Linear Motion

1
Linear Motion

• Holt Physics, pages 39 - 81

2
Distinguish displacement from distance

• Distance is a scalar quantity, which means that
  is only needs a magnitude. It is how far you are
  from a reference point.
• Displacement is a vector quantity, which means it
  needs both magnitude and direction. It is how
  far you ended up from where you started and in
  what direction. Therefore, distance is the
  magnitude of a displacement vector.

3
Distinguish displacement from distance (contd)

• For instance, if you leave home and drive from
  Houston to Dallas, your displacement is
  approximately 200 mi, North. When you return to
  Houston at the end of the trip, your displacement
  is now zero because you are now back at the place
  you started.

4
Distinguish velocity from speed

• Speed is a scalar quantity. It is how fast you
  are traveling. When you look at the speedometer
  on your car, it tells you the speed you are
  traveling at that moment.
• Velocity is a vector quantity. It is how fast
  you are going and in what direction. If you are
  traveling to Dallas, you may be traveling at 70
  mi/hr, North.

5
Define acceleration

• Acceleration is a vector quantity. It is the
  rate at which the velocity is changing. Since
  velocity is dependent on either speed or
  direction, if either of those change, your
  velocity changes and you have acceleration.
• Deceleration is a special name given to an
  acceleration that is in a direction opposite to
  the direction of motion of the object.

6
Hot Wheels Track
7
Define frame of reference.

• A frame of reference is necessary to fully
  explain motion. It is the position from which
  the motion is being observed.
• The frame of reference is assumed to be
  stationary.
• Your frame of reference affects the motion you
  perceive.

8
Give an example of a frame of reference.

• If you are looking at Spring H.S. from across the
  street, it appears to be sitting still but if you
  were viewing it though a telescope from the moon,
  it would appear to be rotating along with the
  earth which is turning on its axis.

9
Define the concept of relativity of velocities.

• Velocity is also affected by the frame of
  reference. This is known as relative velocity.
  It is the velocity of one object with respect to
  another object.

10
Use the concept of relativity of velocities.
(contd)

• If you are riding on the bus talking to your
  friend, his apparent speed is 0 m/s.
• However if you are standing on the street corner
  and the bus drives by without stopping, his
  apparent speed is now 40 mi/hr (the speed of the
  bus).
• If you are in your moms car going 50 mi/hr when
  you pass your friend on the bus that is going 40
  mi/hr then your friend appears to be going 10
  mi/hr (or 10 mi/hr in the opposite direction).

11
Define average velocity

• Average velocity is the total change in position
  divided by the time interval over which it
  occurred.

v average velocity in m/s ?d displacement in
meters ?t time in seconds
12
Define average velocity (contd)

• If two objects have the same average velocity
  means that both objects have the same
  displacement in the same time.

13
Sample Problem

• Suppose a car travels at a constant 10 m/s. How
  far would it move in one minute?

14
Sample Problem

• You drive a car 2 hours at 40 km/h, then 2 hours
  at 60 km/h. What is your average velocity? Do
  you get the same answer if you drive 100 km at
  each of these two speeds?

Answer 50 km/h no, 48 km/h
15
Average vs. Instantaneous Speed
16
Plot and interpret position-time graphs

• A position time graph shows an objects change in
  position over a period of time.

17
Plot and interpret position-time graphs

• An object which is not moving would have the same
  position over the period of time, so the graph
  would be a horizontal line. The y-intercept of
  this line indicates the distance of the object
  from a stationary reference point.

18
Plot and interpret position-time graphs (contd)

• An object which is moving at a constant positive
  velocity would cover equal distances in equal
  amount of time and its graph would appear as an
  upwardly sloping diagonal line.

19
Plot and interpret position-time graphs (contd)

• An object which is moving with a constant
  negative velocity would also cover equal
  distances in equal amounts of time but its
  distance from the reference point would be
  decreasing. Its graph would be a downward
  sloping diagonal line.

20
Plot and interpret position-time graphs (contd)

• An object which is moving with a constant
  positive acceleration will cover larger and
  larger distances in successive equal time
  intervals. Its position time graph would appear
  as a curve.

21
Plot and interpret position-time graphs (contd)

• An object which is moving with a constant
  negative acceleration will cover smaller and
  smaller distances in successive equal time
  intervals.

22
Calculate the velocity of an object from a
position time graph

• The slope of the position time graph is the
  velocity of the object.

23
Calculate the velocity of an object from a
position time graph
24
Plot and interpret position-time graphs

• The faster an object is moving, the steeper the
  line on the position time graph.

25
Passing Lane DT Graphs
26
Define instantaneous velocity

• Instantaneous velocity is the velocity at any
  instant.
• When an object is moving at a constant velocity,
  the instantaneous velocity and the average
  velocity are ALWAYS the same.
• When an object is accelerating, the instantaneous
  velocity and the average velocity are NOT the
  same at all times.

27
Define instantaneous velocity (contd)

• To find the instantaneous velocity of an
  accelerating object from a position time graph,
  you can find the slope of the line that is
  tangent to the curve at the point in time for
  which you want the instantaneous velocity.

28
Plot and interpret a velocity-time graph

• An object which is not moving would have no
  velocity over the entire period of time, so the
  graph would be a horizontal line along the
  x-axis.

29
Plot and interpret a velocity-time graph

• An object which is moving at a constant positive
  velocity would maintain the same velocity over
  the entire time interval and its graph would
  appear as a horizontal line above the x-axis.
  The y-intercept indicates the constant velocity
  of the object.

30
Passing Lane VT Graphs
31
Plot and interpret a velocity-time graph

• An object which is moving at a constant negative
  velocity would maintain the same velocity over
  the entire time interval and its graph would
  appear as a horizontal line below the x-axis.
  The y-intercept indicates the constant velocity
  of the object.

32
Plot and interpret a velocity-time graph

• An object which is moving with a constant
  positive acceleration will have a velocity time
  graph that is an upwardly sloping diagonal line.
  The y-intercept is the initial velocity of the
  object.

33
The Stoplight
34
Plot and interpret a velocity-time graph

• An object which is moving with a constant
  negative acceleration will have a velocity time
  graph that is an downwardly sloping diagonal
  line. The y-intercept is the initial velocity of
  the object.

35
Motion of a Two-Stage Rocket
36
Calculate the acceleration of an object from a
velocity time graph

• The slope of the velocity time graph is the
  acceleration of the object.

37
Calculate the acceleration of an object from a
velocity time graph
38
Calculate the displacement from a velocity time
graph

• The area between the velocity time graph and the
  x-axis is the displacement of the object during
  that time interval.

39
Sample Problem

• What is the displacement of the object during the
  first 5 seconds?

40
Sample Problem (contd)
41
Sample Problem (contd)
42
Sample Problem (contd)

• To find the total displacement of the object
  during the 25 seconds, you would break the shape
  into easily calculated areas as shown, calculate
  the individual areas then add them together.

43
Plot and interpret an acceleration-time graph

• An object which is not moving would have no
  acceleration over the entire period of time, so
  the graph would be a horizontal line along the
  x-axis.

44
Plot and interpret an acceleration-time graph

• An object which is moving with either a constant
  positive or negative velocity would have no
  acceleration over the entire period of time since
  the velocity is not changing. So the graph would
  be a horizontal line along the x-axis.

45
Plot and interpret an acceleration-time graph

• An object which is moving with constant positive
  acceleration would graph as a horizontal line
  above the x-axis. The y-intercept would be equal
  to the value of the acceleration.

46
Plot and interpret an acceleration-time graph

• An object which is moving with constant negative
  acceleration would graph as a horizontal line
  below the x-axis. The y-intercept would be equal
  to the value of the acceleration.

47
Calculate the velocity from an acceleration time
graph

• The area between the acceleration time graph and
  the x-axis is the velocity of the object.

48
Sample Problem

• What is the velocity of the object after 10
  seconds?

49
Sample Problem (contd)

• To calculate the velocity, you would find the
  area of the shaded region.

50
Sample Problem (contd)
51
Constant Positive Velocity
52
Constant Negative Velocity
53
Positive Velocity with Positive Acceleration
54
Positive Velocity with Negative Acceleration
55
Negative Velocity with Positive Acceleration
56
Negative Velocity with Negative Acceleration
57
Hyperlink to TI interactive Graphs
58
Acceleration Equations

• a acceleration in m/s2
• ?v - change in velocity in m/s
• vf final velocity in m/s
• vi initial velocity in m/s
• ?t or t time interval in seconds
• d displacement in m

59
Acceleration equations

• Since velocity, displacement, and acceleration
  are all vector quantities, you must keep
  direction in mind when you substitute values into
  the equations. Up or to the right are considered
  to be positive directions. Down or to the left
  are considered to be negative directions.

60
Sample Problem

• It takes 4.8 seconds for a cars speed to
  increase by 10 m/s. What is its acceleration?

Given ?v 10m/s t 4.8 sec a ?
Answer a 2.08 m/s2
61
Sample Problem

• A rocket is capable of accelerating at 800 m/s2.
  How long after lift off will the rocket reach 500
  m/s?

Given a 800 m/s2 vf 500 m/s vi 0 m/s t ?
Answer t 0.625 sec
62
Sample Problem

• A car is traveling at 50 m/s must slow down to 30
  m/s in the next 10 m. What deceleration must the
  car have?

Given vi 50 m/s vf 30 m/s d 10 m a ?
63
Sample Problem

• A car moves at 12 m/s and coasts up a hill with a
  uniform acceleration of -1.6 m/s2. How far has
  it traveled after 6 sec? How far has it gone
  after 9 sec?

Answer 43 m 43 m
64
Sample Problem

• An engineer must design a runway to accommodate
  airplanes that must reach a ground velocity of 61
  m/s before they take off. These planes are
  capable of being accelerated uniformly at the
  rate of 2.5 m/s2. How long will it take the
  planes to reach takeoff speed? What must be the
  minimum length of the runway?

Answer 24 sec 740 m
65
Sample Problem

• As a traffic light turns green, a waiting car
  starts with a constant acceleration of 6 m/s2.
  At the instant the car begins to accelerate, a
  truck with a constant velocity of 21 m/s passes
  in the next lane. How far will the car travel
  before it overtakes the truck? How fast will the
  car be traveling when it overtakes the truck?
  HINT Set the two distances equations equal to
  each other.

66
Acceleration due to gravity

• If an object is dropped or thrown, it will fall
  under the acceleration of gravity and its
  acceleration is 9.8 m/s2, down. The entire time
  the object is in the air, it has an acceleration
  of 9.8 m/s2, down, but its velocity is constantly
  changing. On the way upward, the speed is
  decreasing and on the way down, the speed is
  increasing.

67
Acceleration due to Gravity

• Keep in mind that acceleration is a vector
  quantity so when it is used in an equation, you
  use a or - sign to indicate whether the
  direction is up or down. Since the acceleration
  due to gravity is 9.8 m/s2, down when it is
  substituted into the equations you will use -9.8
  m/s2.

68
Gravity equations

• You will use the same equations however, you will
  use the value of -9.8 m/s2 for the acceleration
  when an object is in freefall.

69
Sample Problem

• A stone falls freely from rest for 8 seconds.
  What is the stones velocity after 8 seconds?
  What is the stones displacement during this time?

Answer -78 m/s -310 m
70
Sample Problem

• Kyle is flying a helicopter that is rising at 5
  m/s when he releases a bag. After 2 seconds,
  what is the bags velocity? How far has the bag
  fallen? How far below the helicopter is the bag?

Answer -15 m/s -10 m 20 m