Kinematics in 2 dimension

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Vectors and
Kinematics in 2D
Physics Rapid Learning Series
2/48
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© Rapid Learning Inc. All rights reserved.
Wayne Huang, Ph.D.
Keith Duda, M.Ed.
Peddi Prasad, Ph.D.
Gary Zhou, Ph.D.
Michelle Wedemeyer, Ph.D.
Sarah Hedges, Ph.D.
© Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 1

Learning Objectives
By completing this tutorial, you will:
„ Understand vectors and
scalars
scalars.
„ Complete basic vector
operations.
„ Describe motion in 2
dimensions.
„ C l l t i titi
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Calculate various quantities
involving projectile motion.
Concept Map
Physics
Studies
Previous content
New content
Vectors
Motion
Described by
Kinematic
Constant
V l it
and
Math Skills
Constant
Described by
4/48
Velocity Equations Acceleration
Also apply in
May occur in May occur in
2 Dimensions

Vectors and Scalars
It is important to be able to distinguish
between a vector and a scalar
5/48
scalar.
Vector Definition
Any measurement or quantity can be categorized
into one of two types:
A vector is a
quantity that
A scalar is a
has magnitude
quantity that has
(size), and
only magnitude
direction.
(size).
6/48
Ex: Velocity,
displacement,
acceleration
Ex: speed,
distance,
temperature, mass

Vector Examples
All of the following measurements involve
magnitude and direction, thus they are vectors.
10 m/s East
-9.8 m/s2
The negative
sign indicates
the downward
direction
7/48
Vector Notation
Due to their directional nature, vectors are usually
drawn with arrows to signify their direction.
10km Southeast
Occasionally they are also drawn to scale, similar
t th l
8/48
to the scale on maps.
1 cm = 100 m

Textbook Notation
In textbooks, vectors must be distinguished from
scalars. This can be accomplished in a variety of
ways.
A vector is usually abbreviated by a letter that
identifies it as a vector in one of the following ways:
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A
Bold letter
A G
Arrow above
A
Line above
Scalar Examples
All of the following measurements involve only a
size or magnitude. They are directionless scalars.
10 seconds
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15 kg 35 miles per hour
(no direction)

Vector Addition
In order to add vectors, certain rules
must be followed
11/48
followed.
Scalar Addition vs Vector Addition
Just as scalar quantities can be added,
2 + 2 = 4
Vector quantities can be added also.
However, their rules are slightly different.
2 2
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2
2
+ doesn’t
equal 4!

Vector Addition Rule
When vectors are graphically added, they are
drawn head to tail.
This may also be described as placing the
arrowhead of one vector next to the tail end of
another vector:
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A
G G
+
B A B
Resultant
When these vectors are added in this way, the
sum, or resultant, is drawn from the tail of the
first vector to the tip of the last vector.
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A
G G G
+ =
B A B C

Calculation of the Resultant
Mathematics can be used to calculate the
magnitude and/or direction of the resultant. For
the magnitude, use the Pythagorean theorem:
G G G
+ = c2 = a2 +b2
B=
A B C
c = a2 +b2
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A= 2m
2m
c = 22 + 22
c = 8 = 2.83m
Direction of the Resultant
Simple trig functions can be used to calculate
the direction of the resultant. G G G
B
tanθ = opp
+ = adj
B
=
A B C
1
tanθ = 2m =
2m
A
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θ θ = 45D
A= 2m
2m

Vectors Added in a Line
Sometimes, vectors are added along the same
line. This simplifies things greatly.
400 mi/hr
-50 mi/hr +350 mi/hr
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•The airplane flies due west at +400 mi/hr (or mph)
•The headwind blows due east at 50 mi/hr (negative)
•Adding the vectors tip to tail gives a resultant of
+350 mi/hr.
Multiple Vectors Added
When multiple vectors are added, they are still
drawn head to tail.
C
B D
E
18/48
A
G G G G G
A +B+C+D+E =Resultant

Motion in 2
Dimensions
Often, objects don’t move at nice neat
angles There are ways to simplify these
19/48
angles. situations.
Diagonal Motions
A cannon is fired at some angle with the horizontal.
Its diagonal motion could be simplified by showing
its motion only in the horizontal and vertical
vertical
component
directions.
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θ
horizontal component

Defintion - Vector Component
Vector Component -
The parts into which a
single vector can be
separated and that act
in different directions
from the vector.
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Swimming Across a River
Resultant velocity of swimmer
Notice that he swimmers effort
moves in both
directions!
river current
i t
22/48
Swimming across the river involves motion in
two separate perpendicular directions.

Projectile
Motion
A projectile is any object moving through
the air or space that is only acted on by
23/48
gravity.
Projectile Examples
A projectile could take many forms:
• A cannon ball
• A high jumping athlete
• A ball thrown through the air
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Classic Projectile Question
A ball is dropped vertically, and one is launched
horizontally, each from the same height. Which
one hits the ground first?
v
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Same time!
The Explanation
Why does it work this way?
They both hit at
the same time
because they
both fall the
same vertical
ti l
distance and
experience the
same
acceleration due
to gravity.
Although the
thrown one
has an
additional
h i t l
Breaking an
objects
motion into
horizontal
26/48
horizontal
velocity, that
doesn’t
affect its
vertical
motion.
and vertical
components
is very
useful!!

Further Explanation
In the vertical direction, both the balls
experience the usual acceleration from
gravity. Notice the increasing falling
distance and speed in the vertical
direction.
In the horizontal direction, the projected ball
27/48
has a constant velocity. Thus, it covers
regular, consistent horizontal distances.
Thought Question Explanation
Why does it work this way?
Both fall the
same vertical
distance, and
are
accelerated
l t d
the same by
gravity, thus
the times
would be
equal.
The fired
bullet simply
has a very
large extra
This is
usually never
observed
since it
28/48
horizontal
component.
would
require a
large flat
open area.

Thought Question:
At the same time that a high speed bullet is
fired horizontally from a rifle, another bullet is
simply dropped from the same height. Which
bullet strikes the ground first?
Horizontal line of sight
Bullet to be dropped
If ld f
29/48
we could see far
downrange, we would
notice that they both hit
at the same time!
Monkey Explanation
You might think
the bullet would
Since they are in the air
for the same amount of
ti th b ll t hit th
go above the
monkey since he
is falling.
time, the bullet hits the
target!
However, the
bullet is falling
30/48
with the same
vertical
acceleration
too.

The Monkey and the Hunter
A big game hunter spots a monkey in the tree and
aims directly at him. If the monkey releases the
branch exactly as the hunter fires the bullet,
approximately where will the bullet land?
Line of
sight
31/48
Hit!
Formulas
There are no new mathematical
formulas for this tutorial topic.
However, all the old formulas
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,
from motion in 1 dimension
still apply.

Projectile Example
A cannon fires a ball from the top of a 50 m high
cliff. The projectile is fired at 100 m/s. How far
away from the base of the cliff will it land?
100 m/s
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50 m
Projectile Solution
When only the vertical direction is considered:
• Its initial vertical velocity is 0 m/s
• The distance is -50 m (negative refers to down)
• Its acceleration is -9.8 m/s2
• The time in the air, t, is unknown.
• We can use one of the previous kinematic formulas.
d v t at
2
v iv “v” refers to = +
the vertical
direction
34/48 2

Mathematics of Solution
d v t at
2
2
v iv = +
2
Since vi =0 m/s,
that term drops
out.
d at
v =
2
2dv = t
Time isn’t the final answer, but
a
35/48
it will lead us to the distance.
t 3.2 sec
2(-50m)
-9.8m/s
2 = =
,
Obtaining the Horizontal Distance
When only the horizontal motion of the
projectile is considered:
• Since gravity acts only vertically, there is 0
acceleration.
• Horizontally, the projectile moves with a
constant speed.
• We can use one of the previous kinematic
formulas.
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Constant v =
dh
horizontal
h t
velocity.

More Mathematics of Solution
v dh
h =
t
Don’t confuse this
distance with the
vertical distance
previously used.
( ) h h d = t v
d (3.2s)(100m/s) h =
37/48
d = 320m Measured from the base
h of the cliff.
Projectiles at an Angle
Again a projectile is fired at 100 m/s. This
time there is no cliff, but the barrel is
elevated at 30° with the ground. How far
away will this projectile land?
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30o

Separate into Components
Resolve the diagonal motion of the
cannonball into horizontal and vertical
components.
These separate
components will be
used when motion in
only 1 direction is
addressed.
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30o
Calculation of Components
Consider the cannonball velocity and its
components like the sides of a right triangle.
Simple trig functions can be used to calculate
all components.
e side
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30o
Adjacent side
Opposite

Component Values
Vertical component: Horizontal component:
opp
θ adj
hyp
sin θ = sin30D = opp
100m/s
hyp
cos = cos30D = adj
100m/s
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opp = .5 (100m/s)
opp = 50m/s
adj = .87 (100m/s)
adj = 87 m/s
Calculating Time in Air
We now know that the cannonball initially
moves upwards at 50 m/s. (vertical
component)
Thus, it will be moving downward at -50 m/s
when it returns to the Earth.
Next, use our old acceleration definition.
a vf vi −
t
=
−
t
=
vf vi 42/48
a
t -50 m/s −
50 m/s
= t = 10 s
-9.8 m/s2

Calculating Distance Traveled
Finally, since we still have a constant horizontal
velocity, we can use our old constant velocity
formula again.
v dh
h =
t
( ) h h d = t v
Notice the
horizontal
component is
used here.
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d 10.2s (87 m/s) h =
d 887 m h =
Additional Formula
With some algebraic acrobatics, our previous
example can be combined into one formula to
determine the horizontal range of a projectile:
2
Launch velocity
d v sin2θ
g
h =
squared
44/48
Acceleration
from gravity,
assign as a +
here
Double the
angle of launch
measured from
the horizon.

Launches at Various Angles
Notice how two angles yield the
same horizontal distance.
These are complimentary
Vertical distance
angles; they add up to 90°.
The larger angle of
each pair simply puts
the projectile higher
into the air and for a
l ti
V
45/48
longer time.
Horizontal distance
Without air resistance, an angle of 45° will yield the
maximum range.
Learning Summary
Use the
appropriate
component
Vectors have
magnitude
and direction.
Use old
ki ti
Scalars have
magnitude
only.
kinematics
formulas.
Split vectors into
when describing
motion in a
certain
direction.
Add vectors tip
46/48
components
using
mathematics.
(trigonometry)
to tail. The
resultant is the
sum of two
vectors.

Congratulations
You have successfully completed
the tutorial
Vectors and Kinematics
in 2 Dimensions
Chemistry :: Biology :: Physics :: Math
Wh What’s t’ Next N t
…
Step 1: Concepts – Core Tutorial (Just Completed)
Æ Step 2: Practice – Interactive Problem Drill
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Kinematics in 2 dimension

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