nearly equal to –mbatabat, which suggests that the ball’s force
on the bat is of the same magnitude as the bat’s force on the ball, but in the
opposite direction.

113

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