1. 3.1 Acceleration
How would you describe the motion of the runner in each motion diagram?

2. 3.1 Acceleration
Change in velocity indicated by the change in spacing of the dots and the different lengths of the velocity vectors

3. 3.1 Acceleration
acceleration – the rate at which an object’s velocity changes (units of m/s2)
when the velocity of an object changes at a constant rate, it has a constant acceleration

4. In which of these motion diagrams is the runner accelerating?
3.1 Acceleration
In which of these motion diagrams is the runner accelerating?

5. Where was the cheetah at rest?
3.1 Acceleration
v-t graph for a cheetah
Where was the cheetah at rest?

6. Why not at rest between 2 s and 3 s?
3.1 Acceleration
v-t graph for a cheetah
Why not at rest between 2 s and 3 s?

7. What was happening during the first 2 s?
3.1 Acceleration
v-t graph for a cheetah
What was happening during the first 2 s?

8. What was the acceleration between 2 s and 3 s?
v-t graph for a cheetah
What was the acceleration between 2 s and 3 s?

9. What was the acceleration between 4 s and 6 s?
v-t graph for a cheetah
What was the acceleration between 4 s and 6 s?

10. an object accelerates if its speed, direction, or both change
3.1 Acceleration
acceleration has both direction and magnitude, therefore, it is a vector quantity
an object accelerates if its speed, direction, or both change
a negative acceleration does not necessarily mean the object is slowing down, and a positive acceleration does not necessarily mean it is speeding up!!!

11. Position-time graph for the ostrich
2.3 Position-Time Graphs
Position-time graph for the ostrich
Draw a velocity-time graph for this motion

12. Draw a velocity-time graph for this motion
2.3 Position-Time Graphs
Position-time graph for the ostrich
Draw a velocity-time graph for this motion

13. 3.1 Acceleration

14. 3.1 Acceleration
Graph a negative acceleration, not slowing down
Graph a positive acceleration that is slowing down
(T or F) An object that is slowing down is accelerating

15. this is a graph of a train moving
3.1 Acceleration
this is a graph of a train moving
Velocity in the positive direction is constant (acceleration is 0)
Velocity in the positive direction is decreasing (acceleration is -)
Velocity in the positive direction is increasing (acceleration is +)

16. 3.1 Acceleration
The slope of a v-t graph for an object is the object’s average acceleration - the change in velocity during some measurable time interval divided by that time interval.
indicates an average

17. 3.1 Acceleration
Remember!! The sign of the acceleration does not indicate whether the object is speeding up or slowing down.

18. 3.1 Acceleration
Describe the motion of a ball as it rolls up a slanted driveway. The ball starts at 2.50 m/s, slows down for 5.00 s, stops for an instant, and then rolls back down at an increasing speed. The positive direction is chosen to be up the driveway, and the origin the place where the motion begins.
What is the sign of the ball’s acceleration as it rolls up the driveway?
What is the magnitude of the ball’s acceleration as it rolls up the driveway?

19. 3.1 Acceleration
How can you determine velocity using a position time graph?

20. 3.1 Acceleration
What is the area under the velocity-time graph between t=2.0s and t=4.0 s?

21. 3.1 Acceleration
What is the area under the velocity-time graph between t=2.0s and t=4.0 s?
120 m
What is the change in position on the position-time graph between t=2.0 s and t=4.0 s?
How are these two answers related?

22. 3.1 Acceleration
How can you determine acceleration using the velocity-time graph?

23. 3.1 Acceleration
If the velocity were constant, what would the position-time graph look like?
What would the acceleration-time graph look like?

24. 3.1 Acceleration
Which of the following statements correctly define acceleration?
Acceleration is the rate of change of displacement of an object.
Acceleration is the amount of distance covered in unit time.
Acceleration is the rate of change of velocity of an object.
Acceleration is the rate of change of speed of an object.

25. 3.1 Acceleration
Which of the following statements correctly define acceleration?
Acceleration is the rate of change of displacement of an object.
Acceleration is the amount of distance covered in unit time.
Acceleration is the rate of change of velocity of an object.
Acceleration is the rate of change of speed of an object.

26. 3.1 Acceleration
What happens when the velocity vector (5 m/s) and the acceleration vector (2 m/s2) of an object in motion are in same direction?
The acceleration of the object increases.
The speed of the object increases.
The object comes to rest.
The speed of the object decreases.

27. 3.1 Acceleration
What happens when the velocity vector (5 m/s) and the acceleration vector (2 m/s2) of an object in motion are in same direction?
The acceleration of the object increases.
The speed of the object increases.
The object comes to rest.
The speed of the object decreases.

28. 3.1 Acceleration
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?
-0.5 m/s2
0.5 m/s2
2 m/s2
-2 m/s2

29. 3.1 Acceleration
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?
-0.5 m/s2
0.5 m/s2
2 m/s2
-2 m/s2

30. 3.1 Acceleration
1. What is the average acceleration of each car from 0.0 to 97.0 km/h (in m/s2)?
2. Which car can go from 0.0 to 97 km/h in the shortest time? Does this car have the highest or lowest acceleration?
3. For acceleration from 0.0 to 97 km/h, which direction is the acceleration vector pointing?
4. When a car is braking from 97 km/h to 0.0 km/h, is it positive or negative acceleration?
5. Which car is the safest?

31. 3.1 Acceleration
A hockey player glides along the ice at a constant speed of 1.25 m/s in the positive direction onto a rough section of ice, which slows him. If he stops in 5.0 s, what is the magnitude and direction of his acceleration?

32. 3.2 Motion with Constant Acceleration
If an object is accelerated for some time interval, we can rearrange our equation for average acceleration to determine its final velocity:
Solving for :

33. 3.2 Motion with Constant Acceleration
Similarly, if an object is accelerated for some time interval, we can rearrange our equation for average acceleration to determine its final velocity:
Solving for :

34. 3.2 Motion with Constant Acceleration
Similarly, if an object is accelerated for some time interval, we can rearrange our equation for average acceleration to determine its final velocity:
Solving for :
Kinematics Equation #1

35. With a constant acceleration, is ½ the sum of and
3.2 Motion with Constant Acceleration
If an object is accelerated for some time interval, how does its position change? What is its displacement?
so,
With a constant acceleration,
is ½ the sum of and
but,

36. With a constant acceleration, is ½ the sum of and
3.2 Motion with Constant Acceleration
If an object is accelerated for some time interval, how does its position change? What is its displacement?
so,
With a constant acceleration,
is ½ the sum of and
but,
So,

37. 3.2 Motion with Constant Acceleration
Take
and substitute into

38. 3.2 Motion with Constant Acceleration
Take
and substitute into

39. 3.2 Motion with Constant Acceleration
Take
and substitute into

40. 3.2 Motion with Constant Acceleration

41. 3.2 Motion with Constant Acceleration

42. 3.2 Motion with Constant Acceleration
Kinematics Equation #2

43. 3.2 Motion with Constant Acceleration
Often, it is useful to relate position, velocity, and constant acceleration without including time.
Recall,

44. 3.2 Motion with Constant Acceleration
Often, it is useful to relate position, velocity, and constant acceleration without including time.
Recall,
Solve for

45. 3.2 Motion with Constant Acceleration
Often, it is useful to relate position, velocity, and constant acceleration without including time.
Recall,
Solve for
Recall,

46. 3.2 Motion with Constant Acceleration
Often, it is useful to relate position, velocity, and constant acceleration without including time.
Recall,
Solve for
Recall,

47. 3.2 Motion with Constant Acceleration
Multiply by

48. 3.2 Motion with Constant Acceleration
Multiply by

49. 3.2 Motion with Constant Acceleration
Multiply by

50. Multiply by Kinematics Equation #3
3.2 Motion with Constant Acceleration
Multiply by
Kinematics Equation #3

51. Kinematics Equation #1 Kinematics Equation #2 Kinematics Equation #3
3.2 Motion with Constant Acceleration
Kinematics Equation #1
Kinematics Equation #2
Kinematics Equation #3

52. 3.2 Motion with Constant Acceleration
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?
The displacement in equal time intervals is constant.
The displacement in equal time intervals progressively increases.
The displacement in equal time intervals progressively decreases.
The displacement in equal time intervals first increases, then after reaching a particular point it decreases.

53. 3.2 Motion with Constant Acceleration
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?
The displacement in equal time intervals is constant.
The displacement in equal time intervals progressively increases.
The displacement in equal time intervals progressively decreases.
The displacement in equal time intervals first increases, then after reaching a particular point it decreases.

54. a) What is the velocity after 8.5 s?
3.2 Motion with Constant Acceleration
Starting from rest, a ball rolls down an inclined plane at a constant acceleration of 2.00 m/s2.
a) What is the velocity after 8.5 s?
b) How far does the ball roll in 10.0 s?

55. a) How long will it take a plane to achieve take off speed?
3.2 Motion with Constant Acceleration
An engineer is to design a runway to accommodate airplanes that must gain a ground speed of 60.0 m/s before they can take off. If these planes are capable of being accelerated uniformly at the rate of 1.5 m/s2
a) How long will it take a plane to achieve take off speed?
b) What must be the minimum length of the runway?

56. a) How long does it take the car to come to a stop?
3.2 Motion with Constant Acceleration
A car traveling at 88 km/hr undergoes a constant acceleration of 8.0 m/s2 as the brakes are applied
a) How long does it take the car to come to a stop?
b) How far does the car move after the brakes are applied?

57. a) What is the velocity 10.0 s later?
3.2 Motion with Constant Acceleration
A car rolls down a hill with a uniform acceleration of 2.0 m/s2. At a certain point on the hill its velocity is 25.0 m/s
a) What is the velocity 10.0 s later?
b) How far has the car traveled during that 10.0 s?

58. 3.3 Free Fall
Free fall – the motion of a body when air resistance is negligible and the action can be considered due to gravity alone
Galileo was the first to conclude that, neglecting the effect of air, all objects in free fall have the same acceleration.

59. 3.3 Free Fall
These skydivers are technically NOT in free fall because air resistance is acting upon them

60. In our study of free fall:
Acceleration due to gravity – the acceleration of an object in free fall that results from the influence of the Earth’s gravity
given the symbol, g
g = 9.80 m/s2
In our study of free fall:
Upwards is +
Downwards is -
Therefore, g = m/s2

61. 3.3 Free Fall

62. 3.3 Free Fall
Whether the egg is on the way up or the way down it is in free fall !!!!

63. 3.3 Free Fall
Like #36
An in-line skater first accelerates from 0.0 m/s to 8.5 m/s in 9.0 s, then continues at this constant speed for another 10.0 s. What ist he total distance traveled by the in-line skater?
Like #93
How far does a plane fly in 20 s while it velocity is changing from 160 m/s to 70 m/s at a uniform rate of acceleration?

64. 3.3 Free Fall
g = m/s2
vi = 4.9 m/s
g = m/s2
vi = 16 m/s

65. 3.3 Free Fall
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)
100-kg bag will fall with a greater acceleration.
50-kg bag will fall with a greater acceleration.
Both will fall at the same and constant rate of acceleration.
Both will fall at the same rate of acceleration, which changes equally as time goes.

66. 3.3 Free Fall
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)
100-kg bag will fall with a greater acceleration.
50-kg bag will fall with a greater acceleration.
Both will fall at the same and constant rate of acceleration.
Both will fall at the same rate of acceleration, which changes equally as time goes.

67. After 3.00 s of free fall, what is the velocity of the stone?
A stone is dropped from rest from the top of a tall building. After 3.00 s of free fall, what is the displacement of the stone?
After 3.00 s of free fall, what is the velocity of the stone?

68. a) How long is the coin in the air? b) How high does it go?
3.3 Free Fall
At the coin toss of a football game, the ref tosses the coin straight up with an initial speed of 8.0 m/s.
a) How long is the coin in the air?
b) How high does it go?

69. A ball is thrown vertically upward with a velocity of 11.5 m/s.
3.3 Free Fall
A ball is thrown vertically upward with a velocity of 11.5 m/s.
a) To what height will it rise?
b) How long to fall back to where it was released?

70. 3.3 Free Fall
Like #46
You decide to flip a coin to determine whether to do your physics or English homework first. The coin is flipped straight up.
If the coin reaches a high point 0.53 m above where you released it, what was its initial speed?
If you catch it at the same height as you released it, how much time did it spend in the air?
Like #100
You throw a ball downward from a window at a speed of 4.0 m/s. How fast will it be moving when it hits the sidewalk 8.5 m below?

71. 3.3 Free Fall
Suppose you throw a ball straight up into the air. Describe the changes in the velocity of the ball. Describe the changes in the acceleration of the ball.
Which has the greater acceleration: a car that increases its speed from 50 km/h to 60 km/h, or a bike that goes from 0 km/h to 10 km/h in the same time?
The value of g on the Moon in one-sixth of its value on Earth.
Would a ball that is dropped by an astronaut hit the surface of the Moon with a greater, equal, or lesser speed than that of a ball dropped from the same height to Earth?
Would it take the ball more, less, or equal time to fall?

72. 3.3 Free Fall
When a ball is thrown vertically upward, it continues upward until it reaches a certain position, and then it falls downward. At the highest point, its velocity is instantaneously zero. Is the ball accelerating at the highest point?
Rock A is dropped from a cliff and Rock B is thrown upward from the same position, both at the same time.
When they reach the ground at the bottom of the cliff, which rock has a greater velocity?
Which has a greater acceleration?
Which arrives to the ground first?

73. 3.3 Free Fall
An object is thrown vertically into the air. Which of the following five graphs represents the velocity of the object as a function of the time? The positive direction is taken to be upward.

74. 3.3 Free Fall For a truck moving at a constant velocity, the
distance traveled during the second interval is greater than the distance traveled during the first interval
directions during the first, second, and third intervals must be the same
acceleration is increasing for each second
velocity is increasing for each second

75. 3.3 Free Fall
Consider the following five graphs (note the axes carefully). Which of these represent(s) motion at constant speed?

76. What would the position-time graph for this object look like?
3.2 Motion with Constant Acceleration
What would the position-time graph for this object look like?
What would the acceleration-time graph for this object look like?

77. 3.2 Motion with Constant Acceleration
How long will it take the object in Graph B to reach the velocity of the object in Graph A?

78. 3.3 Free Fall
Jason hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. If the volleyball starts from 2.0 m above the floor, how long will it be in the air before it strikes the floor? Assume that Jason is the last player to touch the ball before it hits the floor.

79. 3.3 Free Fall
A bungee jumper jumps from a bridge with a 75.0 m long bungee cord attached. He will free fall until the cord reaches its full length. How fast will he be going at the time the cord begins to stretch? The cord slows the jumper at the rate of 8.50 m/s2 as it stretches out. What is the minimum height that the bridge must be to successfully complete the jump?

80. 3.3 Free Fall
1. A baseball player hits a pop fly to a height of 40.8 m. After the bat strikes the ball, how much time will there be for a fielder to get into position to make the catch?
2. A stone is dropped from a balloon which is ascending at the rate of 5.80 m/s when the balloon is 350. m above the ground. What time is required for the stone to reach the ground?

81. What else can we see from the v-t graph?
3.2 Motion with Constant Acceleration
The slope of the line on a v-t graph is the average acceleration of the object.
What else can we see from the v-t graph?

82. Galileo’s Inclined Plane
3.2 Motion with Constant Acceleration
Galileo’s Inclined Plane
A board is raised to allow the ball to roll from rest to the .5 m position in the 1st second
The motion diagram represents the ball at 1 second intervals

83. Galileo’s Inclined Plane
3.2 Motion with Constant Acceleration
Galileo’s Inclined Plane
Interval
Start Time (s)
End Time (s)
Start Position (m)
End Position (m)
Distance Traveled (m)
Average velocity (m/s)
Inst. Velocity at start (m/s)
Inst. Velocity at end (m/s)
Ave. Acceleration (m/s2) 1
0.0
1.0
0.5 .5 2
2.0
1.5 3
3.0
4.5
2.5 4
4.0
8.0
3.5 5
5.0
12.5

84. Galileo’s Inclined Plane
3.2 Motion with Constant Acceleration
Galileo’s Inclined Plane
Interval
Start Time (s)
End Time (s)
Start Position (m)
End Position (m)
Distance Traveled (m)
Average velocity (m/s)
Inst. Velocity at start (m/s)
Inst. Velocity at end (m/s)
Ave. Acceleration (m/s2) 1
0.0
1.0
0.5 .5 2
2.0
1.5 3
3.0
4.5
2.5 4
4.0
8.0
3.5 5
5.0
12.5

85. Galileo’s Inclined Plane
3.2 Motion with Constant Acceleration
Galileo’s Inclined Plane
Interval
Start Time (s)
End Time (s)
Start Position (m)
End Position (m)
Distance Traveled (m)
Average velocity (m/s)
Inst. Velocity at start (m/s)
Inst. Velocity at end (m/s)
Ave. Acceleration (m/s2) 1
0.0
1.0
0.5 .5 2
2.0
1.5 3
3.0
4.5
2.5 4
4.0
8.0
3.5 5
5.0
12.5

86. Galileo’s Inclined Plane
3.2 Motion with Constant Acceleration
Galileo’s Inclined Plane
Interval
Start Time (s)
End Time (s)
Start Position (m)
End Position (m)
Distance Traveled (m)
Average velocity (m/s)
Inst. Velocity at start (m/s)
Inst. Velocity at end (m/s)
Ave. Acceleration (m/s2) 1
0.0
1.0
0.5 .5 2
2.0
1.5 3
3.0
4.5
2.5 4
4.0
8.0
3.5 5
5.0
12.5

87. Galileo’s Inclined Plane
3.2 Motion with Constant Acceleration
Galileo’s Inclined Plane
Interval
Start Time (s)
End Time (s)
Start Position (m)
End Position (m)
Distance Traveled (m)
Average velocity (m/s)
Inst. Velocity at start (m/s)
Inst. Velocity at end (m/s)
Ave. Acceleration (m/s2) 1
0.0
1.0
0.5 .5 2
2.0
1.5 3
3.0
4.5
2.5 4
4.0
8.0
3.5 5
5.0
12.5

88. Inst. Velocity at end (m/s)
3.2 Motion with Constant Acceleration
What is the area under this line?
End Time (s)
Inst. Velocity at end (m/s)
End Position (m)
1.0
0.5
2.0
3.0
4.5
4.0
8.0
5.0
12.5
The area under a v-t graph equal the object’s displacement!

89. The area under a v-t graph equal the object’s displacement!
3.2 Motion with Constant Acceleration
The area under a v-t graph equal the object’s displacement!
If an object has no initial velocity becomes

90. What is the area under this line?
3.2 Motion with Constant Acceleration
What is the area under this line?
The area under a v-t graph equal the object’s displacement!

91. The area under a v-t graph equal the object’s displacement!
3.2 Motion with Constant Acceleration
The area under a v-t graph equal the object’s displacement!
If an object has an initial velocity

92. 3.2 Motion with Constant Acceleration
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?
vf = vi + aΔt
vf = vi + 2aΔt
vf2 = vi2 + 2aΔt
vf = vi – aΔt

93. 3.2 Motion with Constant Acceleration
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?
vf = vi + aΔt
vf = vi + 2aΔt
vf2 = vi2 + 2aΔt
vf = vi – aΔt