2. 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 2

3. 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.

4. Chapter Vocabulary acceleration acceleration due to gravity (g)
air resistance
constant acceleration
delta (Δ)
free fall
initial speed
m/s2
term
terminal velocity
time of flight
uniform acceleration

5. Inv 4.1 Acceleration Investigation Key Question:
How does acceleration relate to velocity? 5

6. 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.

7. 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).

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.”
Change in speed (m/sec)
Acceleration
(m/sec2)
a = Dv Dt
Change in time (sec)

13. 4.1 Calculating acceleration
The formula for acceleration can also be written in a form that is convenient for experiments.

14. 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.
You are asked for acceleration.
You are given times and speeds from an experiment.
Use the relationship a = (v2 – v1) ÷ (t2 – t1)
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

15. 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.

16. 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.

17. 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.

18. 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.

20. 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).

21. 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.
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

22. 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 22

23. Inv 4.2 Accelerated Motion
Investigation Key Question:
How does acceleration relate to velocity? 23

24. 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.

25. 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.

26. 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.
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

27. 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.

28. 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.

29. 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).

31. 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 m/s2. What distance up the ramp does the ball travel before it turns around and rolls back?
You are asked for distance.
You are given initial speed and acceleration. Assume an initial position of 0 and a final speed of 0.
Use the relationship v = v0 + at and x = x0 + v0t + 1/2at2
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) = m/s = m/s2 (t) t = 4 s
x = (0) + (2 m/s) ( 4 s) + (0.5) (-0.5 m/s2) (4 s)2 = 4 meters

32. 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.

33. 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?
You are asked to find the length of the ramp.
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.
After canceling terms with zeros, v = at and x = ½ at2
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

34. 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?
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 = s t = 12.1 s

35. 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 35

36. Inv 4.3 Free Fall Investigation Key Question:
What kind of motion is falling? 36

37. 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).

38. 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).

40. 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.

42. 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.

43. 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.
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 = m (The negative sign indicates the height is lower than the initial height)

44. 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).

45. 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.

46. 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.