Acceleration

1
Acceleration
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) ?v change in velocity
(m/s) ?t 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 ( ?v vf vi ) divided by the
change in time ( ?t 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 1 s 2 s 3 s 4 s
5 s 6 s 7 s 8 s
9 s 10 s
0 m/s 3 m/s 6 m/s 9 m/s 12 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)
vf (final velocity)
?v (change in 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 1.5 m/s 3 m/s
4.5 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 1.5 m/s 3 m/s
4.5 m/s
What is the acceleration of object 1? 1.5
m/s2 What is the acceleration of object 2?

Object 1
2 m/s 6 m/s 10 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 1.5 m/s 3 m/s
4.5 m/s
What is the acceleration of object 1? 1.5
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 10 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 1.5 m/s 3 m/s
4.5 m/s
What is the acceleration of object 1? 1.5
m/s2 What is the acceleration of object 2? 4
m/s2 What is the missing velocity? 14
m/s
Object 1
2 m/s 6 m/s 10 m/s
14 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.
0
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.
0 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.
0 2 4 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.
0 2 4 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.
0 2 4 4 1 -2 -5
What is happening to the speed?
26
The meaning of acceleration
Speed (m/s) 0 2 4 4 1 2 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.
0 2 4 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
4 m/s
12 m/s
8 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 1 m/s 2 m/s
3 m/s

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 ?
3 m/s 2 m/s 1 m/s
0 m/s
0 m/s 1 m/s 2 m/s
3 m/s


31
Signs of the acceleration
What is the sign of acceleration in each of these
four possible cases?
2. slowing down in direction -a
1. speeding up in direction a
3 m/s 2 m/s 1 m/s
0 m/s
0 m/s 1 m/s 2 m/s
3 m/s


3. speeding up in - direction ?
-3 m/s -2 m/s -1 m/s
0 m/s
-
32
Signs of the acceleration
What is the sign of acceleration in each of these
four possible cases?
2. slowing down in direction -a
1. speeding up in direction a
3 m/s 2 m/s 1 m/s
0 m/s
0 m/s 1 m/s 2 m/s
3 m/s


3. speeding up in - direction -a
4. slowing down in - direction ?
-3 m/s -2 m/s -1 m/s
0 m/s
0 m/s -1 m/s -2 m/s
-3 m/s
-
-
33
Signs of the acceleration
What is the sign of acceleration in each of these
four possible cases?
2. slowing down in direction -a
1. speeding up in direction a
3 m/s 2 m/s 1 m/s
0 m/s
0 m/s 1 m/s 2 m/s
3 m/s


4. slowing down in - direction a!
3. speeding up in - direction -a
-3 m/s -2 m/s -1 m/s
0 m/s
0 m/s -1 m/s -2 m/s
-3 m/s
-
-
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
cars 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
cars 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

1. 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
Acceleration on the v vs. t graph

• A car moves at a constant speed of 3 m/s for 3
  seconds.

No accelerationconstant velocity
40
Acceleration on the v vs. t graph

1. A car moves at a constant speed of 3 m/s for 3
   seconds.
2. The car accelerates to 6 m/s over the next 3
   seconds. What does this look like?

No accelerationconstant velocity
41
Acceleration on the v vs. t graph

1. A car moves at a constant speed of 3 m/s for 3
   seconds.
2. The car accelerates to 6 m/s over the next 3
   seconds. What does this look like?

No accelerationconstant velocity
Positiveaccelerationchangingvelocity
42
Acceleration on the v vs. t graph

1. A car moves at a constant speed of 3 m/s for 3
   seconds.
2. The car accelerates to 6 m/s over the next 3
   seconds.
3. The car continues at 6 m/s for three more
   seconds. What does this look like?

No accelerationconstant velocity
Positiveaccelerationchangingvelocity
43
Acceleration on the v vs. t graph

1. A car moves at a constant speed of 3 m/s for 3
   seconds.
2. The car accelerates to 6 m/s over the next 3
   seconds.
3. The car continues at 6 m/s for three more
   seconds. What does this look like?

No accelerationconstant velocity
No accelerationconstant velocity
Positiveaccelerationchangingvelocity
44
Acceleration on the v vs. t graph
Acceleration causes the velocity vs. time graph
to have a non-zero slope.
Positiveaccelerationchangingvelocity
45
Acceleration on the v vs. t graph
46
Acceleration on the v vs. t graph
?t
Calculate the acceleration from t 3 to t 6
seconds.
?v
47
Acceleration on the v vs. t graph
?t
Calculate the acceleration from t 3 to t 6
seconds.
?v
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

• 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

1. How does statement (A) relate to the diagram
   below it?What does 5 mph/s mean?
2. 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

1. 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

1. 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

1. Define and describe accelerated motion.

57
Assessment

1. 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

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

63
Assessment

1. 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 1 s 2 s 3 s 4 s
5 s 6 s 7 s 8 s
9 s 10 s
0 m/s 2 m/s 4 m/s 6 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

1. 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

1. 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 Earths 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 -9.8 m/s.
• after 2 seconds its velocity is -19.6 m/s.
• after 3 seconds its velocity is __?___
• after 10 seconds its velocity is __?___

72
Gravity and free fall
Near Earths 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 -9.8 m/s.
• after 2 seconds its velocity is -19.6 m/s.
• after 3 seconds its velocity is -29.4 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.
74
Find your reaction time
Use this equation for free fall to find your own
reaction timethe 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?
75
Gravity and free fall
If an object is dropped from rest then . . .

• after 1 second its velocity is -9.8 m/s.
• after 2 seconds its velocity is -19.6 m/s.
• after 3 seconds its velocity is -29.4 m/s.
• after 4 seconds its velocity is -39.2 m/s.
• . . . . and so on . . . .

76
Gravity and free fall
If an object is dropped from rest then . . .

• after 1 second its velocity is -9.8 m/s.
• after 2 seconds its velocity is -19.6 m/s.
• after 3 seconds its velocity is -29.4 m/s.
• after 4 seconds its velocity is -39.2 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.
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?
81
A skydiving trip
When did the parachute open? at t 28 seconds
82
A skydiving trip
1 2 3
4 5
6
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 boys 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?
98
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 boys 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.
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
balls 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
balls 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?

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