Chapter 4: Accelerated Motion in a
Straight Line
 4.1 Acceleration
 4.2 A Model for Accelerated Motion
 4.3 Free Fall and the Acceleration due to
Gravity
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 4.1 Acceleration
Investigation Key Question:
How does acceleration relate to velocity?
4.1 Acceleration
 Acceleration is the rate of change in the speed of
an object.
 Rate of change means the ratio of the amount of
change divided by how much time the change
takes.
4.1 Acceleration in metric units
 If a car’s speed increases
from 8.9 m/s to 27 m/s, the
acceleration in metric units
is 18.1 m/s divided by 4
seconds, or 4.5 meters per
second per second.
 Meters per second per
second is usually written as
meters per second squared
(m/s2).
4.1 The difference between velocity
and acceleration
 Velocity is fundamentally different from
acceleration.
 Velocity can be positive or negative and is the
rate at which an object’s position changes.
 Acceleration is the rate at which velocity
changes.
4.1 The difference between velocity
and acceleration
 The acceleration of an
object can be in the
same direction as its
velocity or in the
opposite direction.
 Velocity increases when
acceleration is in the
same direction.
4.1 The difference between velocity
and acceleration
 When acceleration and
velocity have the
opposite sign the
velocity decreases, such
as when a ball is rolling
uphill.
4.1 The difference between velocity
and acceleration
 If both velocity and acceleration are negative the speed
increases but the motion is still in the negative
direction.
 Suppose a ball is rolling down a ramp sloped downhill
to the left.
 Motion to the left is defined to be negative so the
velocity and acceleration are both negative.
 The velocity of the ball gets LARGER in the negative
direction, which means the ball moves faster to the left.
4.1 Calculating acceleration
 Acceleration is the change in velocity divided by
the change in time. The Greek letter delta (Δ)
means “the change in.”
Acceleration
(m/sec2)
a = Dv
Dt
Change in speed (m/sec)
Change in time (sec)
4.1 Calculating acceleration
 The formula for acceleration can also be written
in a form that is convenient for experiments.
Calculating acceleration in m/s2
A student conducts an acceleration
experiment by coasting a bicycle down a
steep hill. A partner records the speed of the
bicycle every second for five seconds.
Calculate the acceleration of the bicycle.
1. You are asked for acceleration.
2. You are given times and speeds from an
experiment.
3. Use the relationship
a = (v2 – v1) ÷ (t2 – t1)
4. Choose any two pairs of time and speed data since
the change in speed is constant.


a = (6 m/s 4 m/s) ÷ (3 s – 4 s) = (2 m/s) ÷ (-1 s)
a = −2 m/s
4.1 Constant speed and constant
acceleration
 Constant acceleration is different
from constant speed.
 If an object is traveling at
constant speed in one direction,
its acceleration is zero.
 Motion with zero acceleration
appears as a straight horizontal
line on a speed versus time
graph.
4.1 Uniform acceleration
 Constant acceleration is
sometimes called uniform
acceleration.
 A ball rolling down a straight
ramp has constant acceleration
because its speed is increasing
at the same rate.
 Falling objects also undergo
uniform acceleration.
4.1 Constant negative acceleration
 Consider a ball rolling up a
ramp.
 As the ball slows down,
eventually its speed becomes
zero and at that moment the
ball is at rest.
 However, the ball is still
accelerating because its
velocity continues to change.
4.1 The speed vs. time graph for
accelerated motion
 In this experiment, velocity and acceleration are in the
same direction.
 No negative quantities appear, and the analysis simply
uses speed instead of velocity.
4.1 Slope and Acceleration
 Use slope to recognize
acceleration on speed vs.
time graphs.
 Level sections (A) on the graph
show an acceleration of zero.
 The highest acceleration (B) is
the steepest slope on the
graph.
 Sections that slope down (C)
show negative acceleration
(slowing down).
Calculating acceleration
The graph shows the speed of a bicyclist
going over a hill. Calculate the maximum
acceleration of the cyclist and calculate
when in the trip it occurred.
1.
2.
3.
4.
You are asked for maximum acceleration.
You are given a graph of speeds vs. time.
Use the relationship
a = slope of graph
The steepest slope is between 60 and 70
seconds, when the speed goes from 2 to 9 m/s.


a = (9 m/s – 2 m/s) ÷ (10 s)
a = 0.7 m/s2
Chapter 4: Accelerated Motion in a
Straight Line
 4.1 Acceleration
 4.2 A Model for Accelerated Motion
 4.3 Free Fall and the Acceleration due to
Gravity
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
Chapter 4: Accelerated Motion in a
Straight Line
 4.1 Acceleration
 4.2 A Model for Accelerated Motion
 4.3 Free Fall and the Acceleration due to
Gravity
Inv 4.3 Free Fall
Investigation Key Question:
What kind of motion is falling?
4.3 Free Fall and the Acceleration
due to Gravity
 An object is in free fall if it is moving
under the sole influence of gravity.
 Free-falling objects speed up, or
accelerate, as they fall.
 The acceleration of 9.8 m/s2 is given
its own name and symbol—
acceleration due to gravity (g).
4.3 Free fall with initial velocity
 The motion of an object in free fall is described by the
equations for speed and position with constant acceleration.
 The acceleration (a) is replaced by the acceleration due to
gravity (g) and the variable (x) is replaced by (y).
4.3 Free fall with initial velocity
 When the initial speed is
upward, at first the
acceleration due to gravity
causes the speed to
decrease.
 After reaching the highest
point, its speed increases
exactly as if it were dropped
from the highest point with
zero initial speed.
4.3 Solving problems with free fall
 Most free-fall problems ask you to find either
the height or the speed.
 Height problems often make use of the
knowledge that the speed becomes zero at the
highest point of an object’s motion.
 If a problem asks for the time of flight,
remember that an object takes the same time
going up as it takes coming down.
Calculating height from the time
of falling
A stone is dropped down a well and it takes
1.6 seconds to reach the bottom. How deep is
the well? You may assume the initial speed of
the stone is zero.
1.
2.
3.
4.
You are asked for distance.
You are given an initial speed and time of flight.
Use v = v0 - gt and y = y0 + v0t - ½ gt2
Since y0 and v0 = 0, the equation reduces to x = -½ gt2
 y = - (0.5) (9.8 m/s2) (1.6s)2
 y = -12.5 m (The negative sign indicates the height is
lower than the initial height)
4.3 Air Resistance and Mass
 The acceleration due to gravity does not
depend on the mass of the object which is
falling.
 Air creates friction that resists the motion of
objects moving through it.
 All of the formulas and examples discussed
in this section assume a vacuum (no air).
4.3 Terminal Speed
 You may safely assume that a = g = 9.8 m/sec2 for
speeds up to several meters per second.
 The air resistance from friction
increases as a falling object’s speed
increases.
 Eventually, the rate of acceleration is
reduced to zero and the object falls with
constant speed.
 The maximum speed at which an object
falls when limited by air friction is called
the terminal velocity.
Anti-lock Brakes
 Antilock braking systems (ABS) are
standard on most new cars and trucks.
 If brakes are applied too hard or too
fast, a rolling wheel locks up, which
means it stops turning and the car
skids.
 With the help of constant computer
monitoring, these systems give the
driver more control when stopping
quickly.