# Lecture Presentation Chapter 3 Vectors and Motion in Two Dimensions

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1 Lecture Presentation Chapter 3 Vectors and Motion in Two Dimensions

2 Suggested Videos for Chapter 3 Prelecture Videos Vectors and Motion Projectile Motion Circular Motion Class Videos Motion on a Ramp Acceleration Due to Changing Direction Video Tutor Solutions Vectors and Motion in Two Dimensions Video Tutor Demos Balls Take High and Low Tracks Dropped and Thrown Balls Ball Fired Upward from Moving Cart Ball Fired Upward from Accelerating Cart Ball Fired from Cart on Incline Range of a Gun at Two Firing Angles Ball Leaves Circular Track Slide 3-2

3 Suggested Simulations for Chapter 3 ActivPhysics PhETs Vector Addition Ladybug Motion 2D Maze Game Motion in 2D Projectile Motion Ladybug Revolution Slide 3-3

4 Chapter 3 Vectors and Motion in Two Dimensions Chapter Goal: To learn more about vectors and to use vectors as a tool to analyze motion in two dimensions. Slide 3-4

5 Chapter 3 Preview Looking Ahead: Vectors and Components The dark green vector is the ball s initial velocity. The light green component vectors show initial horizontal and vertical velocity. You ll learn to describe motion in terms of quantities such as distance and velocity, an important first step in analyzing motion. Slide 3-5

6 Chapter 3 Preview Looking Ahead: Projectile Motion A leaping fish s parabolic arc is an example of projectile motion. The details are the same for a fish or a basketball. You ll see how to solve projectile motion problems, determining how long an object is in the air and how far it travels. Slide 3-6

7 Chapter 3 Preview Looking Ahead: Circular Motion The riders move in a circle at a constant speed, but they have an acceleration because the direction is constantly changing. You ll learn how to determine the magnitude and the direction of the acceleration for an object in circular motion. Slide 3-7

8 Chapter 3 Preview Looking Ahead Text: p. 64 Slide 3-8

9 Chapter 3 Preview Looking Back: Free Fall You learned in Section 2.7 that an object tossed straight up is in free fall. The acceleration is the same whether the object is going up or coming back down. For an object in projectile motion, the vertical component of the motion is also free fall. You ll use your knowledge of free fall to solve projectile motion problems. Slide 3-9

10 Section 3.1 Using Vectors

11 Using Vectors A vector is a quantity with both a size (magnitude) and a direction. Figure 3.1 shows how to represent a particle s velocity as a vector. The particle s speed at this point is 5 m/s and it is moving in the direction indicated by the arrow. Slide 3-11

12 Using Vectors The magnitude of a vector is represented by the letter without an arrow. In this case, the particle s speed the magnitude of the velocity vector is v = 5 m/s. The magnitude of a vector, a scalar quantity, cannot be a negative number. Slide 3-12

13 Using Vectors The displacement vector is a straight-line connection from the initial position to the final position, regardless of the actual path. Two vectors are equal if they have the same magnitude and direction. This is regardless of the individual starting points of the vectors. Slide 3-13

14 Vector Addition is the net displacement because it describes the net result of the hiker s having first displacement, then displacement. The net displacement is an initial displacement plus a second displacement : The sum of the two vectors is called the resultant vector. Vector addition is commutative: Slide 3-14

15 Vector Addition The figure shows the tip-to-tail rule of vector addition and the parallelogram rule of vector addition. Slide 3-15

16 QuickCheck 3.1 Given vectors and, what is? Slide 3-16

17 QuickCheck 3.1 Given vectors and, what is? A. Slide 3-17

18 Multiplication by a Scalar Multiplying a vector by a positive scalar gives another vector of different magnitude but pointing in the same direction. If we multiply a vector by zero the product is a vector having zero length. The vector is known as the zero vector. Slide 3-18

19 Multiplication by a Scalar A vector cannot have a negative magnitude. If we multiply a vector by a negative number we reverse its direction. Multiplying a vector by 1 reverses its direction without changing its length (magnitude). Slide 3-19

20 QuickCheck 3.2 Which of the vectors in the second row shows? Slide 3-20

21 QuickCheck 3.2 Which of the vectors in the second row shows? C. Slide 3-21

22 Vector Subtraction Text: p. 67 Slide 3-22

23 QuickCheck 3.3 Given vectors and, what is? Slide 3-23

24 QuickCheck 3.3 Given vectors and, what is? D. Slide 3-24

25 QuickCheck 3.4 Which of the vectors in the second row shows 2? Slide 3-25

26 QuickCheck 3.4 Which of the vectors in the second row shows 2? A. Slide 3-26

27 Section 3.2 Using Vectors on Motion Diagrams

28 Reading Question 3.1 The of a vector is always a positive quantity. A. x-component B. y-component C. Magnitude D. Direction Slide 3-28

29 Reading Question 3.1 The of a vector is always a positive quantity. A. x-component B. y-component C. Magnitude D. Direction Slide 3-29

30 Using Vectors on Motion Diagrams In two dimensions, an object s displacement is a vector: The velocity vector is simply the displacement vector multiplied by the scalar 1/Δt. Consequently the velocity vector points in the direction of the displacement. Slide 3-30

31 Example 3.1 Finding the velocity of an airplane A small plane is 100 km due east of Denver. After 1 hour of flying at a constant speed in the same direction, it is 200 km due north of Denver. What is the plane s velocity? Slide 3-31

32 Example 3.1 Finding the velocity of an airplane A small plane is 100 km due east of Denver. After 1 hour of flying at a constant speed in the same direction, it is 200 km due north of Denver. What is the plane s velocity? PREPARE The initial and final positions of the plane are shown in FIGURE 3.8; the displacement is the vector that points from the initial to the final position. Slide 3-32

33 Example 3.1 Finding the velocity of an airplane (cont.) SOLVE The length of the displacement vector is the hypotenuse of a right triangle: The direction of the displacement vector is described by the angle θ in Figure 3.8. From trigonometry, this angle is Slide 3-33

34 Example 3.1 Finding the velocity of an airplane (cont.) Thus the plane s displacement vector is Because the plane undergoes this displacement during 1 hour, its velocity is ASSESS The plane s speed is the magnitude of the velocity, v = 224 km/h. This is approximately 140 mph, which is a reasonable speed for a small plane. Slide 3-34

35 Using Vectors on Motion Diagrams The vector definition of acceleration is a straightforward extension of the one-dimensional version: There is an acceleration whenever there is a change in velocity. Velocity can change in either or both of two possible ways: 1. The magnitude can change, indicating a change in speed. 2. The direction of motion can change. Slide 3-35

36 Finding the Acceleration Vector Text: p. 69 Slide 3-36

37 QuickCheck 3.6 The diagram shows three points of a motion diagram. The particle changes direction with no change of speed. What is the acceleration at point 2? Slide 3-37

38 QuickCheck 3.6 The diagram shows three points of a motion diagram. The particle changes direction with no change of speed. What is the acceleration at point 2? Acceleration of changing direction B. Slide 3-38

39 Vectors and Circular Motion Cars on a Ferris wheel move at a constant speed but in a continuously changing direction. They are in uniform circular motion. Slide 3-39

40 Section 3.3 Coordinate Systems and Vector Components

41 Reading Question 3.2 A x is positive if is directed ; A y is positive if is directed. A. Right, up B. Left, up C. Right, down D. Left, down Slide 3-41

42 Reading Question 3.2 A x is positive if is directed ; A y is positive if is directed. A. Right, up B. Left, up C. Right, down D. Left, down Slide 3-42

43 Coordinate Systems A coordinate system is an artificially imposed grid that you place on a problem in order to make quantitative measurements. We will generally use Cartesian coordinates. Coordinate axes have a positive end and a negative end, separated by a zero at the origin where the two axes cross. Slide 3-43

44 Component Vectors For a vector A and an xy-coordinate system we can define two new vectors parallel to the axes that we call the component vectors of. You can see, using the parallelogram rule, that is the vector sum of the two component vectors: Slide 3-44

45 Components Text: p. 71 Slide 3-45

46 Components Slide 3-46

47 Components Slide 3-47

48 QuickCheck 3.7 What are the x- and y-components of this vector? A. 3, 2 B. 2, 3 C. 3, 2 D. 2, 3 E. 3, 2 Slide 3-48

49 QuickCheck 3.7 What are the x- and y-components of this vector? A. 3, 2 B. 2, 3 C. 3, 2 D. 2, 3 E. 3, 2 Slide 3-49

50 QuickCheck 3.8 What are the x- and y-components of this vector? A. 3, 4 B. 4, 3 C. 3, 4 D. 4, 3 E. 3, 4 Slide 3-50

51 QuickCheck 3.8 What are the x- and y-components of this vector? A. 3, 4 B. 4, 3 C. 3, 4 D. 4, 3 E. 3, 4 Slide 3-51

52 QuickCheck 3.9 What are the x- and y-components of vector C? A. 1, 3 B. 3, 1 C. 1, 1 D. 4, 2 E. 2, 4 Slide 3-52

53 QuickCheck 3.9 What are the x- and y-components of vector C? A. 1, 3 B. 3, 1 C. 1, 1 D. 4, 2 E. 2, 4 Slide 3-53

54 QuickCheck 3.10 The angle Φ that specifies the direction of vector is A. tan 1 (C x /C y ) B. tan 1 (C y /C x ) C. tan 1 ( C x /C y ) D. tan 1 ( C x / C y ) E. tan 1 ( C y / C x ) Slide 3-54

55 QuickCheck 3.10 The angle Φ that specifies the direction of vector is A. tan 1 (C x /C y ) B. tan 1 (C y /C x ) C. tan 1 ( C x /C y ) D. tan 1 ( C x / C y ) E. tan 1 ( C y / C x ) Slide 3-55

56 QuickCheck 3.11 The following vector has length 4.0 units. What are the x- and y-components? A. 3.5, 2.0 B. 2.0, 3.5 C. 3.5, 2.0 D. 2.0, 3.5 E. 3.5, 2.0 Slide 3-56

57 QuickCheck 3.11 The following vector has length 4.0 units. What are the x- and y-components? A. 3.5, 2.0 B. 2.0, 3.5 C. 3.5, 2.0 D. 2.0, 3.5 E. 3.5, 2.0 Slide 3-57

58 QuickCheck 3.12 The following vector has length 4.0 units. What are the x- and y-components? A. 3.5, 2.0 B. 2.0, 3.5 C. 3.5, 2.0 D. 2.0, 3.5 E. 3.5, 2.0 Slide 3-58

59 QuickCheck 3.12 The following vector has length 4.0 units. What are the x- and y-components? A. 3.5, 2.0 B. 2.0, 3.5 C. 3.5, 2.0 D. 2.0, 3.5 E. 3.5, 2.0 Slide 3-59

60 Example 3.3 Finding the components of an acceleration vector Find the x- and y-components of the acceleration vector shown in FIGURE Slide 3-60

61 Example 3.3 Finding the components of an acceleration vector Find the x- and y-components of the acceleration vector shown in FIGURE PREPARE It s important to draw the vectors. Making a sketch is crucial to setting up this problem. FIGURE 3.18 shows the original vector decomposed into component vectors parallel to the axes. Slide 3-61

62 Example 3.3 Finding the components of an acceleration vector (cont.) SOLVE The acceleration vector = (6.0 m/s 2, 30 below the negative x-axis) points to the left (negative x-direction) and down (negative y-direction), so the components a x and a y are both negative: a x = a cos 30 = (6.0 m/s 2 ) cos 30 = 5.2 m/s 2 a y = a sin 30 = (6.0 m/s 2 ) sin 30 = 3.0 m/s 2 Slide 3-62

63 Example 3.3 Finding the components of an acceleration vector (cont.) ASSESS The magnitude of the y-component is less than that of the x-component, as seems to be the case in Figure 3.18, a good check on our work. The units of a x and a y are the same as the units of vector. Notice that we had to insert the minus signs manually by observing that the vector points down and to the left. Slide 3-63

64 Working with Components We can add vectors using components. Let s look at the vector sum for the vectors shown in FIGURE You can see that the component vectors of are the sums of the component vectors of and. The same is true of the components: C x = A x + B x and C y = A y + B y. Slide 3-64

65 Working with Components Equation 3.18 is really just a shorthand way of writing the two simultaneous equations: In other words, a vector equation is interpreted as meaning: Equate the x-components on both sides of the equals sign, then equate the y-components. Vector notation allows us to write these two equations in a more compact form. Slide 3-65

66 QuickCheck 3.13 A x is the of the vector A x. A. Magnitude B. x-component C. Direction D. Size E. Displacement Slide 3-66

67 QuickCheck 3.13 A x is the of the vector A x. A. Magnitude B. x-component C. Direction D. Size E. Displacement Slide 3-67

68 Tilted Axes For motion on a slope, it is often most convenient to put the x-axis along the slope. When we add the y-axis, this gives us a tilted coordinate system. Finding components with tilted axes is done the same way as with horizontal and vertical axes. The components are parallel to the tilted axes and the angles are measured from the tilted axes. Slide 3-68

69 Section 3.4 Motion on a Ramp

70 Accelerated Motion on a Ramp A crate slides down a frictionless (i.e., smooth) ramp tilted at angle θ. The crate is constrained to accelerate parallel to the surface. Both the acceleration and velocity vectors are parallel to the ramp. Slide 3-70

71 Accelerated Motion on a Ramp We choose the coordinate system to have the x-axis along the ramp and the y-axis perpendicular. All motion will be along the x-axis. The acceleration parallel to the ramp is a component of the free-fall acceleration the object would have if the ramp vanished: Slide 3-71

72 QuickCheck 3.14 A ball rolls up the ramp, then back down. Which is the correct acceleration graph? Slide 3-72

73 QuickCheck 3.14 A ball rolls up the ramp, then back down. Which is the correct acceleration graph? D. Slide 3-73

74 Example 3.6 Maximum possible speed for a skier The Willamette Pass ski area in Oregon was the site of the 1993 U.S. National Speed Skiing Competition. The skiers started from rest and then accelerated down a stretch of the mountain with a reasonably constant slope, aiming for the highest possible speed at the end of this run. During this acceleration phase, the skiers traveled 360 m while dropping a vertical distance of 170 m. What is the fastest speed a skier could achieve at the end of this run? Slide 3-74

75 Example 3.6 Maximum possible speed for a skier (cont.) PREPARE We begin with the visual overview in FIGURE The motion diagram shows the acceleration of the skier and the pictorial representation gives an overview of the problem including the dimensions of the slope. As before, we put the x-axis along the slope. Slide 3-75

76 Example 3.6 Maximum possible speed for a skier (cont.) SOLVE The fastest possible run would be one without any friction or air resistance, meaning the acceleration down the slope is given by Equation The acceleration is in the positive x-direction, so we use the positive sign. What is the angle in Equation 3.20? Figure 3.24 shows that the 360-m-long slope is the hypotenuse of a triangle of height 170 m, so we use trigonometry to find which gives θ = sin 1 (170/360) = 28. Equation 3.20 then gives a x = + g sin θ = (9.8 m/s 2 )(sin 28 ) = 4.6 m/s 2 Slide 3-76

77 Example 3.6 Maximum possible speed for a skier (cont.) For linear motion with constant acceleration, we can use the third of the kinematic equations in Synthesis 2.1: (v x ) f2 = (v x ) i2 + 2a x Δx. The initial velocity (v x ) i is zero; thus This is the fastest that any skier could hope to be moving at the end of the run. Any friction or air resistance would decrease this speed. Slide 3-77

78 Example 3.6 Maximum possible speed for a skier (cont.) ASSESS The final speed we calculated is 58 m/s, which is about 130 mph, reasonable because we expect a high speed for this sport. In the competition noted, the actual winning speed was 111 mph, not much slower than the result we calculated. Obviously, efforts to minimize friction and air resistance are working! Slide 3-78

79 Section 3.5 Relative Motion

80 Reading Question 3.3 The acceleration of a cart rolling down a ramp depends on A. The angle of the ramp. B. The length of the ramp. C. Both the angle of the ramp and the length of the ramp. D. Neither the angle of the ramp or the length of the ramp. Slide 3-80

81 Reading Question 3.3 The acceleration of a cart rolling down a ramp depends on A. The angle of the ramp. B. The length of the ramp. C. Both the angle of the ramp and the length of the ramp. D. Neither the angle of the ramp or the length of the ramp. Slide 3-81

82 Relative Motion Amy, Bill, and Carlos are watching a runner. The runner moves at a different velocity relative to each of them. Slide 3-82

83 Relative Velocity The runner s velocity relative to Amy is (v x ) RA = 5 m/s The subscript RA means Runner relative to Amy. The velocity of Carlos relative to Amy is (v x ) CA =15 m/s The subscript CA means Carlos relative to Amy. Slide 3-83

84 QuickCheck 3.15 A factory conveyor belt rolls at 3 m/s. A mouse sees a piece of cheese directly across the belt and heads straight for the cheese at 4 m/s. What is the mouse s speed relative to the factory floor? A. 1 m/s B. 2 m/s C. 3 m/s D. 4 m/s E. 5 m/s Slide 3-84

85 QuickCheck 3.15 A factory conveyor belt rolls at 3 m/s. A mouse sees a piece of cheese directly across the belt and heads straight for the cheese at 4 m/s. What is the mouse s speed relative to the factory floor? A. 1 m/s B. 2 m/s C. 3 m/s D. 4 m/s E. 5 m/s Slide 3-85

86 Example 3.8 Speed of a seabird Researchers doing satellite tracking of albatrosses in the Southern Ocean observed a bird maintaining sustained flight speeds of 35 m/s nearly 80 mph! This seems surprisingly fast until you realize that this particular bird was flying with the wind, which was moving at 23 m/s. What was the bird s airspeed its speed relative to the air? This is a truer measure of its flight speed. Slide 3-86

87 Example 3.8 Speed of a seabird Researchers doing satellite tracking of albatrosses in the Southern Ocean observed a bird maintaining sustained flight speeds of 35 m/s nearly 80 mph! This seems surprisingly fast until you realize that this particular bird was flying with the wind, which was moving at 23 m/s. What was the bird s airspeed its speed relative to the air? This is a truer measure of its flight speed. Slide 3-87

88 Example 3.8 Speed of a seabird (cont.) PREPARE FIGURE 3.27 shows the wind and the albatross moving to the right, so all velocities will be positive. We ve shown the velocity (v x ) bw of the bird with respect to the water, which is the measured flight speed, and the velocity (v x ) aw of the air with respect to the water, which is the known wind speed. We want to find the bird s airspeed the speed of the bird with respect to the air. Slide 3-88

89 Example 3.8 Speed of a seabird (cont.) SOLVE We ve noted three different velocities that are important in the problem: (v x ) bw, (v x ) aw, and (v x ) ba. We can combine these in the usual way: (v x ) bw = (v x ) ba + (v x ) aw Then, to solve for (v x ) ba, we can rearrange the terms: (v x ) ba = (v x ) bw (v x ) aw = 35 m/s 23 m/s = 12 m/s ASSESS 12 m/s about 25 mph is a reasonable airspeed for a bird. And it s slower than the observed flight speed, which makes sense because the bird is flying with the wind. Slide 3-89

90 Example 3.9 Finding the ground speed of an airplane Cleveland is approximately 300 miles east of Chicago. A plane leaves Chicago flying due east at 500 mph. The pilot forgot to check the weather and doesn t know that the wind is blowing to the south at 100 mph. What is the plane s velocity relative to the ground? Slide 3-90

91 Example 3.9 Finding the ground speed of an airplane Cleveland is approximately 300 miles east of Chicago. A plane leaves Chicago flying due east at 500 mph. The pilot forgot to check the weather and doesn t know that the wind is blowing to the south at 100 mph. What is the plane s velocity relative to the ground? Slide 3-91

92 Example 3.9 Finding the ground speed of an airplane (cont.) PREPARE FIGURE 3.28 is a visual overview of the situation. We are given the speed of the plane relative to the air ( ) and the speed of the air relative to the ground ( ); the speed of the plane relative to the ground will be the vector sum of these velocities: This vector sum is shown in Figure Slide 3-92

93 Example 3.9 Finding the ground speed of an airplane (cont.) SOLVE The plane s speed relative to the ground is the hypotenuse of the right triangle in Figure 3.28; thus: The plane s direction can be specified by the angle θ measured from due east: The velocity of the plane relative to the ground is thus ASSESS The good news is that the wind is making the plane move a bit faster relative to the ground. The bad news is that the wind is making the plane move in the wrong direction! Slide 3-93

94 Example Problem A skydiver jumps out of an airplane 1000 m directly above his desired landing spot. He quickly reaches a steady speed, falling through the air at 35 m/s. There is a breeze blowing at 7 m/s to the west. A. At what angle with respect to vertical does he fall? B. When he lands, what will be his displacement from his desired landing spot? Slide 3-94

95 Section 3.6 Motion in Two Dimensions: Projectile Motion

96 Motion in Two Dimensions: Projectile Motion Projectile motion is an extension to two dimensions of free-fall motion. A projectile is an object that moves in two dimensions under the influence of gravity and nothing else. As long as we can neglect air resistance, any projectile will follow the same type of path. Slide 3-96

97 Motion in Two Dimensions: Projectile Motion The vertical motions of the two balls are identical. The vertical motion of the yellow ball is not affected by the fact that the ball is moving horizontally. The horizontal and vertical components of an object undergoing projectile motion are independent of each other. Slide 3-97

98 QuickCheck 3.16 A heavy red ball is released from rest 2.0 m above a flat, horizontal surface. At exactly the same instant, a yellow ball with the same mass is fired horizontally at 3.0 m/s. Which ball hits the ground first? A. The red ball hits first. B. The yellow ball hits first. C. They hit at the same time. Slide 3-98

99 QuickCheck 3.16 A heavy red ball is released from rest 2.0 m above a flat, horizontal surface. At exactly the same instant, a yellow ball with the same mass is fired horizontally at 3.0 m/s. Which ball hits the ground first? A. The red ball hits first. B. The yellow ball hits first. C. They hit at the same time. Slide 3-99

100 Motion in Two Dimensions: Projectile Motion The vertical component of acceleration a y for all projectile motion is just the familiar g of free fall, while the horizontal component a x is zero. Slide 3-100

101 QuickCheck 3.17 A 100-g ball rolls off a table and lands 2.0 m from the base of the table. A 200-g ball rolls off the same table with the same speed. It lands at distance A. 1.0 m B. Between 1 m and 2 m C. 2.0 m D. Between 2 m and 4 m E. 4.0 m Slide 3-101

102 QuickCheck 3.17 A 100-g ball rolls off a table and lands 2.0 m from the base of the table. A 200-g ball rolls off the same table with the same speed. It lands at distance A. 1.0 m B. Between 1 m and 2 m C. 2.0 m D. Between 2 m and 4 m E. 4.0 m Slide 3-102

103 Analyzing Projectile Motion The angle of the initial velocity above the horizontal (i.e., above the x-axis) is the launch angle. Slide 3-103

104 Analyzing Projectile Motion The ball finishes its motion moving downward at the same speed as it started moving upward. Projectile motion is made up of two independent motions: uniform motion at constant velocity in the horizontal direction and free-fall motion in the vertical direction. Slide 3-104

105 Synthesis 3.1 Projectile Motion Text: p. 82 Slide 3-105

106 Synthesis 3.1 Projectile Motion Text: p. 82 Slide 3-106

107 Synthesis 3.1 Projectile Motion Text: p. 82 Slide 3-107

108 Synthesis 3.1 Projectile Motion Text: p. 82 Slide 3-108

109 Section 3.7 Projectile Motion: Solving Problems

110 Projectile Motion Problems Text: p. 83 Slide 3-110

111 Projectile Motion Problems Text: p. 83 Slide 3-111

112 Example 3.11 Dock jumping In the sport of dock jumping, dogs run at full speed off the end of a dock that sits a few feet above a pool of water. The winning dog is the one that lands farthest from the end of the dock. If a dog runs at 8.5 m/s (a pretty typical speed for this event) straight off the end of a dock that is 0.61 m (2 ft, a standard height) above the water, how far will the dog go before splashing down? Slide 3-112

113 Example 3.11 Dock jumping In the sport of dock jumping, dogs run at full speed off the end of a dock that sits a few feet above a pool of water. The winning dog is the one that lands farthest from the end of the dock. If a dog runs at 8.5 m/s (a pretty typical speed for this event) straight off the end of a dock that is 0.61 m (2 ft, a standard height) above the water, how far will the dog go before splashing down? Slide 3-113

114 Example 3.11 Dock jumping (cont.) PREPARE We start with a visual overview of the situation in FIGURE We have chosen to put the origin of the coordinate system at the base of the dock. The dog runs horizontally off the end of the dock, so the initial components of the velocity are (v x ) i = 8.5 m/s and (v y ) i = 0 m/s. We can treat this as a projectile motion problem, so we can use the details and equations presented in Synthesis 3.1 above. We know that the horizontal and vertical motions are independent. The fact that the dog is falling toward the water doesn t affect its horizontal motion. Slide 3-114

115 Example 3.11 Dock jumping (cont.) When the dog leaves the end of the dock, it will continue to move horizontally at 8.5 m/s. The vertical motion is free fall. The jump ends when the dog hits the water that is, when it has dropped by 0.61 m. We are ultimately interested in how far the dog goes, but to make this determination we ll need to find the time interval Δt that the dog is in the air. Slide 3-115

116 Example 3.11 Dock jumping (cont.) SOLVE We ll start by solving for the time interval Δt, the time the dog is in the air. This time is determined by the vertical motion, which is free fall with an initial velocity (v y ) i = 0 m/s. We use the vertical position equation from Synthesis 3.1 to find the time interval: Rearranging terms to solve for Δt, we find that Δt = 0.35 s Slide 3-116

117 Example 3.11 Dock jumping (cont.) This is how long it takes the dog s vertical motion to reach the water. During this time interval, the dog s horizontal motion is uniform motion at the initial velocity. We can use the horizontalposition equation with the initial speed and Δt = 0.35 s to find how far the dog travels. This is the distance we are looking for: The dog hits the water 3.0 m from the edge of the dock. ASSESS 3.0 m is about 10 feet. This seems like a reasonable distance for a dog running at a very fast clip off the end of a 2- foot-high dock. Indeed, this is a typical distance for dogs in such competitions. Slide 3-117

118 The Range of a Projectile The range of a projectile is the horizontal distance traveled. For smaller objects air resistance is critical, and the maximum range comes at an angle less than 45. Slide 3-118

119 QuickCheck 3.18 Projectiles 1 and 2 are launched over level ground with the same speed but at different angles. Which hits the ground first? Ignore air resistance. A. Projectile 1 hits first. B. Projectile 2 hits first. C. They hit at the same time. D. There s not enough information to tell. Slide 3-119

120 QuickCheck 3.18 Projectiles 1 and 2 are launched over level ground with the same speed but at different angles. Which hits the ground first? Ignore air resistance. A. Projectile 1 hits first. B. Projectile 2 hits first. C. They hit at the same time. D. There s not enough information to tell. Slide 3-120

121 Section 3.8 Motion in Two Dimensions: Circular Motion

122 Reading Question 3.4 The acceleration vector of a particle in projectile motion A. Points along the path of the particle. B. Is directed horizontally. C. Vanishes at the particle s highest point. D. Is directed down at all times. E. Is zero. Slide 3-122

123 Reading Question 3.4 The acceleration vector of a particle in projectile motion A. Points along the path of the particle. B. Is directed horizontally. C. Vanishes at the particle s highest point. D. Is directed down at all times. E. Is zero. Slide 3-123

124 Motion in Two Dimensions: Circular Motion For circular motion at a constant speed, the acceleration vector a points toward the center of the circle. An acceleration that always points directly toward the center of a circle is called a centripetal acceleration. Centripetal acceleration is just the name for a particular type of motion. It is not a new type of acceleration. Slide 3-124

125 QuickCheck 3.19 A car is traveling around a curve at a steady 45 mph. Is the car accelerating? A. Yes B. No Slide 3-125

126 QuickCheck 3.19 A car is traveling around a curve at a steady 45 mph. Is the car accelerating? A. Yes B. No Slide 3-126

127 QuickCheck 3.20 A car is traveling around a curve at a steady 45 mph. Which vector shows the direction of the car s acceleration? E. The acceleration is zero. Slide 3-127

128 QuickCheck 3.20 A car is traveling around a curve at a steady 45 mph. Which vector shows the direction of the car s acceleration? B. E. The acceleration is zero. Slide 3-128

129 Motion in Two Dimensions: Circular Motion Slide 3-129

130 Motion in Two Dimensions: Circular Motion Slide 3-130

131 Motion in Two Dimensions: Circular Motion Slide 3-131

132 QuickCheck 3.21 A toy car moves around a circular track at constant speed. It suddenly doubles its speed a change of a factor of 2. As a result, the centripetal acceleration changes by a factor of A. 1/4 B. 1/2 C. No change since the radius doesn t change. D. 2 E. 4 Slide 3-132

133 QuickCheck 3.21 A toy car moves around a circular track at constant speed. It suddenly doubles its speed a change of a factor of 2. As a result, the centripetal acceleration changes by a factor of A. 1/4 B. 1/2 C. No change since the radius doesn t change. D. 2 E. 4 Slide 3-133

134 Example 3.14 Acceleration in the turn World-class female short-track speed skaters can cover the 500 m of a race in 45 s. The most challenging elements of the race are the turns, which are very tight, with a radius of approximately 11 m. Estimate the magnitude of the skater s centripetal acceleration in a turn. Slide 3-134

135 Example 3.14 Acceleration in the turn World-class female short-track speed skaters can cover the 500 m of a race in 45 s. The most challenging elements of the race are the turns, which are very tight, with a radius of approximately 11 m. Estimate the magnitude of the skater s centripetal acceleration in a turn. Slide 3-135

136 Example 3.14 Acceleration in the turn (cont.) PREPARE The centripetal acceleration depends on two quantities: the radius of the turn (given as approximately 11 m) and the speed. The speed varies during the race, but we can make a good estimate of the speed by using the total distance and time: SOLVE We can use these values to estimate the magnitude of the acceleration: ASSESS This is a large acceleration a bit more than g but the photo shows the skaters leaning quite hard into the turn, so such a large acceleration seems quite reasonable. Slide 3-136

137 Reading Question 3.5 The acceleration vector of a particle in uniform circular motion A. Points tangent to the circle, in the direction of motion. B. Points tangent to the circle, opposite the direction of motion. C. Is zero. D. Points toward the center of the circle. E. Points outward from the center of the circle. Slide 3-137

138 Reading Question 3.5 The acceleration vector of a particle in uniform circular motion A. Points tangent to the circle, in the direction of motion. B. Points tangent to the circle, opposite the direction of motion. C. Is zero. D. Points toward the center of the circle. E. Points outward from the center of the circle. Slide 3-138

139 Summary: General Principles Text: p. 89 Slide 3-139

140 Summary: General Principles Text: p. 89 Slide 3-140

141 Summary: Important Concepts Text: p. 89 Slide 3-141

142 Summary: Important Concepts Text: p. 89 Slide 3-142

143 Summary: Applications Text: p. 89 Slide 3-143

144 Summary: Applications Text: p. 89 Slide 3-144

145 Summary Text: p. 89 Slide 3-145

146 Summary Text: p. 89 Slide 3-146

147 Summary Text: p. 89 Slide 3-147

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