Accelerated Motion

1
Accelerated Motion
Chapter
3
In this chapter you will

• Develop descriptions of accelerated motions.
• Use graphs and equations to solve problems
  involving moving objects.
• Describe the motion of objects in free fall.

2
Table of Contents
Chapter
3
Chapter 3 Accelerated Motion
Section 3.1 Acceleration Section 3.2 Motion
with Constant Acceleration Section 3.3 Free Fall

• Homework
• Read Chapter 3. Complete Study Guide.
• Graphical Analysis Packet
• HW 3 handout

3
Warmup Which Velocity Is It?
Section
3.1
Physics Daily Warmup 16

• There are two types of velocity that we encounter
  in our everyday lives. Instantaneous velocity
  refers to how fast something is moving at a
  particular point in time, while average velocity
  refers to the average speed something travels
  over a given period of time.
• For each use of velocity described below,
  identify whether it is instantaneous velocity or
  average velocity.
• The speedometer on your car indicates you are
• going 65 mph.
• A race-car driver was listed as driving 120 mph
• for the entire race.
• A freely falling object has a speed of 19.6 m/s
  after
• 2 seconds of fall in a vacuum.
• 4. The speed limit sign says 45 mph.

instantaneous
average
instantaneous
instantaneous
4
Acceleration
Section
3.1
In this section you will

• Define acceleration.
• Relate velocity and acceleration to the motion of
  an object.
• Create velocity-time graphs.

5
Acceleration
Section
3.1
Changing Velocity

• You can feel a difference between uniform and
  nonuniform motion.

• When you move in
• nonuniform motion, you feel pushed or pulled.
• In contrast, when you are in
• uniform motion and your eyes are closed, you
  feel as though you are not moving at all.

6
Acceleration
Section
3.1
Changing Velocity

• Consider the motion diagrams below showing the
  distance between successive positions.

7
Acceleration
Section
3.1
Changing Velocity

• There are two major indicators of the change in
  velocity in this motion diagram. The change in
  the spacing of the stick figures or dots and the
  differences in the lengths of the velocity
  vectors indicate the changes in velocity.

8
Acceleration
Section
3.1
Changing Velocity

• If an object speeds up, each subsequent velocity
  vector is longer.
• If the object slows down, each vector is shorter
  than the previous one.
• Both types of motion diagrams give an idea of how
  an objects velocity is changing.

9
Acceleration
Section
3.1
Velocity-Time Graphs
Play ch3_1_movanim.
10
Acceleration
Section
3.1
Average and Instantaneous Acceleration

• The rate at which an objects velocity changes is
  called the acceleration of the object. When the
  velocity of an object changes at a constant rate,
  it has a constant acceleration.

• The average acceleration of an object is the
  change in velocity during some measurable time
  interval divided by that time interval.

• Average acceleration is measured in m/s2.
• The change in velocity at an instant of time is
  called instantaneous acceleration.

11
Acceleration
Section
3.1
Instantaneous Acceleration

• The instantaneous acceleration of an object can
  be found by
• drawing a tangent line on the velocity-time
  graph at the point of time in which you are
  interested.
• The slope of this line is equal to the
  instantaneous acceleration.

Example Tangent line is drawn at t 1.0 s and t
5.0 s
12
Acceleration
Section
3.1
Average Acceleration

• The average acceleration of an object can be
  found by plotting two points which define the
  time interval, connecting the points with a
  straight line, ad finding the slope of the line.

Example slope of the line indicates the average
velocity between 1.0 and 5.0 s.
13
Section
Velocity Time Graph Example
3.1
a) How would you describe the sprinters velocity
and acceleration as shown on the graph? From the
graph, note that the sprinters velocity starts
at zero, increases rapidly for the first few
seconds, and then, after reaching about 10.0 m/s,
remains almost constant.
14
Section
Velocity Time Graph Example
3.1
b) What is his instantaneous acceleration at t
1 second and t 5 seconds? Instantaneous
acceleration Draw a tangent to the curve at t
1.0 s and t 6.0 s. Find the slope of each line.
15
Section
Velocity Time Graph Example
3.1
Solve for acceleration at 1.0 s
Solve for acceleration at 5.0 s
The acceleration is not constant because it
changes from 3.4 m/s2 to 0.03 m/s2 at 5.0 s. The
acceleration is in the direction chosen to be
positive because both values are positive.
16
Section
Velocity Time Graph Example
3.1
c) What is his average acceleration between 1 and
5 seconds? Average acceleration Draw a line
between the points at t 1 s and t 5 s. Find
the slope of the line.
The slope is 4 m/s 5 s 0.8 m/s2
17
Acceleration
Section
3.1
Positive and Negative Acceleration

• These four motion diagrams represent the four
  different possible ways to move along a straight
  line with constant acceleration.

• Positive direction, speeding up
• Positive direction, slowing down
• Negative direction, speeding up
• Negative direction, slowing down

18
Acceleration
Section
3.1
Positive and Negative Acceleration

• When the object is speeding up, the velocity and
  acceleration vectors point in the same direction.
  (case 1 and 3)
• When the object is slowing down, the velocity and
  acceleration vectors point in opposite directions
  (case 2 and 4)
• Both the direction of an objects velocity and
  its direction of acceleration are needed to
  determine whether it is speeding up or slowing
  down.

19
Acceleration
Section
3.1
Positive and Negative Acceleration

• An object has a positive acceleration when the
  acceleration vector points in the positive
  direction and a negative acceleration, when the
  acceleration vector points in the negative
  direction.
• The sign of acceleration does not indicate
  whether the object is speeding up or slowing down.

20
Acceleration
Section
3.1
Determining Acceleration from a v-t Graph

• Velocity and acceleration information also is
  contained in velocity-time graphs.

• Graphs A, B, C, D, and E, as shown on the right,
  represent the motions of five different runners.
• Positive velocity in this graph means the
  direction is east. Negative velocity means the
  direction is west.

21
Section
Determining Acceleration from a v-t Graph
3.1

• Describe the direction of motion, velocity and
  acceleration for
• Graph A
• Zero slope means zero acceleration constant
  velocity towards the east.
• Graph B
• Starting with zero velocity, picking up speed,
  moving towards the east. The straight line
  indicates constant acceleration.
• Graph C
• Moving towards the east while slowing down and
  eventually stopping. Slowing down with a constant
  negative acceleration.

22
Section
Determining Acceleration from a v-t Graph
3.1

• Graph D
• Moving towards the west while slowing down, turns
  around, then moves east while speeding up.
• Graph E
• Moving west with constant velocity, zero
  acceleration.
• Remember, for a velocity-time graph
• - positive velocity occurs when the line is
  anywhere above the x-axis
• - positive velocity means the object is moving
  in the positive direction, which might be
  east, to the right, etc.
• - the slope of the line indicated the
  acceleration
• - a straight line means constant acceleration

23
Acceleration
Section
3.1
Determining Acceleration from a v-t Graph

• The following equation expresses average
  acceleration as the slope of the velocity-time
  graph.

• Average acceleration is equal to the change in
  velocity, divided by the time it takes to make
  that change.

24
Acceleration
Section
3.1
Example Suppose you run a wind sprints back
and forth across the gym. You run at a speed of
4.0 m/s toward the wall, touch, and run back at
the same speed. The whole trip takes 10 seconds.
What is your average acceleration if the positive
direction is toward the wall?
Givens vi 4 m/s vf -4 m/s ?t 10
s Unknown a Equation Substitute and
Solve Sense The negative sign means the
acceleration is away from the wall. Change in
direction of motion results in acceleration.
25
Section Check
Section
3.1
Question 1

• Which of the following statements correctly
  define acceleration?

1. Acceleration is the rate of change of
   displacement of an object.
2. Acceleration is the rate of change of velocity of
   an object.
3. Acceleration is the amount of distance covered in
   unit time.
4. Acceleration is the rate of change of speed of an
   object.

26
Section Check
Section
3.1
Answer 1

• Answer B

Reason The rate at which an objects velocity
changes is called acceleration of the object.
27
Section Check
Section
3.1
Question 2

• What happens when the velocity vector and the
  acceleration vector of an object in motion are in
  same direction?

1. The acceleration of the object increases.
2. The speed of the object increases.
3. The object comes to rest.
4. The speed of the object decreases.

28
Section Check
Section
3.1
Answer 2

• Answer B

Reason When the velocity vector and the
acceleration vector of an object in motion are in
same direction, the speed of the object increases.
29
Section Check
Section
3.1
Question 3

• On the basis of the velocity-time graph of a car
  moving up a hill, as shown on the right,
  determine the average acceleration of the car?

1. 0.5 m/s2
2. -0.5 m/s2

1. 2 m/s2
2. -2 m/s2

30
Section Check
Section
3.1
Answer 3

• Answer B

Reason Average acceleration of an object is the
slope of the velocity-time graph.
vf 0 vi 25 m/s tf 50 s ti 0 a vf
vi 0 25 m/s - 0.5 m/s2 tf ti 50
s - 0
31
Section Check
Section
3.2
Practice Problems, p. 64 6, 7, 9, 10.
Motion with Constant Acceleration
Steel Ball Race, p. 58
32
Motion with Constant Acceleration
Section
3.2
In this section you will

• Interpret position-time graphs for motion with
  constant acceleration.
• Determine mathematical relationships among
  position, velocity, acceleration, and time.
• Apply graphical and mathematical relationships to
  solve problems related to constant acceleration.

33
Motion with Constant Acceleration
Section
3.2
Velocity with Average Acceleration

• If an objects average acceleration during a time
  interval is known, then it can be used to
  determine how much the velocity changed during
  that time.
• The definition of average acceleration

can be rewritten as follows
34
Motion with Constant Acceleration
Section
3.2
Velocity with Average Acceleration

• The equation for final velocity with average
  acceleration can be written as follows

• The final velocity is equal to the initial
  velocity plus the product of the average
  acceleration and time interval.

vi
35
Motion with Constant Acceleration
Section
3.2
Velocity with Average Acceleration

• In cases in which the acceleration is constant,
  the average acceleration, a, is the same as the
  instantaneous acceleration, a.
• The equation for final velocity can be rearranged
  to find the time at which an object with constant
  acceleration has a given velocity.
• It also can be used to calculate the initial
  velocity of an object when both the velocity and
  the time at which it occurred are given.

36
Motion with Constant Acceleration
Section
3.2
Position with Constant Acceleration

• The position data at different time intervals for
  a car with constant acceleration are shown in the
  table.
• The data from the table are graphed as shown on
  the next slide.

37
Motion with Constant Acceleration
Section
3.2
Position with Constant Acceleration

• The graph shows that the cars motion is not
  uniform the displacements for equal time
  intervals on the graph get larger and larger.

• The slope of a position-time graph of a car
  moving with a constant acceleration gets steeper
  as time goes on.

38
Motion with Constant Acceleration
Section
3.2
Position with Constant Acceleration

• The slopes from the position time graph can be
  used to create a velocity-time graph as shown on
  the right.
• Note that the slopes shown in the position-time
  graph are the same as the velocities graphed in
  velocity-time graph.

39
Motion with Constant Acceleration
Section
3.2
Position with Constant Acceleration

• A velocity-time graph does not contain any
  information about the objects position.
• However, the velocity-time graph does contain
  information about the objects displacement.
• Recall that for an object moving at a constant
  velocity,

40
Motion with Constant Acceleration
Section
3.2
Position with Constant Acceleration

• On the graph shown on the right, v is the height
  of the plotted line above the t-axis, while ?t is
  the width of the shaded rectangle. The area of
  the rectangle, then, is v?t, or ?d. Thus, the
  area under the v-t graph is equal to the objects
  displacement.
• The area under the v-t graph is equal to the
  objects displacement.

41
Motion with Constant Acceleration
Section
3.2
Finding the Displacement from a v-t Graph Example
1
The v-t graph below shows the motion of an
airplane. Find the displacement of the airplane
at ?t 1.0 s and at ?t 2.0 s.
42
Motion with Constant Acceleration
Section
3.2
Finding the Displacement from a v-t Graph
The displacement is the area under the v-t graph.
The time intervals begin at t 0.0.
43
Motion with Constant Acceleration
Section
3.2
Finding the Displacement from a v-t Graph
Identify the given and unknown variables.
Given v 75 m/s ?t 1.0 s ?t 2.0 s
Unknown ?d ?
44
Motion with Constant Acceleration
Section
3.2
Finding the Displacement from a v-t Graph
Solve for displacement during ?t 1.0
s. Equation
Substitute v 75 m/s, ?t 1.0 s Solve
45
Motion with Constant Acceleration
Section
3.2
Finding the Displacement from a v-t Graph
Solve for displacement during ?t 2.0 s.
Substitute v 75 m/s, ?t 2.0 s
46
Motion with Constant Acceleration
Section
3.2
Finding the Displacement from a v-t Graph Sense

• Are the units correct?
• Displacement is measured in meters.
• Do the signs make sense?
• The positive sign agrees with the graph.
• Is the magnitude realistic?
• Moving a distance to about one football field is
  reasonable for an airplane.

47
Motion with Constant Acceleration
Section
3.2
An Alternative Expression

• Often, it is useful to relate position, velocity,
  and constant acceleration without including time.
• The three equations for motion with constant
  acceleration are summarized in the table.

48
Section Check
Section
3.2
Question 1

• A position-time graph of a bike moving with
  constant acceleration is shown on the right.
  Which statement is correct regarding the
  displacement of the bike?

1. The displacement in equal time interval is
   constant.
2. The displacement in equal time interval
   progressively increases.

1. The displacement in equal time interval
   progressively decreases.
2. The displacement in equal time interval first
   increases, then after reaching a particular point
   it decreases.

49
Section Check
Section
3.2
Answer 1

• Answer B

Reason You will see that the slope gets steeper
as time goes, which means that the displacement
in equal time interval progressively gets larger
and larger.
50
Section Check
Section
3.2
Question 2

• A car is moving with an initial velocity of vi
  m/s. After reaching a highway, it moves with a
  constant acceleration of a m/s2, what will be the
  velocity (vf) of the car after traveling for t
  seconds?

1. vf vi at
2. vf vi 2at
3. vf2 vi2 2at
4. vf vi at

51
Section Check
Section
3.2
Answer 2

• Answer A

Reason Since a ?v/?t vf - vi a (tf -
ti) Also since car is starting from rest, ti
0 Therefore vf vi at (where t is the total
time)
52
Section
Section Check
3.2
Question 3
If you were given initial and final velocities
and the constant acceleration of an object, and
you were asked to find the displacement, what
equation would you use?

1. vf vi at
2. df di vi t ½ at2
3. vf2 vi2 2a(df - di)
4. vf vi at

53
Section
Section Check
3.2
Answer 3
Answer C
Reason Kinematics equation number 3 does not
require time. You are not given time in the
problem.
54
Motion with Constant Acceleration
Section
3.2
Practice Problems p.65 20, 21. Section Review
p.71 34, 39.
55
Free Fall
Section
3.3
Free Fall
In this section you will

• Define acceleration due to gravity.
• Solve problems involving objects in free fall.

56
Free Fall
Section
3.3
Acceleration Due to Gravity

• free fall the motion of a body when air
  resistance is negligible and the motion can be
  considered due to the force of gravity alone.
• After a lot of observation, Galileo concluded
  that, neglecting the effect of the air, all
  objects in free fall had the same acceleration.
• It didnt matter what they were made of, how
  much they weighed, what height they were dropped
  from, or whether they were dropped or thrown.
• The acceleration of falling objects, given a
  special symbol, g, is equal to 9.80 m/s2.
• The acceleration due to gravity is the
  acceleration of an object in free fall that
  results from the influence of Earths gravity.

57
Free Fall
Section
3.3
Acceleration Due to Gravity
ch 3_4_movanim
58
Free Fall
Section
3.3
Acceleration Due to Gravity

• At the top of the flight, the balls velocity is
  0 m/s. What would happen if its acceleration were
  also zero? Then, the balls velocity would not be
  changing and would remain at 0 m/s.
• If this were the case, the ball would not gain
  any downward velocity and would simply hover in
  the air at the top of its flight.
• Because this is not the way objects tossed in the
  air behave on Earth, you know that the
  acceleration of an object at the top of its
  flight must not be zero. Further, because you
  know that the object will fall from that height,
  you know that the acceleration must be downward.

59
Free Fall
Section
3.3
Acceleration Due to Gravity

• Amusement parks use the concept of free fall to
  design rides that give the riders the sensation
  of free fall.
• These types of rides usually consist of three
  parts the ride to the top, momentary suspension,
  and the plunge downward.
• When the cars are in free fall, the most massive
  rider and the least massive rider will have the
  same acceleration.

60
Free Fall
Section
3.3
Acceleration Due to Gravity

• Example Suppose the free-fall ride at an
  amusement park starts at rest and is in free fall
  for 1.5 s. What would be its velocity at the end
  of 1.5 s? How far would it fall?
• Choose a coordinate system with a positive axis
  upward and the origin at the initial position of
  the car. Because the car starts at rest, vi would
  be equal to 0.0 m/s.
• Givens vi 0.0 m/s, di 0.0 m, ti 0.0 s, tf
  1.5 s, a -9.8 m/s2
• Unknown vf

61
Free Fall
Section
3.3
Acceleration Due to Gravity

• To calculate the final velocity, use the equation
  for velocity with constant acceleration.
• Equation
• Substitute Solve
• Sense Negative velocity means down units are
  OK.

62
Free Fall
Section
3.3
Acceleration Due to Gravity

• How far does the car fall? Use the equation for
  displacement when time and constant acceleration
  are known.
• Equation
• Substitute Solve
• Sense The displacement is negative because it
  fell, and the units are meters. Looks good!

63
Section Check
Section
3.3
Question 1

• What is free fall?

64
Section Check
Section
3.3
Answer 1

• Free Fall is the motion of the body when air
  resistance is negligible and the action can be
  considered due to gravity alone.

65
Section Check
Section
3.3
Question 2

• If a stone is thrown vertically upwards with a
  velocity of 25 m/s, what will be the velocity of
  the stone after 1 second?

1. 9.8 m/s
2. 15.2 m/s
3. 25 m/s
4. 34.8 m/s

66
Section Check
Section
3.3
Answer 2

• Answer B

Reason Since the ball is thrown upwards, the
velocity and acceleration are in opposite
directions, therefore the speed of the ball
decreases. After 1 s, the balls velocity is
reduced by 9.8 m/s (as acceleration due to
gravity is 9.8 m/s2), so it is now traveling at
25 m/s 9.8 m/s 15.2 m/s.
67
Section Check
Section
3.3
Question 3

• If a 50-kg bag and a 100-kg bag are dropped from
  a height of 50 m. Which of the following
  statement is true about their acceleration?
  (Neglect air resistance)

1. 100-kg bag will fall with a greater acceleration.
2. 50-kg bag will fall with a greater acceleration.
3. Both will fall at the same and constant rate of
   acceleration.
4. Both will fall at the same rate of acceleration,
   which changes equally as time goes.

68
Section Check
Section
3.3
Answer 3

• Answer C

Reason Any body falling freely towards Earth,
falls with a same and constant acceleration of
9.8 m/s2. It doesnt matter how much it weighed
and what height it was dropped from.
69
Free Fall
Section
3.3

• Exit Ticket Please write in full sentences.
• Describe the velocity and acceleration of a ball
  that is tossed in the air and comes back down.

70
Free Fall
Section
3.3

• Practice Problems p.74 42, 44.
• HW 3 handout.

71
Chapter
Physics Chapter 2 3 Test Information The test
is worth 46 points total. Matching 12
questions, 12 points total Problems 7 questions,
34 points total Know - vocabulary for both
chapters - how to interpret and draw
position-time and velocity-time graphs - how
to use the 3 kinematics equations to solve
problems - how to express answers with correct
units and sig figs
3