Chapter 2
Motion
Motion
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Want to describe bodies in motion
and how they behave in time
Speed vs velocity
• Speed is a scalar
• Velocity is a vector
• examples
Speed
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The average speed of an object is
defined as the total distance traveled
divided by the total time elapsed
total distance
Average speed 
total time
d
v 
t
• Speed is a scalar quantity
• SI units are m/s
Velocity
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It takes time for an object to
undergo a displacement
The average velocity is rate at
which the displacement occurs
Displaceme nt
Velocity average 
time

SI units are m/s
x
v
t
Example
Car travels 120 km 45° N of E in 3
hours.
Conversion Factors

Multiply by “1” until you get the right
units
• Length: in, ft, m, mile, km, light year,
• Time: s, min, h, day,
More on speed and velocity
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Motion by graphs
Slope
Motion with constant speed (velocity)
Motion with changing velocity
Average velocity
Instantaneous velocity
Instantaneous speed is the Mag. Of
instantaneous velocity
Acceleration
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Changing velocity (non-uniform)
means an acceleration is present
Acceleration is the rate of change of
the velocity
change in velocit y
v

v
F
I
accelerati on 
a

time taken for the change
t
Units are m/s² (SI), cm/s² (cgs), and
ft/s² (US Cust)
Acceleration
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Vector quantity
Can be constant
Will mainly study uniform
acceleration
Negative acceleration means the
object is slowing down (when v is
positive)
Graphs
Example
Car accelerates uniformly from 0 to
60 miles/hour (East) in 6 s.
Uniformly Acceleration Motion
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Along a line (one dimension)
Simplest case
Free fall
A lot of examples
Extend to projectile motion (2 dim)
Linear Motion Summary
vI  vF
x  vt 
t
2

(1)

(2)
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(3)
vF  vI  at

(4)
1 2
x  vI t  at
2
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(5)
vF2  vI2  2ax
vI  vF
v
2
Example
Car moving at 25 m/s hits the brakes
& stops in a distance of 25 m
1. What was the car’s (uniform)
deceleration?
2. How long did it take to stop?
Example
A car is moving at a constant speed of
90 km/h when it begins to pass a
train moving in the opposite
direction at 60 km/h. If the train is
200 m long,
• How long does it take the car to pass
the train?
• How far did the car travel?
• How far did the train travel?
Free Fall
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All objects moving under the influence of
gravity only are said to be in free fall
• Free fall does not depend on the object’s
original motion
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All objects falling near the earth’s surface
fall with a constant acceleration
The acceleration is called the acceleration
due to gravity, and indicated by g
Acceleration due to Gravity
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Symbolized by g
g = 9.80 m/s²
• When estimating, use g 10 m/s2
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acc is always directed downward
• toward the center of the earth
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Ignoring air resistance and assuming
g doesn’t vary with altitude over
short vertical distances, free fall is
constantly accelerated motion
Free Fall – an object dropped
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Initial velocity is
zero
Let up be positive
Use the equations
• Generally use y
instead of x since
vertical
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Acceleration is g =
9.80 m/s2
vo= 0
a=-g
Free Fall – an object thrown
downward
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a = -9.80 m/s2
Initial velocity  0
• With upward being
positive, initial
velocity will be
negative
Free Fall -- object thrown
upward
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Initial velocity is
upward, so positive
The instantaneous
velocity at the
maximum height is
zero
a = - 9.80 m/s2
everywhere in the
motion
v=0
Thrown upward, cont.
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The motion may be symmetrical
• Then tup = tdown
• Then v = -vI
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The motion may not be symmetrical
• Break the motion into various parts
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Generally up and down
Free Fall (g=9.8 m/s²)
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(1)

(2)
vI  vF
y  vt 
t
2
vI  vF
v
2
(3)
vF  vI  gt

(4)
1 2
y  vI t  gt
2

(5)
vF2  vI2  2 gy
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y axis up
a  g
Example
A ball is launched straight upward with
an initial speed of 20 m/s.
• How long does it take to return?
• How high did it go?
Projectile Motion
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Example of motion in 2-dim
An object may move in both the x
and y directions simultaneously
• It moves in two dimensions
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The form of two dimensional motion
we will deal with is called projectile
motion
vI and 0
Projectile Motion (2-dim)
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Horizontal motion is independent of
vertical motion
Break motion into 2 separate parts
• Horizontal: x component
• Vertical: y component
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Vertical motion: free fall
Horizontal motion: constant velocity
Assumptions of Projectile
Motion
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We may ignore air friction
We may ignore the rotation of the
earth
With these assumptions, an object in
projectile motion will follow a
parabolic path
Rules of Projectile Motion
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The horizontal motion (x) and vertical of
motion (y) are completely independent of
each other
The x-direction is uniform motion
• ax = 0

The y-direction is free fall
• ay = -g
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The initial velocity can be broken down
into its x- and y-components
•
vIx  vI cos  0
vIy  vI sin  0
Projectile Motion
Projectile Motion at Various
Initial Angles
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Complementary
values of the
initial angle result
in the same range
• The heights will be
different

The maximum
range occurs at a
projection angle
of 45o
Velocity of the Projectile
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The velocity of the projectile at any
point of its motion is the vector sum
of its x and y components at that
point
2
x
v  v v
2
y
and
  tan
1
vy
vx
• Remember to be careful about the
angle’s quadrant
Example
A cannon fires a cannonball with an
initial velocity of 200 m/s at an angle
of 40° above ground. How far does the
c.b. travel horizontally (range) before it
hits the ground?
Example
A hose is held 2 m off the ground such
that water shoots out horizontally &
hits the ground 1.5 m away. What is
the initial speed?
Example
What is the range of a projectile fired
across level ground with a velocity of
100 m/s @ an angle of 30° above
horizontal?