4.2 A Model for Accelerated Motion
Chapter Objectives

Calculate acceleration from the change in speed and the change in
time.

Give an example of motion with constant acceleration.

Determine acceleration from the slope of the speed versus time graph.

Calculate time, distance, acceleration, or speed when given three of
the four values.

Solve two-step accelerated motion problems.

Calculate height, speed, or time of flight in free fall problems.

Explain how air resistance makes objects of different masses fall with
different accelerations.
Chapter Vocabulary
 acceleration
 initial speed
 acceleration due to gravity 
(g)

 air resistance

 constant acceleration

 delta (Δ)

 free fall
m/s2
term
terminal velocity
time of flight
uniform acceleration
Inv 4.2 Accelerated Motion
Investigation Key Question:
How does acceleration relate to velocity?
4.2 A Model for Accelerated Motion
 To get a formula for solve for the speed of an
accelerating object, we can rearrange the
experimental formula we had for acceleration.
4.2 The speed of an accelerating object
 In physics, a piece of an equation is called a term.
 One term of the formula is the object’s starting
speed, or its initial velocity (v0)
 The other term is the amount the velocity changes
due to acceleration.
Calculating speed
A ball rolls at 2 m/s off a level surface and
down a ramp. The ramp creates an
acceleration of 0.75 m/s2. Calculate the speed
of the ball 10 s after it rolls down the ramp.
1.
2.
3.
4.
You are asked for speed.
You are given initial speed, acceleration and time.
Use the relationship v = v0 + at
Substitute values
 v = 2 m/s + (0.75 m/s2)(10 s)
 v = 9.5 m/s2
4.2 Distance traveled in accelerated motion
 The distance traveled by
an accelerating object can
be found by looking at the
speed versus time graph.
 The graph shows a ball
that started with an initial
speed of 1 m/s and after
one second its speed has
increased.
4.2 Distance traveled in accelerated motion
 The area of the
shaded rectangle is
the initial speed v0
multiplied by the time
t, or v0t.
 The second term is
the area of the shaded
triangle.
4.2 A Model for Accelerated Motion
 It is possible that a moving object may not start
at the origin.
 Let x0 be the starting position.
 The distance an object moves is equal to its
change in position (x – x0).
Calculating position from speed
and acceleration
A ball traveling at 2 m/s rolls up a ramp.
The angle of the ramp creates an acceleration of - 0.5 m/s2.
What distance up the ramp does the ball travel before it turns
around and rolls back?
1.
2.
You are asked for distance.
You are given initial speed and acceleration. Assume an initial
position of 0 and a final speed of 0.
3.
Use the relationship v = v0 + at and x = x0 + v0t + 1/2at2
4.
At the highest point the speed of the ball must be zero. Substitute
values to solve for time, then use time to calculate distance.
 0 = 2 m/s + (- 0.5 m/s2)(t) = - 2 m/s = - 0.5 m/s2 (t)
t=4s
 x = (0) + (2 m/s) ( 4 s) + (0.5) (-0.5 m/s2) (4 s)2 = 4 meters
4.2 Solving motion problems with
acceleration
 Many practical problems involving accelerated
motion have more than one step.
 List variables
 Cancel terms that are zero.
 Speed is zero when it starts from rest.
 Speed is zero when it reaches highest point
 Use another formula to find the missing piece
of information.
Calculating position from time
and speed
A ball starts to roll down a ramp with zero initial speed.
After one second, the speed of the ball is 2 m/s. How long
does the ramp need to be so that the ball can roll for 3
seconds before reaching the end?
1. You are asked to find the length of the ramp.
2. You are given v0 = 0, v = 2 m/s at t = 1 s, t = 3 s at the
bottom of the ramp, and you may assume x0 = 0.
3. After canceling terms with zeros, v = at and x = ½ at2
4. This is a two-step problem. First, calculate acceleration,
then you can use the position formula to find the length of
the ramp.


a = v ÷ t = (2 m/s ) ÷ (1 s ) = 2 m/s2
x = ½ at2 = (0.5)(2 m/s )(3 s )2 = 9 meters
Calculating time from distance
and acceleration
A car at rest accelerates at 6 m/s2. How long
does it take to travel 440 meters, or about a
quarter-mile, and how fast is the car going at
the end?
1.
2.
3.
4.


You are asked to find the time and speed.
You are given v0 = 0, x = 440 m, and a = 6 m/s2; assume
x0 = 0.
Use v = v0 + at and x = x0 + v0t + ½ at2
Since x0 and v0 = 0, the equation reduces to x = ½at2
440 m = (0.5)(6 m/s2) (t)2
t2 = 440 ÷ 3 = 146.7 s t = 12.1 s