1. Acceleration
This lesson defines acceleration, its signs and its units, and provides conceptual, graphical, and quantitative examples. Students use an interactive calculator to solve problems involving velocity and acceleration, and learn that acceleration is equal to the slope of the velocity versus time graph.

2. Objectives Define and describe acceleration.
Describe and analyze motion in one dimension using equations for acceleration.
Determine acceleration from the slope of the velocity versus time graph.

3. Physics terms
velocity
acceleration
rate of change

4. Equations
Acceleration is the change in velocity divided by the change in time.

5. Brainstorm: An accelerating object has a changing velocity.
Can you share an example of something accelerating?

6. Examples a coaster making a turn a ball rolling uphill
a baseball falling

7. What is acceleration? 4 m/s 4 m/s 4 m/s 4 m/s
Two balls are moving to the right. Their velocity at each second is shown.
Which ball is accelerating?
What is its acceleration?
Ball 1
0 m/s
4 m/s
2 m/s
Ball 2
6 m/s

8. What is acceleration? 4 m/s 4 m/s 4 m/s 4 m/s
Two balls are moving to the right. Their velocity at each second is shown.
Which ball is accelerating?
What is its acceleration?
Ball 2 is accelerating at +2 m/s per second:
a = +2 m/s2
Ball 1
0 m/s
4 m/s
2 m/s
Ball 2
6 m/s

9. The meaning of acceleration
Acceleration is the rate at which velocity changes.
a = acceleration (m/s2)
Δv = change in velocity (m/s)
Δt = change in time (s)

10. Units of acceleration
The units for acceleration are units of speed divided by units of time.

11. Units of acceleration
The acceleration tells you how many meters per second your velocity changes in each second.
These units are usually written as meters per second squared.

12. The meaning of acceleration
The equation defines acceleration as the change in velocity ( Δv = vf – vi ) divided by the change in time ( Δt = tf – ti ).

13. Example
A car is initially at rest. Ten seconds later it is moving at 30 m/s:
An acceleration of +3.0 m/s2 means that +3.0 m/s is added to the velocity each second.

14. Example
A car is initially at rest. Ten seconds later it is moving at 30 m/s:
0 s s s s s s s s s s s
0 m/s 3 m/s 6 m/s m/s m/s 15 m/s 18 m/s 21 m/s 24 m/s 27 m/s 30 m/s

15. Understanding the concept
If an object has an acceleration of -4 m/s2, its velocity changes by -4 m/s each second.
For example, if it has an initial velocity of +8 m/s and accelerates for one second . . .

16. Understanding the concept
If an object has an acceleration of -4 m/s2, its velocity changes by -4 m/s each second.
For example, if it has an initial velocity of +8 m/s and accelerates for one second, its vf is +4 m/s. vi
(initial velocity) Δv
(change in velocity) vf
(final velocity)

17. The meaning of acceleration
This chart shows the position and speed of an object moving to the right for three consecutive seconds.
0 m/s m/s m/s m/s
What is the acceleration of object 1?
Object 1
Ball 2

18. The meaning of acceleration
This chart shows the position and speed of an object moving to the right for three consecutive seconds.
0 m/s m/s m/s m/s
What is the acceleration of object 1? m/s2
What is the acceleration of object 2?
Object 1
2 m/s 6 m/s m/s
Object 2
Ball 2

19. The meaning of acceleration
This chart shows the position and speed of an object moving to the right for three consecutive seconds.
0 m/s m/s m/s m/s
What is the acceleration of object 1? m/s2
What is the acceleration of object 2? +4 m/s2
What is the missing velocity?
Object 1
2 m/s 6 m/s m/s ?
Object 2
Ball 2

20. The meaning of acceleration
This chart shows the position and speed of an object moving to the right for three consecutive seconds.
0 m/s m/s m/s m/s
What is the acceleration of object 1? m/s2
What is the acceleration of object 2? +4 m/s2
What is the missing velocity? m/s
Object 1
2 m/s 6 m/s m/s m/s
Object 2
Ball 2

21. The meaning of acceleration
Between 0 and 2 seconds the acceleration is +2 m/s2.
Between 2 and 3 seconds the acceleration is zero.
Between 3 and 6 seconds the acceleration is -3 m/s2.
Fill in the missing velocities at each time step.

22. The meaning of acceleration
Between 0 and 2 seconds the acceleration is +2 m/s2.
Between 2 and 3 seconds the acceleration is zero.
Between 3 and 6 seconds the acceleration is -3 m/s2. 2 4
An acceleration of +2 m/s2 means you add 2 m/s to the velocity for each second for the first two seconds.

23. The meaning of acceleration
Between 0 and 2 seconds the acceleration is +2 m/s2.
Between 2 and 3 seconds the acceleration is zero.
Between 3 and 6 seconds the acceleration is -3 m/s2. 2 4
An acceleration of 0 m/s2 means the velocity does not change between 2 seconds and 3 seconds.

24. The meaning of acceleration
Between 0 and 2 seconds the acceleration is +2 m/s2.
Between 2 and 3 seconds the acceleration is zero.
Between 3 and 6 seconds the acceleration is -3 m/s2. 2 4 1 -2 -5
An acceleration of -3 m/s2 means you subtract 3 m/s from the velocity every second.

25. The meaning of acceleration
Between 0 and 2 seconds the acceleration is +2 m/s2.
Between 2 and 3 seconds the acceleration is zero.
Between 3 and 6 seconds the acceleration is -3 m/s2. 2 4 1 -2 -5
What is happening to the speed?

26. The meaning of acceleration
Speed
(m/s) 2 4 1 5
Between 0 and 2 seconds the acceleration is +2 m/s2.
Between 2 and 3 seconds the acceleration is zero.
Between 3 and 6 seconds the acceleration is -3 m/s2. 2 4 1 -2 -5
increases
constant
decreases
increases
Positive AND negative acceleration can cause speed to increase OR decrease – depending on the direction of the velocity!

27. Signs of the acceleration
Positive acceleration of +4 m/s2 adds +4 m/s of velocity each second.
0 m/s
4 m/s
8 m/s
12 m/s
16 m/s

28. Signs of the acceleration
Positive acceleration of +4 m/s2 adds +4 m/s of velocity each second.
Negative acceleration of -4 m/s2 adds -4 m/s of velocity each second.
0 m/s
4 m/s
8 m/s
12 m/s
16 m/s
16 m/s
12 m/s
8 m/s
4 m/s
0 m/s

29. Signs of the acceleration
What is the sign of acceleration in each of these four possible cases?
1. speeding up in + direction: ?
0 m/s m/s m/s m/s +
Ask students to give an example of this.

30. Signs of the acceleration
What is the sign of acceleration in each of these four possible cases?
1. speeding up in + direction: +a
2. slowing down in + direction: ?
0 m/s m/s m/s m/s
+3 m/s m/s m/s 0 m/s + +
Ask students to give an example of this.

31. Signs of the acceleration
What is the sign of acceleration in each of these four possible cases?
1. speeding up in + direction: +a
2. slowing down in + direction: -a
0 m/s m/s m/s m/s
+3 m/s m/s m/s 0 m/s + +
3. speeding up in - direction: ?
-3 m/s m/s m/s m/s
Ask students to give an example of this. -

32. Signs of the acceleration
What is the sign of acceleration in each of these four possible cases?
1. speeding up in + direction: +a
2. slowing down in + direction: -a
0 m/s m/s m/s m/s
+3 m/s m/s m/s 0 m/s + +
3. speeding up in - direction: -a
4. slowing down in - direction: ?
-3 m/s m/s m/s m/s
0 m/s m/s m/s m/s
Ask students to give an example of this. - -

33. Signs of the acceleration
What is the sign of acceleration in each of these four possible cases?
1. speeding up in + direction: +a
2. slowing down in + direction: -a
0 m/s m/s m/s m/s
+3 m/s m/s m/s 0 m/s + +
3. speeding up in - direction: -a
4. slowing down in - direction: +a!
-3 m/s m/s m/s m/s
0 m/s m/s m/s m/s
Ask students to give an example of this. - -

34. Test your knowledge
A car is headed west (the negative direction) on a long straight road. The driver sees a red light up ahead and slows to a stop.
Is the car’s acceleration positive or negative?

35. Test your knowledge
A car is headed west (the negative direction) on a long straight road. The driver sees a red light up ahead and slows to a stop.
Is the car’s acceleration positive or negative?
Slowing down in the negative direction is +a!

36. Test your knowledge What does -1 m/s2 mean?
Can you describe two ways that an object could have a negative acceleration?

37. Test your knowledge What does -1 m/s2 mean?
Can you describe two ways that an object could have a negative acceleration?
The velocity changes by -1 m/s every second.
It could slow down in the positive direction.
It could speed up in the negative direction.

38. Acceleration on the v vs. t graph
A car moves at a constant speed of 3 m/s for 3 seconds.
What does this look like on the velocity vs. time graph?

39. No acceleration constant velocity
Acceleration on the v vs. t graph
A car moves at a constant speed of 3 m/s for 3 seconds.
No acceleration constant velocity

40. No acceleration constant velocity
Acceleration on the v vs. t graph
A car moves at a constant speed of 3 m/s for 3 seconds.
The car accelerates to 6 m/s over the next 3 seconds. What does this look like?
No acceleration constant velocity

41. Acceleration on the v vs. t graph
A car moves at a constant speed of 3 m/s for 3 seconds.
The car accelerates to 6 m/s over the next 3 seconds. What does this look like?
No acceleration constant velocity
Positive acceleration changing velocity

42. Acceleration on the v vs. t graph
A car moves at a constant speed of 3 m/s for 3 seconds.
The car accelerates to 6 m/s over the next 3 seconds.
The car continues at 6 m/s for three more seconds What does this look like?
No acceleration constant velocity
Positive acceleration changing velocity

43. Acceleration on the v vs. t graph
A car moves at a constant speed of 3 m/s for 3 seconds.
The car accelerates to 6 m/s over the next 3 seconds.
The car continues at 6 m/s for three more seconds What does this look like?
No acceleration constant velocity
Positive acceleration changing velocity
No acceleration constant velocity

44. Positive acceleration changing velocity
Acceleration on the v vs. t graph
Acceleration causes the velocity vs. time graph to have a non-zero slope.
Positive acceleration changing velocity

45. Acceleration on the v vs. t graph

46. Acceleration on the v vs. t graph
Calculate the acceleration from t = 3 to t = 6 seconds. Δv

47. Acceleration on the v vs. t graph
Calculate the acceleration from t = 3 to t = 6 seconds. Δv
6 m/s
3 m/s
+1 m/s2
3 s

48. Position vs. time An object starting from rest accelerates at 1 m/s2.
Its velocity increases with time, making a linear v vs. t graph.
What does the position vs. time graph look like?

49. Position vs. time An object starting from rest accelerates at 1 m/s2.
Its velocity increases with time, making a linear v vs. t graph.
What does the position vs. time graph look like?
As the velocity increases the slope must change!

50. Position vs. time The graph is a curve.
An object starting from rest accelerates at 1 m/s2.
Its velocity increases with time, making a linear v vs. t graph.
What does the position vs. time graph look like?
As the velocity increases the slope must change!
The graph is a curve.

51. Curves vs. lines
Acceleration creates a sloped line on a v vs. t graph.
Acceleration creates a curve on an x vs. t graph.

52. Curves vs. lines The x vs. t graph curves upward for +a, like a smile!
The x vs. t graph curves downward for -a, like a frown!

53. Understanding acceleration
How does statement (A) relate to the diagram below it? What does “± 5 mph/s” mean?
What does the word “rate” mean in the context of statement (B)? How do the arrows represent the idea of a rate of changing velocity?

54. Understanding acceleration
How does the equation represent statement (C)? Translate the symbols and operations into English. Give numerical examples of a change in velocity divided by a change in time.

55. Understanding acceleration
How does the diagram in (D) represent the text of statement (D)? What does the shaded triangle represent?
What does it mean that the lines on the velocity vs. time graph in diagram (D) go up then down? How is that reflected in the concept of acceleration?

56. Assessment
Define and describe accelerated motion.

57. Assessment Define and describe accelerated motion.
Acceleration is the rate of change of velocity.
For motion along a line, an object that is accelerating is speeding up or slowing down.

58. Assessment
A car changes its velocity from 0 to 20 m/s in 4.0 seconds. What is its acceleration?

59. Assessment
A car changes its velocity from 0 to 20 m/s in 4.0 seconds. What is its acceleration?
A change of +20 m/s over 4.0 seconds is an acceleration of +5.0 m/s2.

60. Assessment
A car initially traveling at 25 m/s comes to a stop in 3.0 s. What is its acceleration?

61. Assessment
A car initially traveling at 25 m/s comes to a stop in 3.0 s. What is its acceleration?
A change of -25 m/s over 3 seconds is an acceleration of -8.3 m/s2.

62. Assessment
An object starts from rest and accelerates at 2.0 m/s2 for 10 seconds. What is its final velocity?

63. Assessment
An object starts from rest and accelerates at 2.0 m/s2 for 10 seconds. What is its final velocity?
The final velocity is 20 m/s.
0 s s s s s s s s s s s
0 m/s 2 m/s 4 m/s m/s 8 m/s 10 m/s 12 m/s 14 m/s 16 m/s 18 m/s 20 m/s

64. Assessment Velocity vs. time t (s)
The motion of a particle along a straight line is depicted in this graph.
What is the acceleration of the particle from 7 seconds to 10 seconds? V
(m/s)
t (s)

65. Assessment Velocity vs. time t (s)
The motion of a particle along a straight line is depicted in this graph.
What is the acceleration of the particle from 7 seconds to 10 seconds? V
(m/s)
answer: 40 m/s2
t (s)

66. Gravity and free fall

67. Objectives Define the conditions for free fall.
Describe and analyze the motion of objects in free fall using the equations for constant acceleration.

68. Physics terms
acceleration
quadratic equation
free fall

69. Equations

70. What is free fall?
An object is in free fall whenever it moves solely under the influence of gravity, regardless of its direction.
A ball falling down, with negligible air resistance
A ball thrown up, with negligible air resistance
A ball launched at ANY angle, as long as there is negligible air resistance

71. Gravity and free fall
Near Earth’s surface, free-falling objects have a downward acceleration of 9.8 m/s2.
If an object is dropped from rest, then . . .
after 1 second its velocity is m/s.
after 2 seconds its velocity is m/s.
after 3 seconds its velocity is __?___
after 10 seconds its velocity is __?___

72. Gravity and free fall
Near Earth’s surface, free-falling objects have a downward acceleration of 9.8 m/s2.
If an object is dropped from rest, then . . .
after 1 second its velocity is m/s.
after 2 seconds its velocity is m/s.
after 3 seconds its velocity is m/s.
after 10 seconds its velocity is -98 m/s.

73. Describe free fall with equations
The free fall equations are identical to the equations for motion with constant acceleration:
The only difference is that you already know the acceleration because it is always 9.8 m/s2 downward.
Point out that the value of g actually depends on location, and will be different on the Moon, for example. It is 9.8 m/s/s for events close to the Earth’s surface.

74. Find your reaction time
Use this equation for free fall to find your own reaction time—the time to catch a falling ruler.
Make a prediction first:
Will your reaction time be in seconds? Tenths of a second? Hundredths of a second?
Students should complete the student assignment sheet while doing this experiment.

75. Gravity and free fall If an object is dropped from rest then . . .
after 1 second its velocity is m/s.
after 2 seconds its velocity is m/s.
after 3 seconds its velocity is m/s.
after 4 seconds its velocity is m/s.
and so on

76. Gravity and free fall REALLY?
If an object is dropped from rest then . . .
after 1 second its velocity is m/s.
after 2 seconds its velocity is m/s.
after 3 seconds its velocity is m/s.
after 4 seconds its velocity is m/s.
and so on
REALLY?
Do falling objects REALLY keep moving faster and faster?

77. Gravity and free fall
Do falling objects REALLY keep moving faster and faster?
No! In real life there is air resistance.
As falling objects speed up, the force of air resistance increases.
When the air resistance gets as strong as the force of gravity, the falling object stops accelerating.
After showing this slide, ask the students “if the acceleration becomes zero, does that mean the velocity is zero also?”

78. Terminal velocity
Most objects reach this terminal velocity within a few seconds of being dropped.
Terminal velocity is the final maximum velocity an object reaches because of air resistance.
A falling human has a terminal velocity of about 140 miles per hour (or about 60 m/s).

79. Terminal velocity Parachutes increase air resistance.
Opening a parachute changes the terminal velocity from a fast, deadly speed to a low, safe speed.

80. A skydiving trip When did the parachute open?
Ask the students when the parachuter is at terminal velocity (at both C and E).

81. A skydiving trip When did the parachute open? at t = 28 seconds
Ask the students when the parachuter is at terminal velocity (at both C and E).

82. A skydiving trip
What is happening to the acceleration during each of these time segments?

83. A skydiving trip
What is happening to the acceleration during each of these time segments?

84. When can motion be treated as free fall?
Free fall is NOT a good approximation for light objects, or an object with a large surface area compared to its weight (like a parachute).

85. When can motion be treated as free fall?
Free fall is a very good approximation for solid, dense objects dropped from ten meters or so.
For these situations, air resistance can be ignored.

86. When can motion be treated as free fall?
Free fall is a very good approximation for solid, dense objects dropped from ten meters or so.
For these situations, air resistance can be ignored.
The symbol g is often used when the acceleration of an object is due only to gravity.

87. Solving free fall problems
Define your coordinate system:
If you decide up is positive, g = -9.8 m/s2
If you decide down is positive, g = +9.8 m/s2
Write the equations of motion, substituting g for a.
Eliminate any terms that are zero.
Work out a solution strategy.

88. Example free fall problem
From what height should you drop a ball if you want it to hit the ground in exactly 1.0 second?
Asked: x
Given: t v0
Relationship:
Solution:

89. Example free fall problem
From what height should you drop a ball if you want it to hit the ground in exactly 1.0 second?
Asked: x
Given: t = 1.0 s, g = -9.8 m/s2
(assume v0 = 0 m/s and x0 = 0 m)
Relationship:
Solution:

90. Example free fall problem
From what height should you drop a ball if you want it to hit the ground in exactly 1.0 second?
Asked: x
Given: t = 1.0 s, g = -9.8 m/s2
(assume v0 = 0 m/s and x0 = 0 m)
Relationship:
Solution:

91. Example free fall problem
From what height should you drop a ball if you want it to hit the ground in exactly 1.0 second?
Asked: x
Given: t = 1.0 s, g = -9.8 m/s2
(assume v0 = 0 m/s and x0 = 0 m)
Relationship:
Solution:

92. Example free fall problem
From what height should you drop a ball if you want it to hit the ground in exactly 1.0 second?
Asked: x
Given: t = 1.0 s, g = -9.8 m/s2
(assume v0 = 0 m/s and x0 = 0 m)
Relationship:
Solution:
The negative sign means that the final position is 4.9 m below the initial position.
4.9 m
high

93. Another free fall problem
How far does an object have to fall to reach a speed of 10 m/s (neglecting friction)?
Asked:
Given: v
Relationships:
Solution:

94. Another free fall problem
How far does an object have to fall to reach a speed of 10 m/s (neglecting friction)?
Asked: x
Given: v, a (assume v0 = 0 m/s and x0 = 0 m)
Relationships:
Solution:

95. Another free fall problem
How far does an object have to fall to reach a speed of 10 m/s (neglecting friction)?
Asked: x
Given: v, a (assume v0 = 0 m/s and x0 = 0 m)
Relationships:
Solution:

96. Another free fall problem
How far does an object have to fall to reach a speed of 10 m/s (neglecting friction)?
Asked: x
Given: v, a (assume v0 = 0 m/s and x0 = 0 m)
Relationships:
Solution:

97. An object thrown upward
This ball thrown upward is in free fall as soon as the person is no longer touching it.
If the ball leaves the boy’s hand with an upward velocity of 15 m/s, how fast is it moving one second later?
Think: What is the sign of v0?
What is the sign of a?
Ask the students to make a prediction. Make sure that they recognize that the ball should slow down.

98. This makes sense. The ball must lose 9.8 m/s each second!
An object thrown upward
This ball thrown upward is in free fall as soon as the person is no longer touching it.
If the ball leaves the boy’s hand with an upward velocity of 15 m/s, how fast is it moving one second later?
This makes sense. The ball must lose 9.8 m/s each second!

99. An object thrown upward
Here is the velocity-time graph for a ball thrown up at +15 m/s.
The slope of the velocity-time graph equals the acceleration.
Point out that the slope equals one g. It is a constant straight line

100. An object thrown upward
When does the ball reach its highest height?
How do you know?

101. An object thrown upward
When does the ball reach its highest height?
at 1.5 seconds
How do you know?
Its velocity is zero for an instant.
What is the ball’s acceleration at that instant?

102. An object thrown upward
When does the ball reach its highest height?
at 1.5 seconds
How do you know?
Its velocity is zero for an instant.
What is the ball’s acceleration at that instant?
It is NOT zero! It is -9.8 m/s2.

103. An object thrown upward
Here is the position-time graph for the ball thrown up at +15 m/s.
What is the highest height the ball reaches?
How do you know?

104. An object thrown upward
Here is the position-time graph for the ball thrown up at +15 m/s.
What is the highest height the ball reaches?
about 11 meters
How do you know?
This is where it is farthest from the origin (at 1.5 s).

105. Assessment
A pitcher on a baseball team throws a high lob across home plate. For each part of this event described below, is the ball in free fall with a constant acceleration of 1 g?
The outfielder is winding up to throw the ball.
The ball is in the air, rising to the top of its arc.
The ball is in the air, descending toward the plate.
The bat is connecting with the ball.

106. Assessment
A pitcher on a baseball team throws a high lob across home plate. For each part of this event described below, is the ball in free fall with a constant acceleration of 1 g?
The outfielder is winding up to throw the ball No
The ball is in the air, rising to the top of its arc Yes
The ball is in the air, descending toward the plate. Yes
The bat is connecting with the ball No

107. Assessment A ball is thrown straight upward at 15 m/s.
How long does it take to reach its highest point?
What height does it reach, assuming it started at zero height?

108. Assessment A ball is thrown straight upward at 15 m/s.
How long does it take to reach its highest point?
asked: time
given: v0 = 15 m/s, v = 0 m/s, a = g = -9.8 m/s2
relationship:
solution:

109. Assessment A ball is thrown straight upward at 15 m/s.
What height does it reach, assuming it started at zero height?
Point out to the students that the time t was found in part a, so it is now a “given”.

110. Assessment A ball is thrown straight upward at 15 m/s.
What height does it reach, assuming it started at zero height?
asked: the height, which is x.
given: t = 1.5 s, v0 = 15 m/s, v = 0 m/s, a = g = -9.8 m/s2
relationship:
solution:
Point out to the students that the time t was found in part a, so it is now a “given”.