1. Linear Motion

2. Jumpstart
Calculate the acceleration of a car that is traveling from 80 m/s to 0/s in 20 seconds.

3. Kinematics
Branch of mechanics that describes the motion of objects without necessarily discussing what causes the motion.

4. Motion is relative Measured in reference to another object or point.
Depends on Frame of Reference.
How fast is your body moving right now?

5. How far did you go? Distance Needs no frame of reference
Separation between two points
Only a length, no direction
Scalar Quantity
Units:
Meters, kilometers, centimeters

6. What is your new position?
Displacement
Change in position.
Where are you relative to some starting point?
Magnitude and direction
Vector Quantity
Express direction with a sign or direction
45m to the North
-82 cm A B C D

7. The difference between distance and displacement. . . . .

8. Speed How fast something is moving. Speed = distance/time
Units: m/s, km/h, mph. . .
Scalar Quantity
Instantaneous speed
Average speed
Over the course of a trip, can instantaneous speed and average speed be different from each other?

9. Approximate Speeds in Different Units
Miles per hour
Meters per second
Kilometers per hour
12 mph
6 m/s
20 km/h
25 mph
11 m/s
40 km/h
37 mph
17 m/s
60 km/h
50 mph
22 m/s
80 km/h
62 mph
28 m/s
100 km/h
75 mph
33 m/s
120 km/h

10. Velocity Speed in a given direction. Vector quantity
Magnitude AND direction
Velocity = displacement/time
V = d/t
Units = m/s, km/h, mph. . .
Direction expressed with
Signs: + or –
Direction: North, South, East, West, Left, Right, Forward, Backward. . .

11. A change in velocity is called. .
Constant Velocity
constant speed and direction
Changes in velocity are due to
Increase/decrease in speed
Change in direction
If you drive around a circular track with a constant speed of 60 km/h, is your velocity also constant?
A change in velocity is called. .

12. Velocity
Instantaneous Velocity
Average Velocity

13. Example Problems
A team begins play on the 50 yard line; they lose 5 yards on a play and then gain 15 yards on the next play.
What distance did the team travel?
What was the team’s displacement?
A truck travels 300km north in 10 hours. What was the truck’s average velocity?

14. Example Problems
How long would it take you to drive 15 km a rate of 30 km/h?

15. Acceleration The rate of change of velocity.
Equal to the change in velocity divided by time.
a = (vf – vi)/t OR a = Δv/t
Units:
Meters per second per second (m/s/s)
Meters per second squared (m/s2)
If a car can accelerate from 0-60mph in 15 seconds, what is its acceleration?
(60mph – 0 mph) / 15s = 4 mph / s2

16. Acceleration Vector quantity Can refer to:
Has both magnitude and direction
Direction expressed with a sign (+/-)
Can refer to:
Increase in speed
Decrease in speed
Change in direction

17. Acceleration
If speed increases while you are moving forward, acceleration is positive.
If speed decreases while you are moving forward, acceleration is negative
Example 1: You are driving and increase your speed from 50 m/s to 75 m/s in 10 seconds. What is your acceleration?
75 m/s – 50 m/s = 25 m / s = (2.5 m / s2)
10 s 10 s

18. Acceleration
Example 2: You are riding your bike and slow from 10 m/s to 5 m/s in 2 seconds. What is your acceleration?
Example 3: How long will it take to accelerate from 20 m/s to 40 m/s if your velocity changes at a rate of 4 m/s2?

19. Position-Time Graphs Position on the y-axis, time on the x-axis.
Position is measured with respect to a reference point; therefore position can increase or decrease.
Slope of the line is velocity.

20. Position-Time Graphs Time (seconds) Distance (meters) 10 5 20 30 15 4010 5 20 30 15 40 50 25 60 70 35 80

21. Position Time Graphs Time (seconds) Position (meters) 5 10 20 15 30 405 10 20 15 30 40 50 60 70 80 90
100
110
120
130

22. Position Time Graphs Time (seconds) Position (meters) 5 10 20 15 30 405 10 20 15 30 40 50 60 70
-10 80
-20 90
100
-15
110
120 -5
130

23. Position-Time Graphs Time (seconds) Distance (meters) 1 2 4 3 9 16 51 2 4 3 9 16 5 25 6 36 7 49 8 64

24. Position-Time Graphs Time (seconds) Distance (meters) 1 2 4 3 9 16 51 2 4 3 9 16 5 25 6 36 7 49 8 64
Average Velocity
Instantaneous Velocity
Instantaneous Velocity
Instantaneous Velocity

25. Position-Time Graphs Curve = changing velocity
Constant Acceleration = Curve
Straight Line = constant velocity

26. Position Time Graphs Constant + v, slow Constant + v, fast

27. Position-Time Graphs
Positive acceleration (increasing velocity) in the negative direction
Negative acceleration (decreasing velocity) in the negative direction

28. Position-Time Graphs

29. Velocity-Time Graphs Time on the x-axis (s)
Velocity on the y-axis (m/s)
Slope is acceleration (m/s2)
Area under graph represents displacement.
It line is curved – the acceleration is changing

30. Velocity-Time Graphs Slope of the graph is important:
Slope = 0 (Horizontal line)
Velocity is constant
Positive slope
Velocity is increasing
Negative slope
Velocity is decreasing
Shape of the graph is important
Straight line: acceleration = 0
Velocity may be 0
Velocity may be constant
Curve
Acceleration is changing

31. Velocity-Time Graphs Increasing Velocity Constant +Acceleration
Time (seconds)
Velocity (m/s) 1 10 2 20 3 30 4 40 5 50 6 60 7 70 8 80
What was this object’s displacement?

32. Velocity-Time Graphs Decreasing Velocity Constant -Acceleration
Time (seconds)
Velocity (m/s) 80 1 70 2 60 3 50 4 40 5 30 6 20 7 10 8

33. Velocity-Time Graphs
Increasing Negative Velocity Constant -Acceleration
Time (seconds)
Velocity (m/s) 1
-10 2
-20 3
-30 4
-40 5
-50 6
-60 7
-70 8
-80

34. What do you think?
If you are given a table of velocities of an object at various times, how would you determine if the acceleration of the object was constant?
Time (seconds)
Velocity (m/s) 1
-10 2
-20 3
-30 4
-40 5
-50 6
-60 7
-70 8
-80

35. Velocity-Time Graphs
Decreasing Negative Velocity Constant +Acceleration
Time (seconds)
Velocity (m/s)
-80 1
-70 2
-60 3
-50 4
-40 5
-30 6
-20 7
-10 8

36. Velocity-Time Graphs Wed. 10/10/12
HOW COULD YOU DETERMINE:
Average Velocity?
Instantaneous acceleration?
Velocity-Time Graphs Wed. 10/10/12
Time (seconds)
Velocity (m/s) 1 2 4 3 9 16 5 25 6 36 7 49 8 64

37. Velocity-Time Graphs Increasing Velocity Positive Acceleration
Time (seconds)
Velocity (m/s) 1 2 4 3 9 16 5 25 6 36 7 49 8 64

38. Velocity-Time Graphs Time (seconds) Velocity (m/s) 1 2 4 3 9 16 5 25 61 2 4 3 9 16 5 25 6 36 7 49 8 64
Average Acceleration
Instantaneous Acceleration
Instantaneous Acceleration
Instantaneous Acceleration

39. Velocity-Time Graphs Constant velocity (v=40m/s) Zero Acceleration

40. Acceleration-Time Graphs

41. Acceleration-Time Graphs

42. Acceleration-Time Graphs

43. Constant Rightward Velocity
Website Animation

44. Constant Leftward Velocity

45. Rightward Velocity/Rightward Acceleration

46. Rightward Velocity/Leftward Acceleration

47. (10/10/12) Wednesday
Sketch a position vs. time , velocity vs. time and an acceleration vs. time graph for each of the following two situations.
Leftward velocity and rightward acceleration
Leftward velocity and leftward acceleration

48. Leftward Velocity/Rightward Acceleration

49. Leftward Velocity/Leftward Acceleration

50. And now for the calculations . . .
Assume acceleration is constant
Either zero or some non zero value.
Variables
a = acceleration
v0 = initial velocity
v = final velocity
d = displacement
t = time

51. Recap: Displacement – change in position Δx = x2-x1
Velocity – a measure of how fast something moves from one point to another.
Average Velocity - change in position
change in time
Vavg = Δx or vavg = (vf + vi) Δt

52. DERIVING THE BIG 4 KINEMATIC EQUATIONS
Velocity of an object with constant acceleration (condition often true if air resistance ignored)
Basic formula for acceleration
Solve for vf (final velocity)
Given any three of these variables, you can solve for the fourth.
(vf – vi) = a t
vf – vi = at
vf = vi + at

53. The Big 4 Kinematic Equations
vf = vi + at

54. Velocity and Displacement under average acceleration
DERIVING THE BIG 4
Velocity and Displacement under average acceleration
Δx = Vavg Δt
Vavg = Δx AND Vavg = (vf + vi) Δt

55. Combine the equations for displacement and average velocity:
DERIVING THE BIG 4
Combine the equations for displacement and average velocity:
Δx= vt AND Vavg = (vf + vi) 2
vt = ½(vf + vi)t
Δx = ½(vf + vi)t
Sub in Vavg
substitute Δx for tv

56. The Big 4 Kinematic Equations
vf = vi + at
Δx = ½(vf + vi)t

57. Displacement when acceleration and time are known.
vf = vi + at
Δx = ½(vf + vi)t
Δx = ½(vi + at + vi)t
Δx = ½ t (2vi + at)
Δx = 2vit + at2
Δx = vit + ½at2
plug first formula into second formula for vf
Simplify the resulting formula
(ΔX is displacement)

58. Review 3 Kinematic equations from our earlier derivation
vf = vi + at
Δx = ½(vf + vi)t
Δx = vit + ½at2

59. Displacement when velocity and acceleration are known
Δx = ½(vf + vi)t v = vi + at
t = (vf– vi)/a
Δx = ½(vf + vi)(vf – vi)/a
Δx = ½(vf2 – vi2)/a
Δx = (vf2 – vi2) 2a
(re-arrange to solve for Vf2 )
vi2 + 2aΔx = vf2

60. The Big 4 Kinematic Equations
vf = vi + at
Δx = ½(vf + vi)t
Δx = vit + ½at2
vf2 = vi2 + 2aΔx

61. The Big Four Kinematic Equations v = vi + at x = ½(v + vi)t
x = vit + ½at2
(starts at rest it becomes:)
x = ½ at2
v2 = vi2 + 2ax
Variables
v, v0, a, t
x, v, v0, t
x, v0, t, a
v, a, x, v0

62. Using the big 4 derived from the definition of displacement, velocity and acceleration – you can solve almost any 1D problem under constant acceleration.

63. How to solve a problem
Read the problem carefully – try to visualize situation. Make a sketch.
Define your variables.
Write out your knowns and unknowns.
Decide which formula contains each of these variables.
Solve the formula for the variable you are trying to solve for.
Plug in numbers and solve the problem
Check your answers – to you have units? Do they make sense – did you answer the question?
(seconds for time – meters for distance etc…)

64. Example 1
A race car starts at rest and accelerates at a constant m/s2 for 10 seconds.
What is the rider’s displacement during this time?
What are the knowns?
V0 =
a =
t =
x =

65. Example 2
Starting from rest, a race car moves 100 m in the first 5.0 s of uniform acceleration. What is the car’s acceleration?
What are the knowns?
V0 =
a =
t =
x =

66. Example 3
A car accelerates at a constant rate from 15 m/s to 25 m/s while it travels 125 m. How long does this motion take?

67. Example 4
A pilot stops a plane in 484 m using a constant acceleration of -8.0 m/s2. How fast was the plan moving before braking began?

68. What about the direction?
Airplanes …
Speed is how fast is something going.
Speed = distance crossed / time elapsed
Velocity is the speed and direction of an object.
Velocity is a vector (has magnitude and direction).
Speed is a scalar (has magnitude only)

69. Speed and Velocity
Moving all the time equally fast (with respect to the ground) - called?
Constant speed
Moving all the time equally fast and also in the same direction?
Constant velocity
Do we always move equally fast?
Do we always move in the same direction?

70. Average and Instantaneous Speed Trip from home to college
Things to Consider:
Stop signs
Traffic lights
Speeds allowed
What does the car speedometer measure?
What is the police concerned about?

71. Average and Instantaneous Speed
Things to Consider for the each trip:
Average speed?
Maximum speed?
Time
What is the trip choice based on?
Unit conversion
2.8 mil in 5 min
1.8 mil in 6 min
Video 1

72. Average Velocity Things to Consider: Average Velocity?
Distance and Displacement
Average Velocity?
Avg. Velocity same or different for both trips?
Avg. Velocity after returning home
2.8 mil in 5 min
Straight Shot 1 mile
1.8 mil in 6 min

73. Average and Instantaneous Velocity
Things to Consider:
Average Velocity and Instantaneous Velocity
2.8 mil in 5 min
Straight Shot 1 mile
1.8 mil in 6 min

74. Reminder (Previous cycle): Distance and Displacement
START
FINISH
Need to distinguish how long we traveled from how far away (and in what direction) we traveled.

75. Distance and displacement
Distance and displacement are two quantities which may seem to mean the same thing, yet have distinctly different definitions and meanings.
Distance is a scalar quantity (has magnitude only) which refers to "how much ground an object has covered" during its motion.
Displacement is a vector quantity (has magnitude and direction) which refers to "how far out of place an object is"; it is the object's change in position.

76. Direction of the Average Velocity
START
FINISH

77. Direction of the Average Velocity
Displacement is a vector - Distance is not
Velocity is a vector - Speed is not
START
FINISH

78. Make sense: Velocity and speed
How would you determine the average speed of an object on these paths:
START
FINISH
START
FINISH
Average speed vs. Average velocity

79. Velocity is a vector Velocity is a vector
Vectors are quantities that have magnitude and direction
Speed is the magnitude of a velocity.

80. Average vs. Instantaneous Velocity

81. Gravity On earth, all objects fall with an acceleration of -9.8m/s2.
varies slightly with location
Pike’s Peak = m/s2
Maine = m/s2
Any object with mass will fall with this acceleration—if air resistance is ignored.

82. Gravity As an object falls, it gains 9.8m/s every second.
As a thrown object rises, it loses 9.8 m/s every second.
v = vo + gt

83. Draw the p-t, v-t, and a-t graphs for this situation
Gravity
Because the object’s velocity is increasing, it covers more distance each second.
x = vot + ½gt2

84. Gravity The same uniform acceleration formulas apply to gravity
Instead of a, use “g”
v = vo + gt
x = ½(v + vo)t
x = vot + ½gt2
v2 = vo2 + 2gx

85. Example 1 Suppose a ball is dropped from a height of 100m.
How long will it take for the ball to hit the ground?
What will the ball’s velocity be when it hits the ground?

86. What do you think?
A baseball is thrown straight up into the air. When the ball reaches its maximum height its velocity is equal to ______?

87. What do you think?
What is the total displacement of the ball upon landing? (from previous slide)

88. What do you think?
Ignoring air resistance, if a 2 kg ball and a 5 ton SUV were dropped from the top of the Empire State building (assume all pedestrians below had been cleared) – the acceleration of the 2 kg ball would be _______ the acceleration of the SUV.
Greater than
Less than
Equal To

89. Example 2
You drop a stone off the side of a cliff. You hear it hit the ground 4 seconds later. How high is the cliff?

90. Example 3
You throw a ball straight up in the air. The ball leaves your hand with an upward velocity of 30 m/s. Draw a diagram showing the path of the ball and its velocity at the following times:
0.0s, 1.0s, 2.0s, 3.0s, 4.0s, 5.0s, 6.0s