1. Unit 2 Linear Motion

2. Any measurement of position, distance, or speed must be made with respect to a reference frame.
For example, if you are sitting on a train and someone walks down the aisle, their speed with respect to the train is a few miles per hour, at most. Their speed with respect to the ground is much higher.

3. Distance and Displacement:
Distance is a measure of “how far an object has moved” and is independent of direction.
Example: If a person travels 50m due east, turns and travels 10m due south, then turns and travels 50m due west the distance traveled is 110m
50 meters
10 meters
50 meters

4. Distance and Displacement:
Displacement refers to both the distance and direction of an object’s change in position from the starting point of origin.
Example: If a person travels 50m due east, turns and travels 10m due south, then turns and travels 50m due west the total displacement is 10m south.
50 meters
10 m
10 meters
50 meters

5. Dd = df - di Words Have Specific Meaning
Distance – The total length that an object travels
Displacement – How far an object is from where it started including direction
Dd = df - di
Distance and Displacement are not always equal

6. Displacement – Time Graph shows the displacement an object travels over some time interval. The slope of the displacement time graph is the velocity of the object. Velocity is constant as long as slope does not change directions or steepness. 10 5 25
d (m)
t (s)
Stationary – No change in position – velocity is 0.
Moving away with a positive velocity
Moving back toward origin – velocity negative here

7. We make a distinction between distance and displacement.
Displacement (blue line) is how far the object is from its starting point, regardless of how it got there.
Distance traveled (dashed line) is measured along the actual path.

8. The displacement is written:
Right:
Displacement is negative.
Left:
Displacement is positive.

9. Vectors and Scalars A vector has magnitude as well as direction.
Some vector quantities: displacement, velocity, force, momentum
A scalar has only a magnitude.
Some scalar quantities: mass, time, temperature

10. Speed
Speed is how fast something is going.
It is a measure of the distance covered divided by the unit of time it took to cover the distance.
Speed is calculated by dividing the distance traveled by the time it took to travel the distance. d
Speed is measured in meters per second ( m/s) s t

11. Types of speed
Instantaneous speed is “the speed at a specific instant”
A speedometer measures instantaneous speed. .
Average speed is “the total distance covered in a particular time period”

12. But What is Velocity? d v t

13. Velocity
Velocity can change even if the speed stays constant.
Example:
A racecar on a circular track moving at a constant speed of 100 km/h has a constantly changing velocity because of a changing direction of travel

14. Different Kinds of Velocity
Instantaneous Velocity – Velocity at any moment in time and is represented by a point on a velocity-time graph
Average Velocity – Total displacement divided by the total time interval
Constant Velocity – Velocity that does not change over a time interval
Velocity is the slope of the displacement-time graph

15. I seem to have forgotten something……?
Quiz time!!!!! (not really)
1) If you ride your bicycle down a straight road for 500 m then turn around and ride back, your distance is ____ your displacement.
(A) greater than
(B) less than
(C) Equal
(D) Can’t determine

16. Quiz time!!!!! 2) The speed you read on a speedometer is ____.
(A) instantaneous speed
(B) average speed
(C) constant speed
(D) velocity

17. Quiz time!!!!!
3) 153 m/s north is an example of a(n) ____. a. speed c. position b.velocity d. acceleration

18. Quiz time!!!!!
4) A merry-go-round horse moves at a constant speed but at a changing ____.
a. velocity c.force
b.balanced force d.displacement

19. Quiz time!!!!!
5) A car travels at a speed of 50m/s for 20 seconds. What distance did it cover?
a. 2.5m c. 400m
b.1000m d. 0.4m

20. Distance time Graph
The slope of a graph showing distance plotted over time gives the velocity ( or speed).
Because Change in distance over time is velocity!
The steeper the line the greater the velocity

21. Distance versus Time Graph
Straight line represents constant (uniform) speed

22. Practice problem = 201.1 time 3.1 hours
What is the time it would take to travel 125 miles if you were traveling at a rate of 65 km/hour?
Speed = distance/time miles m 1 km mile m km is your distance =
time
3.1 hours

23. What Is Change in Velocity (or Speed) over a period of time called?
Acceleration!

24. Acceleration
Acceleration = Change in velocity (m/s) (m/s2 ) Time (s)

25. Remember that the slope of a a displacement - time graph gives the velocity

26. If we plot the velocity of an object over a period of time:

27. Constant Velocity Slope = 0
Slope = Dy/Dx
Constant Velocity Slope = 0
The slope of the velocity-time graph is acceleration. The area under the curve is the objects displacement.

28. Acceleration

29. Because velocity involves both speed and direction, either a change in speed (slowing down or speeding up) or a change in direction causes acceleration.

30. Negative Acceleration ( Deceleration) is seen when an object is slowing down.
This means that its velocity is decreasing with time.

31. No Acceleration is seen when an object is neither speeding up nor slowing down
This means that its velocity is constant

32. Displacement and Constant Acceleration
When an object moves with constant velocity, the distance the object moves is the same for every equal time interval. 0s 0m 1s 2m 2s 4m 3s 6m 4s 8m

33. But when an object is under constant acceleration, the distance traveled for equal time intervals IS NOT the same. But since the increase in velocity is constant, the change in displacement changes the same amount 0s 0m 1s 1m 2s 4m 3s 9m 4s
16m

34. Which of these graphs relate to the time vs position graph?

36. Quiz time!!!!! 6) Acceleration is rate of change of ____.
a. position c. velocity
b. time d. force

37. A curved line on the position-time graph shows that the object has ____ speed whereas a horizontal line on the position-time graph shows that the object has ____ speed.
a. constant, changing b. changing, no c. no, constant d. changing, constant e. constant, no

38. Quiz time!!!!!
7) A horizontal line on a velocity/time graph shows ____ acceleration.
a. positive c. changing
b. negative d. zero

39. Quiz time!!!!!
8) If you ride your bike up a hill, then ride down the other side, your acceleration is ____.
a) first positive, then negative
b) all negative
c) first negative, then positive
d) all positive

40. REVIEW Average Velocity
Speed: how far an object travels in a given time interval
Velocity includes directional information:

41. Examples

42. Acceleration
Acceleration is the rate of change of velocity.

43. Acceleration
Acceleration is a vector, although in one-dimensional motion we only need the sign.
The previous image shows positive acceleration; here is negative acceleration:

44. Acceleration
There is a difference between negative acceleration and deceleration:
Negative acceleration is acceleration in the negative direction as defined by the coordinate system.
Deceleration occurs when the acceleration is opposite in direction to the velocity.

45. Quiz time!!
A motorcycle traveling at 25 m/s accelerates at a rate of 7.0 m/s2 for 6.0 seconds. What is the final speed of the motorcycle?
7 = Vf-25 6

46. Practice problems
With a time of 6.92 s, Irina Privalova of Russia holds the women’s record for running 60 m. Suppose she ran this distance with a constant acceleration, so that she crossed the finish line with a speed of m/s. Assuming she started at rest, what was the magnitude of Privalova’s average acceleration.
The solid-fuel rocket boosters used to launch the space shuttle are able to lift the shuttle 45 kilometers above Earth’s surface. During the 2.00 min that the boosters operate, the shuttle accelerates from rest to a speed of nearly 7.50 × 102 m/s.What is the magnitude of the shuttle’s average acceleration?
In 1970, Don “Big Daddy” Garlits set what was then the world record for drag racing.With an average acceleration of 16.5 m/s2, Garlits started at rest and reached a speed of km/h. How much time was needed for Garlits to reach his final speed?

47. Practice problems answers

48. Quiz Time!!!
1. You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement?
1) yes
2) no

49. What is the Unit for Acceleration?
Since acceleration is the change in velocity over time, and the fundamental SI unit for velocity is m/s, and for time is s, then the fundamental SI unit for acceleration is m/ss or m/s2.
But what does m/s2 mean? It is the change in position per second per second, or the change in velocity per second. Every second that the object is accelerating the velocity changes by the value of the acceleration.
Example: If an object is accelerating at 10 m/s2 then the speed of the object will change by 10 m/s every second.

50. Quiz time!!
Yes, you have the same displacement. Since you and your dog had the same initial position and the same final position, then you have (by definition) the same displacement.

51. Quiz time! 1) velocity 2) speed 3) both 4) neither
2. Does the speedometer in a car measure velocity or speed?
1) velocity
2) speed
3) both
4) neither

52. Quiz time!!
The speedometer clearly measures speed, not velocity. Velocity is a vector (depends on direction), but the speedometer does not care what direction you are traveling. It only measures the magnitude of the velocity, which is the speed.

53. Quiz time!! 1) yes 2) no 3) it depends
3. If the average velocity is non-zero over some time interval, does this mean that the instantaneous velocity is never zero during the same interval?
1) yes
2) no
3) it depends

54. Quiz time!!
No!!! For example, your average velocity for a trip home might be 60 mph, but if you stopped for lunch on the way home, there was an interval when your instantaneous velocity was zero, in fact!

55. Quiz time!! 1) yes 2) no 3) depends on the velocity
4. If the velocity of a car is non-zero can the acceleration of the car be zero?
1) yes
2) no
3) depends on the velocity

56. Quiz time!
Sure it can! An object moving with constant velocity has a non-zero velocity, but it has zero acceleration since the velocity is not changing.

57. Quiz time!!
5. You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration?
1) its acceleration is constant everywhere
2) at the top of its trajectory
3) halfway to the top of its trajectory
4) just after it leaves your hand
5) just before it returns to your hand on the way down

58. Quiz time!!
The ball is in free fall once it is released. Therefore, it is entirely under the influence of gravity, and the only acceleration it experiences is g=9.8 m/s2, which is constant at all points.

59. Motion at Constant Acceleration
The average velocity of an object during a time interval t is
The acceleration, assumed constant, is

60. Motion at Constant Acceleration
In addition, as the velocity is increasing at a constant rate, we know that
Combining these last three equations, we find:

61. Motion at Constant Acceleration
We can also combine these equations so as to eliminate t:
We now have all the equations we need to solve constant-acceleration problems.

63. Solving Problems
Read the whole problem and make sure you understand it. Then read it again.
Decide on the objects under study and what the time interval is.
Draw a diagram and choose coordinate axes.
Write down the known (given) quantities, and then the unknown ones that you need to find.
What physics applies here? Plan an approach to a solution.

64. Solving Problems
6. Which equations relate the known and unknown quantities? Are they valid in this situation? Solve algebraically for the unknown quantities, and check that your result is sensible (correct dimensions).
7. Calculate the solution and round it to the appropriate number of significant figures.
8. Look at the result – is it reasonable? Does it agree with a rough estimate?
9. Check the units again.

65. Practice problems
A truck traveling at 10 m/s accelerates at 3 m/s2 for a period of 5 seconds. What is the velocity at the end of its period of acceleration?

66. Practice problems
Steve is cruising on his rollerblades at 4 m/s. He starts accelerating at 0.5 m/s2. After he has skated for 4 seconds, he slows down at a rate of 1 m/s2 for 1.5 sec. How fast is he going at the end?
This is a 2 part problem…..find the Vf for the first 4 seconds then use that V as the Vi in the 2nd part of the problem.

68. Review problems
You are traveling at 80 mph in a 45 mph zone. You see a cop ahead, step on the brakes and
slow down at a rate of 8 mph/s. If you pass the cop 2.5 seconds later, do you get a ticket?

69. John is driving along Highland Ave at 32m/s when he sees an errant pedestrian walk into the street 16
meters ahead of him. John applies his brakes, which provide an acceleration of ‐8m/s2. Is John able to
avoid hitting the pedestrian without swerving?

71. A ball (starting from rest) is rolling down a hill with an acceleration of 3.3 m/s2. If it accelerates for 7 seconds,
a. How fast will it be going?
b. How far will it move?

73. Ima Hurryin is approaching a stoplight moving with a velocity of 30
Ima Hurryin is approaching a stoplight moving with a velocity of 30.0 m/s. The light turns yellow, and Ima applies the brakes and skids to a stop. If Ima's acceleration as she slows is 8.00 m/s2 determine the displacement of the car during the skidding process.
vi = m/s vf = 0 m/s a = m/s2 d=?
vf2 = vi2 + 2 • a • d

74. vf2 = vi2 + 2 • a • d (0 m/s)2 = (30.0 m/s)2 + 2 • (-8.00 m/s2) • d
(16.0 m/s2) • d = 900 m2/s2 - 0 m2/s2
(16.0 m/s2)*d = 900 m2/s2
d = (900 m2/s2)/ (16.0 m/s2)
d = 56.3 m

75. Will Elsie, the cow, make it
Will Elsie, the cow, make it? Al Einstein is speeding down a country road in Palatine at a constant 32 m/s (73 mph) when he crests a hill and spots a cow (Elsie) standing in the middle of the road just 45 m in front of him. Can he stop in time to avoid hitting the cow?
The facts:
Mr. E’s response time is 0.32 seconds
The corvette has a braking acceleration of 20 m/s/s when the brakes are applied.
a. Calculate how far the car will travel while Al moves his foot from the gas pedal to the brake.
b. Calculate how far the car will travel while it is coming to a complete stop (stopping distance).

78. 6. A jetliner starts from rest and needs to attain a speed of 75
6. A jetliner starts from rest and needs to attain a speed of 75.0 m/s to leave the ground. If the jet can accelerate at a rate of 12.5 m/s/s, then…
a. How much runway is required for the jet to take off?
b. How much time elapses from the time the jet begins to accelerate until it takes off?

80. Falling Objects
The acceleration due to gravity at the Earth’s surface is approximately 9.80 m/s2.

81. Acceleration of the Sun, Planets and the Moon in m/s2
Jupiter 24.92
Neptune 11.15
Saturn 10.44
Earth 9.798
Uranus 8.87
Venus 8.87
Mars 3.71
Mercury 3.7
Moon 1.62
Pluto 0.58

82. Falling Objects
In the absence of air resistance, all objects fall with the same acceleration, although this may be hard to tell by testing in an environment where there is air resistance.

83. 1 D Kinematics – Part Two Solving Kinematics Problems
For problems with freely falling objects, a coordinate
system should be implemented.
If the object is moving upward ___.
a. s (distance/displacement) is positive (+)
b. v (velocity) is positive (+)
c. a (acceleration) is negative (–)
– 9.8 m s2
(on the earth at sea level)

84. 1 D Kinematics – Part Two Solving Kinematics Problems
If the object is moving downward ___.
a. s (distance/displacement) is negative (–)
b. v (velocity) is negative (–)
c. a (acceleration) is negative (–)
– 9.8 m s2
(on the earth at sea level)
(speeding up in the (–) direction)

85. Falling Objects
Near the surface of the Earth, all objects experience approximately the same acceleration due to gravity.
This is one of the most common examples of motion with constant acceleration.

86. One-dimensional Motion With Constant Acceleration
If acceleration is uniform
Shows velocity as a function of acceleration and time

87. All objects are accelerated by gravity by the same rate
The acceleration of gravity, near the surface of the earth is about – 9.81 m/s2
The symbol “g” is substituted for “a” for the special case of gravitational acceleration
The gravitational acceleration is CONSTANT for all parts of the falling objects path

88. At every point in the path of the object, the acceleration of gravity remains constant, even though the velocity changes
Velocity = 0 m/s g = m/s2
Initial Velocity = 5 m/s g = m/s2
Final Velocity = -5 m/s g = m/s2

90. Below is a Velocity-Time graph for a thrown upward and then falling
Initial Velocity Up
Top of Path V = 0
Velocity Back at Original Height

91. The previous formulas will be used but…
The previous formulas will be used but…. Non Free Fall Equations (a ≠ g) vf = vi + a t d = vi t + ½ a t vf 2 = vi a d Free Fall Equations (a = g) vf = vi + g t d = vi t + ½ g t vf 2 = vi g d

92. A balloon is rising at 29. 4 m/s and a stone falls from it
A balloon is rising at 29.4 m/s and a stone falls from it. If the stone takes 20.0 s to reach the ground, how high is the balloon when the stone is dropped?

93. 2-5. Free Fall
A ball thrown into the air will slow down, stop, and then begin to fall with the acceleration due to gravity. When it passes the thrower, it will be traveling at the same rate at which it was thrown.

94. Notes on the equations
Gives displacement as a function of velocity and time
Gives displacement as a function of time, velocity and acceleration
Gives velocity as a function of acceleration and displacement

95. One-dimensional Motion With Constant Acceleration
Used in situations with uniform acceleration
Velocity changes
uniformly!!!

96. The formulas change the a in the formula to g
VF 2 = Vo g(Δd)
Δd = Vo Δt ½ g (t2)
Vf = Vo + gΔt

97. Dropped objects Initial velocity is 0
How high is the cliff if it falls for 3.0 s?
What is its velocity when it reaches the bottom?

98. ???????? #2
A bird sits on a branch of a tree 6.8 meters above the ground. How long does it take for the worm in his mouth to fall to the ground?

99. Start by writing down what you have and what you need.
Distance= 6.8
G = a since it is falling 9.8 m/s2
T= ???
VF 2 = Vo g(Δd)
Δd = Vo Δt ½ g (t2)
Vf = Vo + gΔt

100. Δd = vo t + ½ gΔt2 Solved with Vo if the work is spit etc.
Vo= 0 so rearrange the formula 2d = t g
2 (-6.8) = s
-9.8

101. Free Fall -- an Object Droppedy
Initial velocity is zero
Frame: let up be positive
Use the kinematic equations
Generally use y instead
of x since vertical x
vo= 0
a = g

102. Free Fall -- object thrown upward
Initial velocity is upward, so positive
The instantaneous velocity at the maximum height is zero
a = g everywhere in the motion
g is always downward, negative
v = 0

103. Problem Solving tips for free fall
1. Acceleration is always -9.8 m/s/s
2. Even at the top
3. v at the top is zero
If something starts and ends at same elevation
4. 1/2 time up, 1/2 time down
5. v = -v
Make a v vs. time graph for an object thrown straight up from leaving my hand, to landing back in my hand
(draw axes, endpoints)
TOC

104. Thrown upward The motion may be symmetrical
then tup = tdown
then vf = -vo
The motion may not be symmetrical
Break the motion into various parts
generally up and down

105. Does our formula work? Time of free fall (s) Distance of fall (m) 11
4.9 2
19.6 3
44.1 4 80 5
125 * t
½ (9.8) t 2

106. Question 3
A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true: 1. velocity is zero and acceleration is zero velocity is not zero and acceleration is zero velocity is zero and acceleration is not zero velocity is not zero and acceleration is not zero
correct
The velocity vector changes from moment to moment, buts its acceleration vector does not change. Though the velocity at the top is zero, the acceleration is still constant because the velocity is changing.

107. Question #4
A ball falls freely from rest for 15.0 s. Calculate the ball's velocity after 15.0 s.  a.  -147 m/s     b.  147 m/s      c.  78 m/s      d.  -78 m/s 

108. -147 m/s Acceleration due to gravity is -9.80 m/s2 (downward).
Formula Vf = Vo + gt

109. A pebble is dropped down a well and hits the water 1.5 seconds later.
a. Determine how fast the pebble was moving when it hit the water.
b. Determine how far the pebble fell.

110. Now a pebble is thrown downward with a speed of 5 m/s from the top of a building. It takes the pebble 3 seconds to hit the ground.
c. How tall is the building?
d. How fast was the pebble moving when it hit the ground?

111. A person standing on top of a building that is 50
A person standing on top of a building that is 50.0 m tall throws a stone vertically downward with a velocity of 15.0 m/s
a. What is the velocity of the stone right before it strikes the ground?
b. How long does it take for the stone to reach the ground?

113. A conveyor belt moves packages on a horizontally at 2. 0 m/s
A conveyor belt moves packages on a horizontally at 2.0 m/s. When the packages reach the end of the belt, they fall into a box. If the conveyor belt is 3.0 m above the box, how far should the middle of the box be placed from the end of the conveyor belt so the boxes will fall into the middle of the box?
A squirrel is running along a horizontal branch when it drops its acorn that it is carrying. If the branch is 10 m above the ground and the acorn hits the ground 5.0 m in front of where the squirrel dropped the acorn, how fast was the squirrel running?