Kinematics-linear-motion equation and derivation

Describing motion
(along a line)
a.k.a. ‘the kinematics of linear motion’

Learning outcomes
• define speed and acceleration, instantaneous and average values
• explain the difference between relevant scalar and vector quantities
• apply Galilean relativity to motions in inertial frames of reference
• present a historical ‘thought experiment’ to illustrate physics thinking
• establish concepts qualitatively (using proportional reasoning) before
introducing quantitative relationships (equations)
• choose contexts for teaching kinematics that motivate student learning
• understand basic algebra and use it to rearrange kinematic equations
• draw and interpret graphs of position, velocity, acceleration
• translate information about uniform motions between words, pictures,
graphs and equations
• begin to develop a strategy for solving quantitative problems

Starting points
Misconceptions:
Heavier objects are commonly thought to fall faster than lighter objects.
Teaching challenges:
• Concepts: Some students fail to grasp the distinction between velocity
and acceleration – to them it’s simply ‘motion’. Acceleration is not
simple idea: it is the rate of change of velocity, and velocity itself is the
rate a change of distance (making acceleration the rate of change of a
rate of change).
• Graphs: Most students have difficulty with drawing and interpreting
graphs representing motion (distinguishing s - t graphs from v - t
graphs; appreciating significance of area under a v - t graph, of
gradients of s - t and v - t graphs).
• Equations: Students need help understanding that some equations
constitute definitions and that other equations apply only when there is
constant acceleration.

Kinematics – describing motion
Object is treated as a particle (a point-like concentration of
matter that has no size, no shape and no internal structure).
Questions to ask:
• Where is the particle?
• How fast is it moving?
• How rapidly is it speeding up or slowing down?
This is modelling.
Restricted to motion along a line.

Contexts
In pairs:
List other examples of real motion that might be
modelled as a particle moving along a line.
• Include some examples that can motivate students.

Uniform motion
Galileo (1638) Dialogue concerning two new sciences
Definition:
By steady or uniform motion, I mean one in which the
distances traversed by the moving particle during any
equal intervals of time, are themselves equal.

Galileo’s Two new sciences
Axioms
I The distance traversed during a _______ interval of time is greater
than the distance travelled during a _______ interval of time.
II The time required to traverse a _______ distance is longer than the
time required for a _______ distance.
III Over the same time interval, the distance traversed at a greater
speed is _______ than the distance traversed at a ______ speed.
IV The speed required to traverse a longer distance is greater than that
required to traverse a ________ distance during the same time
interval.

Galileo’s Two new sciences
Theorems
I If a moving particle, carried at a constant speed, traverses two
distances, the time intervals required are to each other in the
ratio of these distances.
II If a moving particle traverses two distances in equal intervals of
time, these distances will bear to each other the same ratio as
the speeds. And conversely, if the distances are as the speeds,
then the times …
III In the case of unequal speeds, the time intervals required to
traverse a given space are to each other inversely as the
speeds.

We say …
In symbols,
taken
time
travelled
distance
speed 
t
s
v 

Other essential ingredients
• a coordinate system
• units: metres, seconds
• scalar or vector?
– distance, displacement
– speed, velocity

Measuring distances & times
For class experiments & demonstrations
• metre rules and stopwatches
• ticker timers
• light gates
• sonic (ultrasound) sensor
• video capture
Discuss in small groups: What do you use?

Graphical representation
Uniform motion
a) List the objects below in order of increasing
speed.
b) Which of the objects have positive velocity?
c) List the objects in order of increasing velocity.

Ticker timers
Running on mains, they make 50 ticks each second.
Time between ticks is therefore 1/50 s = 0.02 s

A car is driven along a straight road. The graph shows how the
velocity of the car changes from the moment the driver sees a
very slow moving queue of traffic ahead.
Use the graph to calculate the distance the car travels while it is
slowing down.
Show clearly how you work out your answer.
Area under a v-t graph

Finding an average speed
1 If you are not already familiar with ticker timers, first
do the experiment Using the ticker-timer to measure
time
2 Do one of these two experiments.
Timing a trolley on a slope
Pupil speed

Naturally accelerated motion
Aristotle: objects fall at constant speed; the more
massive, the faster they fall.
Galileo’s thought experiment.
(Ignoring air resistance) All objects fall the same way,
getting faster and faster.
• a dramatic experimental test of this idea.
http://www.physics.ucla.edu/demoweb/demomanual/mechanics/gra
vitational_acceleration/guinea_and_feather_tube.html

Free fall
Galileo’s findings, modelled with chains.
If the time of fall is twice as long, how much further does
an object fall?
• the v–t graph gives the answer.
So what happens when an object is thrown vertically
upwards?

Graphical representation
Constant acceleration
units: km/h/s, m/s/s or m/s2
t
u
v
a



Walking the graph
Using an ultrasound sensor to generate graphs.
Every picture tells a story.

Equations of uniform motion
Two definitions:
Four relationships derived from these:
t
u
v
a
t
s
v



as
u
v
at
ut
s
t
v
u
s
at
u
v
2
2
1
2
2
2
2









 




Solving quantitative problems
A standard approach
1. Write down what you know, using conventional symbols.
2. Decide what equation to use and write it down.
3. Re-arrange the equation to make the unknown its subject.
4. Substitute values and find the unknown quantity.
5. Write answer to correct number of significant figures, with
units.

Using equations of motion
A worked example
Some problems to solve

Experiments about motion
In fours: do a few of these experiments:
Compensating for friction
Investigating free fall with a light gate
Measurement of g using an electronic timer
Finding average acceleration with a ticker-timer
Measurement of acceleration using light gates
Building a reaction tester
[from the Practical Physics website]

Using simulation software
Physlets: http://physics.bu.edu/~duffy/classroom.html
PhET: http://phet.colorado.edu/simulations/
Video analysis:
Multimedia Motion II, from Cambridge Science Media
Discuss, in small groups: What do you use?

Geogebra exercises
Distance - time simulator
Velocity Distance Time 1
Velocity Distance Time 2

Galilean relativity
… and the idea of an inertial frame of reference

Endpoints
In small groups:
Review the main ideas.
Identify anything that is not clear and clarify it by discussion.
Individually:
Decide what you need to do to consolidate any or all of this
material.

Kinematics-linear-motion equation and derivation

More Related Content

Similar to Kinematics-linear-motion equation and derivation (20)

Recently uploaded (20)

Kinematics-linear-motion equation and derivation

Editor's Notes