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Acceleration. 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. Define and describe accelerated motion.

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slide2

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.
assessment

Define and describe accelerated motion.

  • A car changes its velocity from 0 to 20 m/s in 4.0 seconds. What is its acceleration?
  • A car initially traveling at 25 m/s comes to a stop in 3.0 s. What is its acceleration?
  • An object starts from rest and accelerates at 2.5 m/s2 for 10 seconds. What is its final velocity?

Assessment

assessment1

Velocity vs. time

  • The motion of a particle along a straight line is depicted in this graph.

Assessment

What is the acceleration of the particle from 7 seconds to 10 seconds?

V

(m/s)

t (s)

physics terms

velocity

  • acceleration
  • rate of change

Physics terms

slide7

Brainstorm:

An accelerating object has a changing velocity.

Can you share an example of something accelerating?

slide8

Examples

a coaster making a turn

slide9

Examples

a coaster making a turn

a ball rolling uphill

slide10

Examples

a coaster making a turn

a ball rolling uphill

a baseball falling

slide11

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

slide12

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

slide13

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)

slide14

Units of acceleration

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

slide15

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.

slide16

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 ).

slide17

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.

slide18

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

slide19

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 . . .

slide20

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)

slide21

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

slide22

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

slide23

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

slide24

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

slide25

Exploring the ideas

Click on this interactive calculator on page 110.

slide26

Engaging with the concepts

A car starts at rest and accelerates to a velocity of 10 m/s in 5 seconds. What is its acceleration?

10 0

10

Enter the values and click [Run] to see the car perform the motion.

5

5 0

Acceleration

slide27

Engaging with the concepts

A car starts at rest and accelerates to a velocity of 10 m/s in 5 seconds. What is its acceleration?

a = 2 m/s2

10 0

10

2

5

5 0

Acceleration

slide28

Engaging with the concepts

A race car accelerates from 0 m/s to 27 m/s (60 mph) in 4 seconds.

What is its acceleration?

27 0

27

4

4 0

Acceleration

slide29

Engaging with the concepts

A race car accelerates from 0 m/s to 27 m/s (60 mph) in 4 seconds.

What is its acceleration?

27 0

27

a= 6.75 m/s2

6.75

4

4 0

Acceleration

slide30

Engaging with the concepts

Can you make the car go forward and then backward? How?

0 0

0

0

0 0

Acceleration

slide31

Engaging with the concepts

Can you make the car go forward and then backward? How?

Use positive viand negative a

0 0

0

0

Can you start at +5 m/s and finish 10 seconds later with a velocity of -5 m/s? What is this acceleration?

0 0

Acceleration

slide32

Engaging with the concepts

Can you make the car go forward and then backward? How?

Use positive viand negative a

-5.0 5.0

-10

-1

10

Can you start at +5 m/s and finish 10 seconds later with a velocity of -5 m/s? What is this acceleration?

10 0

Acceleration

Yes. -1 m/s2

slide33

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.

slide34

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.

slide35

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.

slide36

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.

slide37

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?

slide38

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!

slide39

Signs of the acceleration

Positive acceleration of +4 m/s2 adds +4 m/s of velocityeach second.

0 m/s

4 m/s

8 m/s

12 m/s

16 m/s

slide40

Signs of the acceleration

Positive acceleration of +4 m/s2 adds +4 m/s of velocityeach second.

Negativeacceleration of -4 m/s2 adds -4 m/s of velocityeach 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

slide41

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

+

slide42

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

+

+

slide43

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

-

slide44

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

-

-

slide45

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: +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

-

-

slide46

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?

slide47

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!

slide48

Test your knowledge

What does -1 m/s2 mean?

Can you describe two ways that an object could have a negative acceleration?

slide49

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.
slide50

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?

slide51

Acceleration on the v vs. t graph

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

No accelerationconstant velocity

slide52

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 accelerationconstant velocity

slide53

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 accelerationconstant velocity

Positiveaccelerationchangingvelocity

slide54

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 accelerationconstant velocity

Positiveaccelerationchangingvelocity

slide55

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 accelerationconstant velocity

No accelerationconstant velocity

Positiveaccelerationchangingvelocity

slide56

Acceleration on the v vs. t graph

Acceleration causes the velocity vs. time graph to have a non-zero slope.

Positiveaccelerationchangingvelocity

slide58

Acceleration on the v vs. t graph

Δt

Calculate the acceleration from t= 3 to t= 6 seconds.

Δv

slide59

Acceleration on the v vs. t graph

Δt

Calculate the acceleration from t = 3 to t = 6 seconds.

Δv

6 m/s

3 m/s

+1 m/s2

3 s

slide60

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?

slide61

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!
slide62

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.
slide63

Curves vs. lines

Acceleration creates a sloped line on a v vs. t graph.

Acceleration creates a curveon an xvs. tgraph.

slide64

Curves vs. lines

The x vs. tgraph curves upward for +a, like a smile!

The x vs. t graph curves downward for -a, like a frown!

slide65

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?
understanding acceleration

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.
understanding acceleration1

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?

assessment3

Define and describe accelerated motion.

Assessment

Acceleration is the rate of change of velocity.

For motion along a line, an object that is accelerating is speeding up or slowing down.

assessment5

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

Assessment

A change of +20 m/s over 4.0 seconds is an acceleration of +5.0 m/s2.

assessment7

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

Assessment

A change of -25 m/s over 3 seconds is an acceleration of -8.3 m/s2.

assessment9

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

Assessment

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

assessment10

Velocity vs. time

  • The motion of a particle along a straight line is depicted in this graph.

Assessment

What is the acceleration of the particle from 7 seconds to 10 seconds?

V

(m/s)

t (s)

assessment11

Velocity vs. time

  • The motion of a particle along a straight line is depicted in this graph.

Assessment

What is the acceleration of the particle from 7 seconds to 10 seconds?

V

(m/s)

answer: 40 m/s2

t (s)