2. Lectures by James L. Pazun 2 Motion Along a Straight Line

3. 2 Chapter 2: Motion Along a Line Position & Displacement Speed & Velocity Acceleration Describing motion in 1D Free Fall

4. Copyright © 2012 Pearson Education, Inc. publishing as Addison-Wesley Motion Motion is divided into two areas of study: – Kinematics This will be our focus in chapter 2. Kinematics describes the movement of the object. – Dynamics Will come in Chapter 4 and after. Dynamics answers the “Why is this object moving?” question.

5. Kinematics One Dimensional Motion

6. Position and Displacement  Position: coordinate of position – distance to the origin (m) O x P x

7. Example Position 6 0 x (cm) 2 1 22 11 The position of the ball is The + indicates the direction to the right of the origin.

8. Displacement Δx Displacement: change in position x 0 = original (initial) location x = final location Δx = x - x 0 Example: x 0 = 1m, x = 4m, Δx = 4m – 1m = 3m 1m4m

9. Displacement Δx Displacement: change in position x 0 = original (initial) location x = final location Δx = x - x 0 Example: x 0 = 4m, x = 1 m, Δx = 1m – 4m = - 3m 1m4m

10. Displacement, Distance Distance traveled usually different from displacement. Distance always positive. Previous example: always 3 m.

11. Distance and Speed: Scalar Quantities Distance is the path length traveled from one location to another. It will vary depending on the path. Distance is a scalar quantity—it is described only by a magnitude.

12. What is the displacement in the situation depicted bellow? a) 3 m b) 6 m c) -3 m d) 0 m 1m4m

13. What is the distance traveled in the situation depicted bellow? a) 3 m b) 6 m c) -3 m d) 0 m 1m4m

14. Position, Displacement

15. One-Dimensional Displacement and Velocity: Vector Quantities Displacement is a vector that points from the initial position to the final position of an object.

16. Velocity: Rate of Change of Position Average velocity = displacement/time interval v av = (x-x o )/(t-t o )=Δx/Δt displacement = average velocity x time interval Δx = v av Δt Position x. Since Δx = x – x o : x = x o + v av Δt Note that an object’s position coordinate may be negative, while its velocity may be positive; the two are independent. (Equation of a straight line)

17. Example

18. Velocity and Position In general, the average velocity is the slope of the line segment that connects the positions at the beginning and end of the time interval Section 2.2 Average velocity = displacement/time interval

19. Constant Velocity v = 8m/2s = 4 m/s velocity = slope of graph position vs. time

20. Graphs of Velocity Different ways of visualizing uniform velocity:

21. Instantaneous Velocity Average velocity doesn’t tell us anything about details during the time interval To look at some of the details, smaller time intervals are needed The slope of the curve at the time of interest will give the instantaneous velocity at that time Will be referred to as velocity in the text Section 2.2

22. Instantaneous Velocity Instantaneous Velocity = Slope of the tangent line Example v = 5.2 m/s

23. One-Dimensional Displacement and Velocity: Vector Quantities This object’s velocity is not uniform. Does it ever change direction, or is it just slowing down and speeding up?

24. Average Speed Average Speed = Distance/Time Distance = Average Speed x Time Average speed is the total distance traveled divided by the elapsed time:

25. Distance and Speed: Scalar Quantities Since distance is a scalar, speed is also a scalar (as is time). Instantaneous speed is the speed measured over a very short time span. This is what a speedometer reads.

26. One-Dimensional Displacement and Velocity: Vector Quantities For motion in a straight line with no reversals, the average speed and the average velocity are the same. Otherwise, they are not; indeed, the average velocity of a round trip is zero, as the total displacement is zero!

27. Example Usein Bolt crossed the 100 m dash race in 9.69 s, ahead of the second athlete by 15 m. How long did it take for the second athlete to complete this race? speed of second athlete = 85m/9.69s = 8.77 m/s time = distance/speed = 100m/8.77m/s = 11.4 s distance traveled by second athlete in 9.69 s = 100 m – 15 m = 85 m Known: distance = 100 m, time = 9.69 s, Δd = 15 m Find: time of second athlete.

28. 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 ConcepTest 2.1Walking the Dog

29. 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 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. ConcepTest 2.1Walking the Dog Follow-up: Have you and your dog traveled the same distance?

30. ConcepTest 2.2Displacement ConcepTest 2.2Displacement Does the displacement of an object depend on the specific location of the origin of the coordinate system? 1) yes 2) no 3) it depends on the coordinate system

31. ConcepTest 2.2Displacement ConcepTest 2.2Displacement difference Since the displacement is the difference between two coordinates, the origin does not matter. 1020304050 3040506070 Does the displacement of an object depend on the specific location of the origin of the coordinate system? 1) yes 2) no 3) it depends on the coordinate system

32. If the position of a car is zero, does its speed have to be zero? 1) yes 2) no 3) it depends on the position ConcepTest 2.3Position and Speed

33. If the position of a car is zero, does its speed have to be zero? 1) yes 2) no 3) it depends on the position No, the speed does not depend on position; it depends on the change of position. Since we know that the displacement does not depend on the origin of the coordinate system, an object can easily start at x = –3 and be moving by the time it gets to x = 0. ConcepTest 2.3Position and Speed

34. Does the odometer in a car measure distance or displacement? Does the odometer in a car measure distance or displacement? 1) distance 2) displacement 3) both ConcepTest 2.4Odometer

35. Does the odometer in a car measure distance or displacement? Does the odometer in a car measure distance or displacement? 1) distance 2) displacement 3) both If you go on a long trip and then return home, your odometer does not measure zero, but it records the total miles that you traveled. That means the odometer records distance. ConcepTest 2.4Odometer Follow-up: How would you measure displacement in your car?

36. Does the speedometer in a car measure velocity or speed? 1) velocity 2) speed 3) both 4) neither ConcepTest 2.5Speedometer

37. Does the speedometer in a car measure velocity or speed? 1) velocity 2) speed 3) both 4) neither 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. ConcepTest 2.5Speedometer Follow-up: How would you measure velocity in your car?

38. You drive for 30 minutes at 30 mi/hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip? 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr ConcepTest 2.6a Cruising Along I

39. You drive for 30 minutes at 30 mi/hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip? 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr It is 40 mi/hr in this case. Since the average speed is distance/time and you spend the same amount of time at each speed, then your average speed would indeed be 40 mi/hr. ConcepTest 2.6a Cruising Along I

40. You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip? 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr ConcepTest 2.6b Cruising Along II

41. You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip? 1) more than 40 mi/hr 2) equal to 40 mi/hr 3) less than 40 mi/hr It is not 40 mi/hr! Remember that the average speed is distance/time. Since it takes longer to cover 4 miles at the slower speed, you are actually moving at 30 mi/hr for a longer period of time! Therefore, your average speed is closer to 30 mi/hr than it is to 50 mi/hr. ConcepTest 2.6b Cruising Along II Follow-up: How much farther would you have to drive at 50 mi/hr in order to get back your average speed of 40 mi/hr?

42. ConcepTest 2.7Velocity in One Dimension 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

43. ConcepTest 2.7Velocity in One Dimension 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. 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

44. 43 The area under a velocity versus time graph (between the curve and the time axis) gives the displacement in a given interval of time. v x (m/s) t (sec) Area Interpretation of Displacement

45. 44 Example :Speedometer readings are obtained and graphed as a car comes to a stop along a straight-line path. How far does the car move between t = 0 and t = 16 seconds? Since there is not a reversal of direction, the area between the curve and the time axis will represent the distance traveled. The rectangular portion has an area of Lw = (20 m/s)(4 s) = 80 m. The triangular portion has an area of ½bh = ½(8 s) (20 m/s) = 80 m. Thus, the total area is 160 m. This is the distance traveled by the car.

46. Acceleration: Rate of Change of Velocity Acceleration is the rate at which velocity changes. v o initial velocity @ time t o v final velocity @ time t

47. Average Acceleration  Average Acceleration = change in velocity/time interval a av = Δv/Δt v t Slope = Δv/Δt = a av ΔtΔt ΔvΔv

48. (Instantaneous) Acceleration  Instantaneous Acceleration = acceleration @ a given moment in time a = Δv/Δt v t ΔtΔt ΔvΔv slope = a

49. 48 Example: The graph shows speedometer readings as a car comes to a stop. What is the acceleration at t = 7.0 s? The slope of the graph at t = 7.0 sec is v

50. Acceleration Acceleration means that the speed of an object is changing, or its direction is, or both.

51. Constant Acceleration  Constant Acceleration a av = a  In the + Ox direction: velocity increases a > 0 acceleration velocity increases a > 0 acceleration velocity decreases a < 0 deceleration velocity decreases a < 0 deceleration a = Δv/Δt

52. Acceleration  Constant Acceleration = slope of line a = Δv/Δt v t ΔtΔt ΔvΔv slope = a

53. Sign of Acceleration Acceleration may result in an object either speeding up or slowing down (or simply changing its direction).

54. Acceleration

55. Acceleration a av =a = (v-v 0 ) / (t-t 0 ) => a(t-t 0 ) = (v-v 0 ) => v = v 0 + a (t-t 0 ) (1) If t o = 0, v = v 0 + at Law of Velocities If v o = 0, v = at

56. Acceleration If the acceleration is constant, we can find the velocity as a function of time: displacement = area under the velocity vs. time graph

57. Deceleration, Example v = v 0 + a (t-t 0 ) (1) v = 28m/s – 5 m/s 2 x (12s – 9s) = 13 m/s

58. Deceleration, Example

59. Acceleration, Example v = v 0 + a (t-t 0 ) (1) v = 6m/s + 2m/s 2 x (8s – 0s) = 22 m/s

60. ConcepTest 2.8aAcceleration I If the velocity of a car is non-zero (v  0), can the acceleration of the car be zero? 1) Yes 2)No 3)Depends on the velocity

61. ConcepTest 2.8aAcceleration I constantvelocity zeroacceleration 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. If the velocity of a car is non-zero (v  0), can the acceleration of the car be zero? 1) Yes 2)No 3)Depends on the velocity

62. When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? 1) both v = 0 and a = 0 2) v  0, but a = 0 3) v = 0, but a  0 4) both v  0 and a  0 5) not really sure ConcepTest 2.8bAcceleration II

63. y At the top, clearly v = 0 because the ball has momentarily stopped. But the velocity of the ball is changing, so its acceleration is definitely not zero! Otherwise it would remain at rest!! When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? 1) both v = 0 and a = 0 2) v  0, but a = 0 3) v = 0, but a  0 4) both v  0 and a  0 5) not really sure ConcepTest 2.8bAcceleration II Follow-up: …and the value of a is…?

64. Law of Motion  Position vs. time v av = (x-x 0 ) / (t-t 0 ) = (v+v 0 )/2 => x = x 0 + (t-t 0 ) (v+v 0 )/2 (*) Use Eq. (1) to replace v in Eq.(*): x = x 0 + (t-t 0 ) (v 0 + a (t-t 0 ) +v 0 )/2 x = x 0 + (t-t 0 ) v 0 + ½ a(t-t 0 ) 2 (2) x = x 0 + (t-t 0 ) v 0 + ½ a(t-t 0 ) 2 (2) If t o = 0, x = x 0 + v 0 t + ½ at 2 If t o = 0, and motion starts from origin (x o = 0) x = v 0 t + ½ at 2 x = v 0 t + ½ at 2 From rest v 0 = 0, x = ½ at 2

65. Example Car   A car, starting from rest, accelerates in a straight-line path at a constant rate of 2.5 m/s 2.   How far will the car travel in 12 seconds?   (a) 180 m   (b) 120 m   (c) 30 m   (d) 15 m   (e) 4.8 m x = ½ at 2 x = ½ (2.5m/s 2 ) x (12s) 2

66. Graph of Motion v o = 2m/s v a = 4 m/s 2 x = 2t + 4t 2 /2

67. Ticker-Tape Diagram 1212 2 3232 4242

68. A car, moving with 10 m/s, accelerates with 2 m/s 2 for 4s. What is the displacement of the car in this time interval? x = x 0 + v 0 t + ½ at 2 x -x 0 = v 0 t + ½ at 2 Δx = v 0 t + ½ at 2 Δx = 10 m/s x 4s + ½ 2m/s 2 x (4s) 2 Δx = 56 m

69. 68 Visualizing Motion Along a Line with Constant Acceleration Motion diagrams for three carts:

70. 69 Graphs of x, v, a for each of the three carts

71. Galileo’s Formula  Plugging in for t:  Solving for t:

72. Kinematic Equations (Constant Acceleration) These are all the equations we have derived for constant acceleration. The correct equation for a problem should be selected considering the information given and the desired result. 1) Law of Velocities 2) Law of Motion 3) Galileo’s Formula

73. 72 Example: A train of mass 55,200 kg is traveling along a straight, level track at 26.8 m/s. Suddenly the engineer sees a truck stalled on the tracks 184 m ahead. If the maximum possible braking acceleration has magnitude 1.52 m/s 2, can the train be stopped in time? Know: a =  1.52 m/s 2, v o = 26.8 m/s, v = 0 Using the given acceleration, compute the distance traveled by the train before it comes to rest. The train cannot be stopped in time.

74. Example Car l A car starts from rest and accelerates at a constant rate in a straight line. In the first second the car covers a distance of 2.0 meters. How fast will the car be moving at the end of the second second? l (a) 4.0 m/s l (b) 2.0 m/s l (c) 8.0 m/s l (d) 16 m/s l (e) 32 m/s v = at v = 4m/s 2 x 2 s

75. Example Take Off l The minimum takeoff speed for a certain airplane is 75 m/s. What minimum acceleration is required if the plane must leave a runway of length 950 m? Assume the plane starts from rest at one end of the runway. l (a) 1.5 m/s 2 l (b) 4.5 m/s 2 l (c) 7.5 m/s 2 l (d) 3.0 m/s 2 l (e) 6.0 m/s 2 v o = 0

76. A train is moving parallel and adjacent to a highway with a constant speed of 35 m/s. Initially a car is 44 m behind the train, traveling in the same direction as the train at 47 m/s, and accelerating at 3 m/s 2. What is the speed of the car just as it passes the train? Example Meeting Example Meeting t o = 0 s, initial time t = final time x o = 44 m v t = 35 m/s, constant v oc = 47 m/s, initial speed of car v c = final speed of car Car’s equation of motion Train’s equation of motion: To meet : x c = x t t = 2.733 s O xoxo x t = xcxc

77. ConcepTest 2.13aGraphing Velocity I t x The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? 1) it speeds up all the time 2) it slows down all the time 3) it moves at constant velocity 4) sometimes it speeds up and sometimes it slows down 5) not really sure

78. ConcepTest 2.13aGraphing Velocity I t x The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? The car moves at a constant velocity because the x vs. t plot shows a straight line. The slope of a straight line is constant. Remember that the slope of x versus t is the velocity! 1) it speeds up all the time 2) it slows down all the time 3) it moves at constant velocity 4) sometimes it speeds up and sometimes it slows down 5) not really sure

79. t x 1) it speeds up all the time 2) it slows down all the time 3) it moves at constant velocity 4) sometimes it speeds up and sometimes it slows down 5) not really sure The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? ConcepTest 2.13bGraphing Velocity II

80. 1) it speeds up all the time 2) it slows down all the time 3) it moves at constant velocity 4) sometimes it speeds up and sometimes it slows down 5) not really sure t x The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? The car slows down all the time because the slope of the x vs. t graph is diminishing as time goes on. Remember that the slope of x vs. t is the velocity! At large t, the value of the position x does not change, indicating that the car must be at rest. ConcepTest 2.13bGraphing Velocity II

81. Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? v t A B 1) decreases 2) increases 3) stays constant 4) increases, then decreases 5) decreases, then increases ConcepTest 2.14av versus t Graphs I

82. Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? v t A B 1) decreases 2) increases 3) stays constant 4) increases, then decreases 5) decreases, then increases In case A, the initial velocity is positive and the magnitude of the velocity continues to increase with time. ConcepTest 2.14av versus t Graphs I

83. Consider the line labeled B in the v versus t plot. How does the speed change with time for line B? Consider the line labeled B in the v versus t plot. How does the speed change with time for line B? v t A B 1) decreases 2) increases 3) stays constant 4) increases, then decreases 5) decreases, then increases ConcepTest 2.14bv versus t Graphs II

84. Consider the line labeled B in the v versus t plot. How does the speed change with time for line B? Consider the line labeled B in the v versus t plot. How does the speed change with time for line B? v t A B 1) decreases 2) increases 3) stays constant 4) increases, then decreases 5) decreases, then increases In case B, the initial velocity is positive but the magnitude of the velocity decreases toward zero. After this, the magnitude increases again, but becomes negative, indicating that the object has changed direction. ConcepTest 2.14bv versus t Graphs II

85. Free Fall An object in free fall has a constant acceleration (in the absence of air resistance) due to the Earth’s gravity. This acceleration is directed downward.

86. Free fall in a vacuum - Figure 2.24 The apple and the feather are in a vacuum chamber and the photograph taken with a strobe light. The effects of air resistance are particularly obvious when dropping a small, heavy object such as an apple, as well as a larger light one such as a feather or a piece of paper. However, if the same objects are dropped in a vacuum, they fall with the same acceleration.

87. Free Fall Here are the constant-acceleration equations for free fall: The positive y -direction has been chosen to be upwards. If it is chosen to be downwards, the sign of g would need to be changed.

88. ConcepTest 2.9aFree Fall I 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

89. free fall only acceleration it experiences is -g 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, which is constant at all points. ConcepTest 2.9a Free Fall I 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

90. ConcepTest 2.9b Free Fall II Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1) Alice’s ball 2) it depends on how hard the ball was thrown 3) neither -- they both have the same acceleration 4) Bill’s ball v0v0v0v0 BillAlice vAvAvAvA vBvBvBvB

91. Both balls are in free fall once they are released, therefore they both feel the acceleration due to gravity (g). This acceleration is independent of the initial velocity of the ball. ConcepTest 2.9b Free Fall II Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1) Alice’s ball 2) it depends on how hard the ball was thrown 3) Neither - they both have the same acceleration 4) Bill’s ball v0v0v0v0 BillAlice vAvAvAvA vBvBvBvB Follow-up: Which one has the greater velocity when they hit the ground?

92. ConcepTest 2.10a Up in the Air I You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? 1) more than 10 m/s 2) 10 m/s 3) less than 10 m/s 4) zero 5) need more information

93. The ball is slowing down on the way up due to gravity. Eventually it stops. Then it accelerates downward due to gravity (again). Since a = g on the way up and on the way down, the ball reaches the same speed when it gets back to you as it had when it left. ConcepTest 2.10a Up in the Air I You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? 1) more than 10 m/s 2) 10 m/s 3) less than 10 m/s 4) zero 5) need more information

94. v0v0v0v0 v0v0v0v0 BillAlice H vAvAvAvA vBvBvBvB Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v 0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are v A and v B. If there is no air resistance, which is true? 1) v A < v B 2) v A = v B 3) v A > v B 4) impossible to tell v0v0v0v0 v0v0v0v0 BillAlice H vAvAvAvA vBvBvBvB ConcepTest 2.10b Up in the Air II

95. it is moving downward with v 0, the same as Alice’s ball Bill’s ball goes up and comes back down to Bill’s level. At that point, it is moving downward with v 0, the same as Alice’s ball. Thus, it will hit the ground with the same speed as Alice’s ball. Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v 0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are v A and v B. If there is no air resistance, which is true? 1) v A < v B 2) v A = v B 3) v A > v B 4) impossible to tell v0v0v0v0 v0v0v0v0 BillAlice H vAvAvAvA vBvBvBvB ConcepTest 2.10b Up in the Air II Follow-up: What happens if there is air resistance?

96. A ball is thrown straight upward with some initial speed. When it reaches the top of its flight (at a height h), a second ball is thrown straight upward with the same initial speed. Where will the balls cross paths? 1) at height h 2) above height h/2 3) at height h/2 4) below height h/2 but above 0 5) at height 0 ConcepTest 2.11 Two Balls in the Air

97. A ball is thrown straight upward with some initial speed. When it reaches the top of its flight (at a height h), a second ball is thrown straight upward with the same initial speed. Where will the balls cross paths? 1) at height h 2) above height h/2 3) at height h/2 4) below height h/2 but above 0 5) at height 0 The first ball starts at the top with no initial speed. The second ball starts at the bottom with a large initial speed. Since the balls travel the same time until they meet, the second ball will cover more distance in that time, which will carry it over the halfway point before the first ball can reach it. ConcepTest 2.11 Two Balls in the Air Follow-up: How could you calculate where they meet?

98. You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? 1) the separation increases as they fall 2) the separation stays constant at 4 m 3) the separation decreases as they fall 4) it is impossible to answer without more information ConcepTest 2.12a Throwing Rocks I

99. You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? 1) the separation increases as they fall 2) the separation stays constant at 4 m 3) the separation decreases as they fall 4) it is impossible to answer without more information At any given time, the first rock always has a greater velocity than the second rock, therefore it will always be increasing its lead as it falls. Thus, the separation will increase. ConcepTest 2.12a Throwing Rocks I

100. You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their velocities? 1) both increase at the same rate 2) the velocity of the first rock increases faster than the velocity of the second 3) the velocity of the second rock increases faster than the velocity of the first 4) both velocities stay constant ConcepTest 2.12bThrowing Rocks II

101. You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their velocities? 1) both increase at the same rate 2) the velocity of the first rock increases faster than the velocity of the second 3) the velocity of the second rock increases faster than the velocity of the first 4) both velocities stay constant Both rocks are in free fall, thus under the influence of gravity only. That means they both experience the constant acceleration of gravity. Since acceleration is defined as the change of velocity, both of their velocities increase at the same rate. ConcepTest 2.12bThrowing Rocks II Follow-up: What happens when air resistance is present?

102. ConcepTest 2.15a ConcepTest 2.15aRubber Balls I v t 1 v t 2 v t 3 v t 4 You drop a rubber ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume your y- axis is pointing up). You drop a rubber ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume your y- axis is pointing up).

103. v t 2 v t 3 v t 4 initial velocity is zero velocity is negativemore and more negative The ball is dropped from rest, so its initial velocity is zero. Since the y- axis is pointing upward and the ball is falling downward, its velocity is negative and becomes more and more negative as it accelerates downwards. ConcepTest 2.15a ConcepTest 2.15aRubber Balls I v t 1

104. v t 4 v t 2 v t 3 v t 1 ConcepTest 2.15b ConcepTest 2.15bRubber Balls II You toss a ball straight up in the air and catch it again. Right after it leaves your hand and before you catch it, which of the above plots represents the v vs. t graph for this motion? (Assume your y-axis is pointing up). You toss a ball straight up in the air and catch it again. Right after it leaves your hand and before you catch it, which of the above plots represents the v vs. t graph for this motion? (Assume your y-axis is pointing up).

105. v t 4 v t 2 v t 3 v t 1 initial velocity that is positive velocity becomes negative more and more negative The ball has an initial velocity that is positive but diminishing as it slows. It stops at the top (v = 0), and then its velocity becomes negative and becomes more and more negative as it accelerates downward. ConcepTest 2.15b ConcepTest 2.15bRubber Balls II

106. v t 1 v t 2 v t 3 v t 4 ConcepTest 2.15c ConcepTest 2.15cRubber Balls III You drop a very bouncy rubber ball. It falls, and then it hits the floor and bounces right back up to you. Which of the following represents the v vs. t graph for this motion? You drop a very bouncy rubber ball. It falls, and then it hits the floor and bounces right back up to you. Which of the following represents the v vs. t graph for this motion?

107. v t 1 v t 2 v t 3 v t 4 negative increasing positive decreasing Initially, the ball is falling down, so its velocity must be negative (if UP is positive). Its velocity is also increasing in magnitude as it falls. Once it bounces, it changes direction and then has a positive velocity, which is also decreasing as the ball moves upward. ConcepTest 2.15c ConcepTest 2.15cRubber Balls III

108. Time up, Maximum Height Alice throws a ball in the air with a velocity of 10 m/s. Take g = 10m/s 2. a) How much time elapses until the ball reaches the maximum height? Known: v o = 10 m/s Find: t up. Use the law of velocities: At maximum height y max, v = 0 m/s, therefore: Solve for t up : b) What is the maximum height of the ball? Use Galileo: At maximum height y max, v = 0 m/s. Also assume y o = 0 m, therefore: Solve for y max :

109. Meeting When Alice’s ball is at the maximum height of 5m, Bill also throws a ball vertically upwards with 15 m/s. c) When and were do the balls meet? Take g = 10m/s 2. Known: y max = 5 m. to = 0s at the moment Bill throws the ball. v ob = 15 m/s, initial velocity of Bill's ball Find: t, y. To meet y a = y b : Use the law of motion: Alice:Bill: Cancel and solve for t: Find height: d) What is the speed of Bill’s ball when they meet? Use the law of velocities:

110. Relative Velocity Velocity may be measured in any inertial reference frame. At top, the velocities are measured relative to the ground; at bottom they are measured relative to the white car.

111. “Against the wind” – Figure 2.29 Good examples include: a plane and a head/tail wind, a boat on a river, and a cruel joke at a stop light.

112. Summary of Chapter 2 Motion involves a change in position; it may be expressed as the distance (scalar) or displacement (vector). A scalar has magnitude only; a vector has magnitude and direction. Average speed (scalar) is distance traveled divided by elapsed time. Average velocity (vector) is displacement divided by total time.

113. Summary of Chapter 2 Instantaneous velocity is evaluated at a particular instant. Acceleration (vector) is the time rate of change of velocity. Kinematic equations for constant acceleration:

114. Example1   At time t = 0 s, an object is observed at x = 0 m; and its position along the x axis follows this expression: x = –3t + t 2, where the units for distance and time are meters and seconds, respectively. What is the object’s displacement Δx between t = 1.0 s and t = 3.0 s?   (a) +20 m (b) –20m (c) +10 m (d) +2 m (e) –2 m Δx = x(3) – x(1) = 0 – (-2) = + 2

115. Example 2-3   Questions 2 and 3 pertain to the situation described below: Peter noticed a bug crawling along a meter stick and decided to record the bug’s position in five-second intervals. After the bug crawled off the meter stick, Peter created the table shown. 2. What is the displacement of the bug between t = 0.00 s and t = 20.0 s? (a) +39.9 cm (b) –39.9 cm (c) +65.7 cm (d) –16.1 cm (e) +16.1 cm 3. What is the total distance that the bug traveled between t = 0.00 s and t = 20.0 s? Assume the bug only changed direction at the end of a five second interval (a) 39.9 cm (b) 65.7 cm (c) 16.1 cm (d) 47.1 cm (e) 26.5 cm

116. Example 4   In which one of the following situations does the car have a westward acceleration?   (a) The car travels westward at constant speed.   (b) The car travels eastward and speeds up.   (c) The car travels westward and slows down.   (d) The car travels eastward and slows down.   (e) The car starts from rest and moves toward the east.

117. Example 5   An elevator is moving upward with a speed of 11 m/s. Three seconds later, the elevator is still moving upward, but its speed has been reduced to 5.0 m/s. What is the average acceleration of the elevator during the 3.0 s interval?   (a) 2.0 m/s2, downward   (b) 2.0 m/s2, upward   (c) 5.3 m/s2, downward   (d) 2.7 m/s2, downward   (e) 5.3 m/s2, upward

118. Example 6   A car is moving at a constant velocity when it is involved in a collision. The car comes to rest after 0.450 s with an average acceleration of 65.0 m/s 2 in the direction opposite that of the car’s velocity.   What was the speed, in km/h, of the car before the collision? (a) 29.2 km/h   (b) 144 km/h   (c) 44.8 km/h   (d) 80.5 km/h   (e) 105 km/h

119. Example 7   Starting from rest, a particle confined to move along a straight line is accelerated at a rate of 5.0 m/s 2.   Which one of the following statements accurately describes the motion of this particle?   (a) The particle travels 5.0 m during each second.   (b) The particle travels 5.0 m only during the first second.   (c) The speed of the particle increases by 5.0 m/s during each second.   (d) The acceleration of the particle increases by 5.0 m/s2 during each second.   (e) The final speed of the particle will be proportional to the distance that the particle covers.

120. Example 8 (a) v 2 = 2ax (b) v = x/t (c) v = xt (d) v = at (e) v = (1/2)at 2

121. Example 8   A car, starting from rest, accelerates in a straight-line path at a constant rate of 2.5 m/s 2.   How far will the car travel in 12 seconds?   (a) 180 m   (b) 120 m   (c) 30 m   (d) 15 m   (e) 4.8 m