1. Graphical Analysis
of Motion

2. Which car/s is/are undergoing an acceleration?

3. Which car experiences the greatest acceleration?

4. Match a Graph
Consider the position-time graphs below. Each one of the 3 lines on the position-time graph corresponds to the motion of one of the 3 cars. Match the appropriate line to the particular color of car.

6. Section 1: Position vs Time Graphs

7. Position vs Time Graphs
position (x) vs time (t) graphs plot position as a function of time
the x-t graph of a constantly moving object is linear

8. If an object is staying in the same position for a time interval, then it’s…
at rest.
stopped.
not moving.
still.
Represented by a horizontal line on a position vs time graph

9. velocity- displacement traveled in a certain amount of time; a vector quantity
can be constant or average
constant velocity- velocity that does not change

10. Constant Velocity

11. Constant Velocity Questions: What is slope?
What is the formula for slope?
Use the slope formula to determine the slope of the line.
What do you notice?

12. Constant Velocity Conclusions:
the slope of a position vs time graph is the velocity
m = rise = ∆y = displacement = v
run ∆x time

13. b c a d e

14. Questions:
Where does the graph have positive slope?
What does this positive slope mean?
Where does the graph have negative slope?
What does this negative slope mean?
Is there any place(s) on the graph where the object is not moving?
How do you know?
What is the slope of the graph at this section?

15. Average Speed vs Average Velocity
Velocity is not the same as speed.
average velocity- the change in position divided by the time it took for that change to occur; a ratio of an object’s change in position (displacement) to the time it takes for that change to occur
vavg = distance traveled
time of travel
vavg = ∆x ∆t

16. Speed or Velocity? speed & velocity are different quantities
speed is the rate distance covered over time
speed is a scalar quantity
velocity is the rate displacement traveled over time
velocity is a vector quantity
speed & velocity equal magnitudes when an object does not change direction

17. Average Velocity
Consider a car moving with a constant velocity of +10 m/s for 5 seconds. The diagram below depicts such a motion.

18. Average Velocity
The position-time graph would look like the graph below:

19. How would this graph look?
Average Velocity
Now consider a car moving at a constant velocity of +5 m/s for 5 seconds, abruptly stopping, & then remaining at rest (v = 0 m/s) for 5 seconds.
How would this graph look?

20. Average Velocity
Check your answer…
                                                                      

21. Average Velocity

22. Instantaneous Speed
speed at any given point in time

23. Can I use one point to find velocity?
Important Note: DO NOT use one point to calculate velocity on a position-time graph!!! b c a d e

24. Example 2—Drawing Graphs from Graphs
Use the position vs time below to draw a velocity vs time graph.

25. Solution Velocity vs Time 6 5 4 3 2 1 -12 -8 -4 12 8 Velocity (m/s)12 8
Velocity (m/s)
Time (s)

26. Position vs Time Graphs
the x-t graph of a constantly moving object is linear

27. Position vs Time Graphs
the slope of an x-t graph is rise (position) over run (time)
for an x-t graph,
m = ∆x ∆t

28. Position vs Time Graphs
the change in position of an object is displacement, ∆x
the rate change in position of an object over an interval of time (the time it takes for the change to occur) is called average velocity, vavg
vavg = ∆x ∆t

29. Position vs Time Graphs

30. Position vs Time Graph

31. Position vs Time Graphs
the slope of an x-t graph describes the velocity of an object in motion
Fast, Rightward(+) Constant Velocity
Slow, Rightward(+) Constant Velocity

32. Position vs Time Graphs
the slope of an x-t graph describes the velocity of an object in motion
Slow, Leftward(-) Constant Velocity
Fast, Leftward(-) Constant Velocity

33. Position vs Time Graphs

34. Position vs Time Graphs
Quadrant I

35. Position vs Time Graphs
Quadrant IV

37. Section 2: Velocity vs Time Graphs

38. Velocity vs Time Graphs
a velocity (v) vs time (t) graph plots velocity as a function of time
the v-t graph of a constantly moving object is a horizontal line
remember, horizontal lines have zero slope

39. Velocity vs Time Graphs
the v-t graph of an object moving at a changing rate is a diagonal line
the velocity is changing by a constant rate

41. Constant, Uniform Acceleration
Bessie took a trip on her bicycle. The data of Bessie’s bike ride below, along with its accompanying graph, is shown. Use the graph to answer the questions that follow:

42. Constant, Uniform Acceleration
Questions:
What do you notice about this position vs time graph?
What does that mean?

43. Constant, Uniform Acceleration
Now, here is the data, along with the resulting velocity vs time graph, of the same bike ride.

44. Constant, Uniform Acceleration
Questions:
What can you say about the slope of this graph?
What does that mean?

45. Constant, Uniform Acceleration
Conclusions:
If an object has constant velocity, then that object has zero acceleration.
Constant velocity  Δv equals zero.
The graph of zero acceleration is a horizontal line.
All horizontal lines have zero slope because Δy equals zero.

46. Constant, Uniform Acceleration

47. Velocity vs Time Graphs
the slope of a v-t graph describes changing velocity over time
m = ∆v ∆t

48. Velocity vs Time Graphs
the rate at which velocity changes over an interval time is known as acceleration, aavg
mathematically,
aavg = ∆v ∆t

49. Velocity vs Time Graphs
the slope of a v-t graph describes the acceleration of the object in motion
Positive Velocity Positive Acceleration

50. Velocity vs Time Graphs
units of acceleration: m/s/s, m/s2
acceleration is a vector quantity
acceleration describes a change in velocity in the positive direction
acceleration also describes a change in direction
deceleration (negative acceleration) describes a change in velocity in the opposite direction

51. Velocity vs Time Graphs

52. Velocity vs Time Graphs

53. Velocity vs Time Graphs

54. Velocity vs Time Graphs

56. Section 2a: Constant Acceleration

57. Example 2—Drawing Graphs from Graphs
Use the position vs time below to draw a velocity vs time graph.

58. Solution Velocity vs Time 6 5 4 3 2 1 -12 -8 -4 12 8 Velocity (m/s)12 8
Velocity (m/s)
Time (s)

59. Solution Acceleration vs Time 6 5 4 3 2 1 -12 -8 -4 12 812 8
Example 3—Drawing Graphs from Graphs
Acceleration (m/s2)
Time (s)

60. Constant, Uniform Acceleration
velocity changes by the same rate
ex: a ball rolling down a ramp

61. Constant, Uniform Acceleration

62. Constant, Uniform Acceleration
Questions:
What is slope?
What is the formula for slope?
Use the slope formula to determine the slope of the line.
What do you notice?

63. Constant, Uniform Acceleration
Conclusions:
the slope of a velocity vs time graph is acceleration
m = rise = ∆y = change in velocity = a
run ∆x time

64. Constant, Uniform Acceleration

66. Section 3: Using Tangent Lines

68. Draw a straight line that is tangent to the curve.
Using Tangent Lines
What does the following graph show?
Draw a straight line that is tangent to the curve.
What does tangent mean?

69. Position vs Time
Position (m)
Time (s)

70. Using Tangent Lines
What does this graph show?

71. Constant Acceleration vs Constant Velocity

73. positive slope  positive acceleration
position vs time graph with increasing speed & its resulting velocity vs time graph:
positive slope  positive acceleration

74. negative slope  negative acceleration
position vs time graph with decreasing speed & its resulting velocity vs time graph:
negative slope  negative acceleration

75. Animations

76. Animations

77. Animations

78. Animations

79. Animations

80. Animations

82. Example: Determine what is happening at each section
constant velocity
increasing acceleration
deceleration
acceleration

84. Section 4: Lines That Intersect

85. Lines That Intersect
Sometimes, a problem will ask you to compare/contrast the motion of 2 objects.
To do this, use the information provided in the problem to construct a position vs time graph.
Then, use the graph to compare/contrast the 2 objects’ motions.

86. Lines That Intersect

87. What do the 2 cars have in common?

88. Do they ever share the same velocity?

89. Example 2—100 meter dash
Harry gives Sam a 30. m head start in the 100. m dash. Harry can run at 10.0 m/s, while Sam only runs at 6.0 m/s.

90. Example
Does Harry win?

91. Solution
Draw a position vs time graph for a 100. m dash, plotting both runners.
Position vs Time
Position (m)
Time (s)
100 50 5 10

92. Example
If Harry wins, at what time & place does he catch up to Sam?

93. Solution y = mx + b Now, you can determine the equation of each line.
Remember, the equation of a line is
Since we are using position, time, & velocity, the equation becomes
y = mx + b

94. Solution
x = vt + b

95. Solution xH = vHtH + bH For Harry… Where, vH = Harry’s velocity
bH = where Harry begins
xH = vHtH + bH

96. Solution
So…
xH = 10t

97. Solution xS = vStS + bS For Sam… Where, vS = Sam’s velocity
bS = where Sam begins
xS = vStS + bS

98. Solution
So…
xS = 6t + 30

99. Solution
Since you are looking for the point where Harry & Sam pass each other, that means the coordinates (tH, xH) & (tS, xS) are equal.
So, set the 2 equations for each line equal to each other & solve for t.

100. Solution
10t = 6t + 30
4t = 30
t = 7.5 sec

101. Solution xH = 10t = (10.0 m/s) (7.5 s) xH = 75 m
Now, that you know at what time Harry catches up to Sam, you can calculate at what place this happens.
Plug the value for time (7.5 sec) into the equation & solve for x.
xH = 10t
= (10.0 m/s) (7.5 s)
xH = 75 m

103. Section 5: Displacement from V-T Graphs

104. Displacement from a Velocity vs Time Graph
in the given v-t graph, the area under the graph is that for a rectangle
area = l* w
= v (t2 – t1)
= ∆x

105. Displacement from a Velocity vs Time Graph
in the given v-t graph, the area under the graph is that for a rectangle ∆x