A diver runs horizontally off the end of a diving board

Problem 98IP IP Referring to Example 4-6 Suppose we change the dolphin's launch angle to 45.0°, but everything else remains the same. Thus, the horizontal distance to the ball is 5.50 m, the drop height is 4.10 m, and the dolphin's launch speed is 12.0 m/s. (a) What is the vertical distance between the dolphin and the ball when the dolphin reaches the horizontal position of the ball? We refer to this as the "miss distance." (b) If the dolphin's launch speed is reduced, will the miss distance increase, decrease, or stay the same? (c) Find the miss distance for a launch speed of 10.0 m/s.

Problem 2CQ A projectile is launched with an initial speed of v0 at an angle ? above the horizontal. It lands at the same level from which it was launched. What was its average velocity between launch and landing? Explain.

Problem 2P A sailboat runs before the wind with a constant speed of 4.2 m/s in a direction 32° north of west. How far (a) west and (b) north has the sailboat traveled in 25 min?

Problem 1P CE Predict/Explain As you walk briskly down the street, you toss a small ball into the air. (a) If you want the ball to land in your hand when it comes back down, should you toss the ball straight upward, in a forward direction, or in a backward direction, relative to your body? (b) Choose the best explanation from among the following: I. If the ball is thrown straight up you will leave it behind. II. You have to throw the ball in the direction you are walking. III. The ball moves in the forward direction with your walking speed at all times.

Problem 3CQ A projectile is launched from level ground. When it Sands, its direction of motion has rotated clockwise through 60°. What was the launch angle? Explain.

Problem 1CQ What is the acceleration of a projectile when it reaches its highest point? What is its acceleration just before and just after reaching this point?

Problem 4CQ In a game of baseball, a player hits a high fly ball to the outfield. (a) Is there a point during the flight of the ball where its velocity is parallel to its acceleration? (b) Is there a point where the ball's velocity is perpendicular to its acceleration? Explain in each case.

A projectile is launched with an initial velocity of \(\mathrm{\vec{v}=(4\ m/s)\hat{x}+(3\ m/s)\hat{y}}\). What is the velocity of the projectile when it reaches its highest point? Explain. ________________ Equation Transcription: Text Transcription: vec{v}=(4 m/s)hat{x}+(3 m/s)hat{y}

A projectile is launched from a level surface with an initial velocity of \(\mathrm{\vec{v}=(2\ m/s)\hat{x}+(4\ m/s)\hat{y}}\). What is the velocity of the projectile just before it lands? Explain. ________________ Equation Transcription: Text Transcription: vec{v}=(2 m/s)hat{x}+(4 m/s)hat{y}

As you walk to class with a constant speed of 1.75 m/s, you are moving in a direction that is 18.0° north of east. How much time does it take to change your displacement by (a) 20.0 m east or (b) 30.0 m north?

A particle passes through the origin with a velocity of \(\mathrm {(6.2\ m/s)\hat{y}}\). If the particle's acceleration is \(\mathrm {(-4.4 m/s^2)\hat{x}}\), (a) what are its and positions after ? (b) What are \(v_x\) and \(v_y\) at this time? (c) Does the speed of this particle increase with time, decrease with time, or increase and then decrease? Explain. ________________ Equation Transcription: Text Transcription: (6.2 m/s)hat{y} (-4.4 m/s^2)hat{x} v_x v_y

Problem 4P Starting from rest, a car accelerates at 2.0 m/s2 up a hill that is inclined 5.5° above the horizontal, How far (a) horizontally and (b) vertically has the car traveled in 12 s?

Problem 6P An electron in a cathode-ray tube is traveling horizontally at 2.10 × 109 cm/s when deflection plates give it an upward acceleration of 5.30 × 1017 cm/s2. (a) How long does it take for the electron to cover a horizontal distance of 6.20 cm? (b) What is its vertical displacement during this time?

Two canoeists start paddling at the same time and head toward a small island in a lake, as shown in Figure 4–12. Canoeist 1 paddles with a speed of 1.35 m/s at an angle of \(45^\circ\) north of east. Canoeist 2 starts on the opposite shore of the lake, a distance of 1.5 km due east of canoeist 1. (a) In what direction relative to north must canoeist 2 paddle to reach the island? (b) What speed must canoeist 2 have if the two canoes are to arrive at the island at the same time? ________________ Equation Transcription: Text Transcription: 45^o 45^o theta

Problem 7CQ Do projectiles for which air resistance is non negligible, such as a bullet fired from a rifle, have maximum range when the launch angle is greater than, less than, or equal to 45°? Explain.

Problem 8P CE Predict/Explain Two divers run horizontally off the edge of a low cliff. Diver 2 runs with twice the speed of diver 1. (a) When the divers hit the water, is the horizontal distance covered by diver 2 twice as much, four times as much, or equal to the horizontal distance covered by diver 1? (b) Choose the best explanation from among the following: I. The drop time is the same for both divers. II. Drop distance depends on t2. III. All divers in free fall cover the same distance.

Problem 8CQ Two projectiles are launched from the same point at the same angle above the horizontal. Projectile 1 reaches a maximum height twice that of projectile 2. What is the ratio of the initial speed of projectile 1 to the initial speed of projectile 2? Explain.

Problem 9P CE Predict/Explain Two youngsters dive off an overhang into a lake. Diver 1 drops straight down, and diver 2 runs off the cliff with an initial horizontal speed v0.(a) is the splashdown speed of diver 2 greater than, less than, or equal to the splashdown speed of diver 1? (b) Choose the best explanation from among the following: I. Both divers are in free fall, and hence they will have the same splashdown speed. II. The divers have the same vertical speed at splashdown, but diver 2 has the greater horizontal speed. III. The diver who drops straight down gains more speed than the one who moves horizontally.

Problem 10CQ Driving down the highway, you find yourself behind a heavily loaded tomato truck. You follow close behind the truck, keeping the same speed. Suddenly a tomato falls from the back of the truck. Will the tomato hit your car or land on the road, assuming you continue moving with the same speed and direction? Explain.

Problem 9CQ A child rides on a pony walking with constant velocity. The boy leans over to one side and a scoop of ice cream falls from his ice cream cone. Describe the path of the scoop of ice cream as seen by (a) the child and (b) his parents standing on the ground nearby.

Problem 10P An archer shoots an arrow horizontally at a target 15 m away. The arrow is aimed directly at the center of the target, but it hits 52 cm lower. What was the initial speed of the arrow?

Problem 11P Victoria Falls The great, gray-green, greasy Zambezi River flows over Victoria Falls in south central Africa. The falls are approximately 108 m high. If the river is flowing horizontally at 3.60 m/s just before going over the falls, what is the speed of the water when it hits the bottom? Assume the water is in free fall as it drops.

Problem 12P A diver runs horizontally off the end of a diving board with an initial speed of 1.85 m/s. if the diving board is 3.00 m above the water, what is the diver's speed just before she enters the water?

Problem 13P An astronaut on the planet Zircon tosses a rock horizontally with a speed of 6.95 m/s. The rock falls through a vertical distance of 1.40 m and lands a horizontal distance of 8.75 m from the astronaut. What is the acceleration of gravity on Zircon?

Problem 11CQ A projectile is launched from the origin of a coordinate system where the positive x axis points horizontally to the right and the positive y axis points vertically upward. What was the projectile's launch angle with respect to the x axis if, at its highest point, its direction of motion has rotated (a) clockwise through 50° or (b) counterclockwise through 30°? Explain.

Problem 14P IP Pitcher's Mounds Pitcher's mounds are raised to compensate for the vertical drop of the ball as it travels a horizontal distance of 18 hi to the catcher, (a) If a pitch is thrown horizontally with an initial speed of 32 m/s, how far does it drop by the time it reaches the catcher? (b) If the speed of the pitch is increased, does the drop distance increase, decrease, or stay the same? Explain, (c) If this baseball game were to be played on the Moon, would the drop distance increase, decrease, or stay the same? Explain.

Playing shortstop, you pick up a ground ball and throw it to second base. The ball is thrown horizontally, with a speed of 22 m/s, directly toward point A (Figure 4–13). When the ball reaches the second baseman 0.45 s later, it is caught at point B. (a) How far were you from the second baseman? (b) What is the distance of vertical drop, AB?

A crow is flying horizontally with a constant speed of 2.70 m/s when it releases a clam from its beak (Figure 4–14). The clam lands on the rocky beach 2.10 s later. Just before the clam lands, what is (a) its horizontal component of velocity, and (b) its vertical component of velocity? (c) How would your answers to parts (a) and (b) change if the speed of the crow were increased? Explain. ________________ Equation Transcription: Text Transcription: v_0

A white-crowned sparrow flying horizontally with a speed of 1.80 m/s folds its wings and begins to drop in free fall, (a) How far does the sparrow fall after traveling a horizontal distance of 0.500 m? (b) If the sparrow's initial speed is increased, does the distance of fall increase, decrease, or stay the same?

A mountain climber jumps a 2.8-m-wide crevasse by leaping horizontally with a speed of 7.8 m/s. (a) If the climber's direction of motion on landing is -45°, what is the height difference between the two sides of the crevasse? (b) Where does the climber land?

Pumpkin Toss In Denver, children bring their old jack-o-lanterns to the top of a tower and compete for accuracy in hitting a target on the ground (Figure 4–15). Suppose that the tower is 9.0 m high and that the bull’s-eye is a horizontal distance of 3.5 m from the launch point. If the pumpkin is thrown horizontally, what is the launch speed needed to hit the bull’s-eye?

Fairgoers ride a Ferris wheel with a radius of 5.00 m (Figure 4–16). The wheel completes one revolution every 32.0 s. (a) What is the average speed of a rider on this Ferris wheel? (b) If a rider accidentally drops a stuffed animal at the top of the wheel, where does it land relative to the base of the ride? (Note:The bottom of the wheel is 1.75 m above the ground.)

If, in the previous problem, a jack-o-lantern is given an initial horizontal speed of 3.3 m/s, what are the direction and magnitude of its velocity (a) 0.75 s later, and (b) just before it lands?

Problem 22P IP A swimmer runs horizontally off a diving board with a speed of 3.32 m/s and hits the water a horizontal distance of 1.78 in from the end of the board, (a) How high above the water was the diving board? (b) If the swimmer runs off the board with a reduced speed, does it take more, less, or the same time to reach the water?

Problem 23P Baseball and the Washington Monument On August 25, 1894, Chicago catcher William Schriver caught a baseball thrown from the top of the Washington Monument (555 ft, 898 steps), (a) If the ball was thrown horizontally with a speed of 5.00 m/s, where did it land? (b) What were the ball's speed and direction of motion when caught?

A basketball is thrown horizontally with an initial speed of 4.20 m/s (Figure 4–17). A straight line drawn from the release point to the landing point makes an angle of \(30.0^\circ\) with the horizontal. What was the release height? ________________ Equation Transcription: Text Transcription: 30.0^o 30.0^o

Problem 25P IP A ball rolls off a table and falls 0.75 m to the floor, landing with a speed of 4.0 m/s. (a) What is the acceleration of the ball just before it strikes the ground? (b) What was the initial speed of the ball? (c) What initial speed must the ball have if it is to land with a speed of 5.0 m/s?

CE A certain projectile is launched with an initial speed \(v_0\). At its highest point its speed is \(\frac{1}{2}v_0\). What was the launch angle of the projectile? A. \(30.0^\circ\) B. \(45^\circ\) ?. \(60^\circ\) D. \(75^\circ\)

Problem 29P A second baseman tosses the ball to the first baseman, who catches it at the same level from which it was thrown. The throw is made with an initial speed of 18.0 m/s at an angle of 37.5° above the horizontal, (a) What is the horizontal component of the ball's velocity just before it is caught? (b) How long is the ball in the air?

CE Three projectiles (A, B, and C) are launched with different initial speeds so that they reach the same maximum height, as shown in Figure 4–19. Rank the projectiles in order of increasing (a) initial speed and (b) time of flight. Indicate ties where appropriate.

CE Three projectiles (A, B, and C) are launched with the same initial speed but with different launch angles, as shown in Figure 4–18. Rank the projectiles in order of increasing (a) hori- zontal component of initial velocity and (b) time of flight. Indicate ties where appropriate.

Problem 30P Referring to the previous problem, what are the y component of the ball's velocity and its direction of motion just before it is caught?

Problem 31P A cork shoots out of a champagne bottle at an angle of 35.0° above the horizontal. If the cork travels a horizontal distance of 1.30 m in 1.25s, what was its initial speed?

Problem 32P A soccer ball is kicked with a speed of 9.85 m/s at an angle of 35.0° above the horizontal. If the ball lands at the same level from which it was kicked, how long was it in the air?

Problem 33P In a game of basketball, a forward makes a bounce pass to the center. The ball is thrown with an initial speed of 4.3 m/s at an angle of 15° below the horizontal. Tt is released 0.80 m above the floor. What horizontal distance does the ball cover before bouncing?

Problem 36P In the previous problem, find the direction of motion of the two snowballs just before they land.

Problem 39P The "hang time" of a punt is measured to be 4.50 s. If the ball was kicked at an angle of 63.0° above the horizontal and was caught at the same level from which it was kicked, what was its initial speed?

Problem 38P What is the highest tree the ball in the previous problem could clear on its way to the longest possible holc-in-one? Reference (previous) Problem: . •• A golfer gives a ball a maximum initial speed of 34.4 m/s. (a) What is the longest possible hole-in-one for this golfer? Neglect any distance the ball might mil on the green and assume that the tee and the green are at the same level, (b) What is the minimum speed of the ball during this hole-in-one shot?

Problem 34P Repeat the previous problem for a bounce pass in which the ball is thrown 15° above the horizontal.

A golfer gives a ball a maximum initial speed of 34.4 m/s. (a) What is the longest possible hole-irt-one for this golfer? Neglect any distance the ball might roll on the green and assume that the tee and the green are at the same level, (b) What is the minimum speed of the ball during this hole-in-one shot?

Snowballs are thrown with a speed of 13 m/s from a roof 7.0 m above the ground. Snowball A is thrown straight down-ward; snowball B is thrown in a direction 25° above the horizontal. (a) Is the landing speed of snowball A greater than, less than, or the same as the landing speed of snowball B? Explain. (b) Verify your answer to part (a) by calculating the landing speed of both snowballs.

Problem 40P In a friendly game of handball, you hit the ball essentially at ground level and send it toward the wall with a speed of 18 m/s at an angle of 32° above the horizontal, (a) How long does it take for the ball to reach the wall if it is 3.8 m away? (b) How high is the ball when it hits the wall?

Problem 41P IP Tn the previous problem, (a) what are the magnitude and direction of the ball's velocity when it stilkes the wall? (b) Has the ball reached the highest point of its trajectory at this time? Explain.

Problem 42P A passenger on the Ferris wheel described in Problem 21 drops his keys when he is on the way up and at the 10 o'clock position. Where do the keys land relative to the base of the ride?

Problem 44P A certain projectile is launched with an initial speed v0. At its highest point its speed is v0/4- What was the launch angle?

On a hot summer day, a young girl swings on a rope above the local swimming hole (Figure 4–20). When she lets go of the rope her initial velocity is 2.25 m/s at an angle of \(35.0^\circ\) above the horizontal. If she is in flight for 0.616 s, how high above the water was she when she let go of the rope? ________________ Equation Transcription: Text Transcription: 35.0^o v_0 35.0^o

Problem 47P A player passes a basketball to another player who catches it at the same level from which it was thrown. The initial speed of the ball is 7.1 m/s, and it travels a distance of 4.6 m. What were (a) the initial direction of the ball and (b) its time of flight?

Problem 46P A dolphin jumps with an initial velocity of 12.0 m/s at an angle of 40.0° above the horizontal. The dolphin passes through the center of a hoop before returning to the water. If the dolphin is moving horizontally when it goes through the hoop, how high above the water is the center of the hoop?

Problem 45P Punkin Chunkin In Sussex County, Delaware, a post-Halloween tradition is "Punkin Chunkin," in which contestants build cannons, catapults, trebuchets, and other devices to launch pumpkins and compete for the greatest distance. Though hard to believe, pumpkins have been projected a distance of 4086 feet in this contest. What is the mnirmum initial speed needed for such a shot?

Problem 48P A golf ball is struck with a five iron on level ground. It lands 92.2 m away 4.30 s later. What were (a) the direction and (b) the magnitude of the initial velocity?

Predict/Explain You throw a ball into the air with an initial speed of 10 m/s at an angle of \(60^\circ\) above the horizontal. The ball returns to the level from which it was thrown in the time T. (a) Referring to Figure 4–21, which of the plots (A, B, or C) best represents the speed of the ball as a function of time? (b) Choose the best explanation from among the following: I. Gravity causes the ball’s speed to increase during its flight. II. The ball has zero speed at its highest point. III. The ball’s speed decreases during its flight, but it doesn’t go to zero. ________________ Equation Transcription: Text Transcription: 60^o frac{1}{2}T

Problem 49P A Record Toss Babe Didrikson holds the world record for the longest baseball throw (296 ft) by a woman. For the following questions, assume that the ball was thrown at an angle of 45.0° above the horizontal, that it traveled a horizontal distance of 296 ft, and that it was caught at the same level from which it was thrown, (a) What was the ball's initial speed? (b) How long was the ball in the air?

Volcanoes on Io Astronomers have discovered several volcanoes on Io, a moon of Jupiter. One of them, named Loki, ejects lava to a maximum height of \(2.00\times10^5\ \mathrm m\). (a) What is the initial speed of the lava? (The acceleration of gravity on Io is \(1.80\ \mathrm {m/s}^2\).) (b) If this volcano were on Earth, would the maximum height of the ejected lava be greater than, less than, or the same as on Io? Explain. ________________ Equation Transcription: Text Transcription: 2.00x10^5 1.80 m/s^2

Problem 50P In the photograph to the left on page 87, suppose the cart that launches the ball is 11 cm high. Estimate (a) the launch speed of the ball and (b) the time interval between successive stroboscopic exposures.

Problem 54P A second soccer ball is kicked with the same initial speed as in Problem 53. After 0.750 s it is at its highest point. What was its initial direction of motion? Problem 53 . • • IP A soccer ball is kicked with an initial Speed of 10.2 m/s in a direction 25.0° above the horizontal. Find the magnitude and direction of its velocity (a) 0.250 s and (b) 0.500 s after being kicked, (c) Is the ball at its greatest height before or after 0.500 s? Explain.

A soccer ball is kicked with an initial speed of 10.2 m/s in a direction 25.0° above the horizontal. Find the magnitude and direction of its velocity (a) 0.250 s and (b) 0.500 s after being kicked, (c) is the ball at its greatest height before or after 0.500 s? Explain.

Problem 55P IP A golfer tees off on level ground, giving the ball an initial speed of 46.5 m/s and an initial direction of 37.5° above the horizontal, (a) How far from the golfer does the ball land? (b) The next golfer in the group hits a ball with the same initial speed but at an angle above the horizontal that is greater than 45.0°. Tf the second ball travels the same horizontal distance as the first ball, what was its initial direction of motion? Explain.

The penguin to the left in the accompanying photo is about to land on an ice floe. Just before it lands, is its speed greater than, less than, or equal to its speed when it left the water?

IP One of the most popular events at Highland games is the hay toss, where competitors use a pitchfork to throw a bale of hay over a raised bar. Suppose the initial velocity of a bale of hay is \(\mathrm {\vec{v}=(1.12\ m/s)\hat{x}+(8.85\ m/s)\hat{y}}\). (a) After what minimum time is its speed equal to (b) How long after the hay is tossed is it moving in a direction that is \(45.0^\circ\) below the horizontal? (c) If the bale of hay is tossed with the same initial speed, only this time straight upward, will its time in the air increase, decrease, or stay the same? Explain. ________________ Equation Transcription: Text Transcription: vec{v}=(1.12 m/s)hat{x}+(8.85 m/s)hat{y} 45.0^o

Problem 57GP CE Child 1 throws a snowball horizontally from the top of a roof; child 2 throws a snowball straight down. Once in flight, is the acceleration of snowball 2 greater than, less than, or equal to the acceleration of snowball 1?

Problem 59GP CE Predict/Explain A person flips a coin into the air and it lands on the ground a few feet away, (a) If the person were to perform an identical coin flip on an elevator rising with constant speed, would the coin's time of flight be greater than, less than, or equal to its time of flight when the person was at rest? (b) Choose the best explanation from among the following: I. The floor of the elevator is moving upward, and hence it catches up with the coin in mid flight. II. The coin has the same upward speed as the elevator when it is tossed, and the elevator's speed doesn't change during the coin's flight. III. The coin starts off with a greater upward speed because of the elevator, and hence it reaches a greater height.

Problem 60GP CEPredict/Explain Suppose the elevator in the previous problem is rising with a constant upward acceleration, rather than constant velocity, (a) In this case, would the coin's time of flight be greater than, less than, or equal to its time of flight when the person was at rest? (b) Choose the best explanation from among the following: I. The coin has the same acceleration once it is tossed, whether the elevator accelerates or not. II. The elevator's upward speed increases during the coin's flight, and hence it catches up with the coin at a greater height than before. III. The coin's downward acceleration is less than before because the elevator's upward acceleration partially cancels it.

Problem 61GP A train moving with constant velocity travels 1.70 m north in 12 s and an undetermined, distance to the west. The speed of the train is 32 m/s. (a) Find the direction of the train's motion relative to north, (b) How far west has the train traveled in this time?

Problem 62GP Referring to Example 4-2, find (a) the x component and (b) the y component of the hummingbird's velocity at the time t = 0.72 s. (c) What is the bird's direction of travel at this time, relative to the positive x axis?

Problem 63GP A racket ball is struck in such a way that it leaves the racket with a speed of 4.87 m/s in the horizontal direction. When the ball hits the court, it is a horizontal distance of 1.95 m from the racket. Find the height of the racket ball when it left the racket.

Problem 68GP When the dried-up seed pod of a scotch broom plant bursts open, it shoots out a seed with an initial velocity of 2.62 m/s at an angle of 60.5° above the horizontal. If the seed pod is 0.455 m above the ground, (a) bow long does it take for the seed to land? (b) What horizontal distance does it cover during its flight?

A particle leaves the origin with an initial velocity \(\mathrm {\vec{v}=(2.40\ m/s)\hat{x}}\), and moves with constant acceleration \(\mathrm {\vec{a}=(-1.90\ m/s^2)\hat{x}+(3.20\ m/s^2)\hat{y}}\). (a) How far does the particle move in the direction before turning around? (b) What is the particle's velocity at this time? (c) Plot the particle's position at \(t=0.500\ \mathrm s\), 1.00 s, 1.50 s, and 200 s. Use these results to sketch position versus time for the particle. ________________ Equation Transcription: Text Transcription: vec{v}=(2.40 m/s)hat{x} vec{a}=(-1.90 m/s^2)hat{x}+(3.20 m/s^2)hat{y} t=0.500 s

Problem 65GP Repeat the previous problem, this time assuming that the balloon is descending with a speed of 2.00 m/s.

Problem 66GP IP A soccer ball is kicked from the ground with an initial speed of 14.0 m/s. After 0.275 s its speed is 12.9 m/s. (a) Give a strategy that will allow you to calculate the ball's initial direction of motion, (b) Use your strategy to find the initial direction.

IP A hot-air balloon rises from the ground with a velocity of \(\mathrm {(2.00\ m/s)\hat y}\). A champagne bottle is opened to celebrate takeoff, expelling the cork horizontally with a velocity of \(\mathrm {(5.00\ m/s)\hat x}\) relative to the balloon. When opened, the bottle is above the ground. (a) What is the initial velocity of the cork, as seen by an observer on the ground? Give your answer in terms of the and unit vectors. (b) What are the speed of the cork and its initial direction of motion as seen by the same observer? (c) Determine the maximum height above the ground attained by the cork. (d) How long does the cork remain in the air?

Problem 69GP Referring to Problem 68, a second seed shoots out from the pod with the same speed but with a direction of motion 30.0° below the horizontal, (a) How long does it take for the second seed to land? (b) What horizontal distance does it cover during its flight?

Problem 71GP Pararescue Jumpers Coast Guard pararescue jumpers are framed to leap from helicopters into the sea to save boaters in distress. The rescuers like to step off their helicopter when it is "ten and ten", which means that it is ten feet above the water and moving forward horizontally at ten knots. What are (a) the speed and (b) the direction of motion as a pararescuer enters the water following a ten and ten jump?

Problem 70GP A shot-putter throws the shot with an initial speed of 12.2 m/s from a height of 5.15 ft above the ground. What is the range of the shot if the launch angle is (a) 20.0°, (b) 30.0°, or (c) 40.0°?

Problem 72GP A ball thrown straight upward returns to its original level in 2.75 s. A second ball is thrown at an angle of 40.0° above the horizontal. What is the initial speed of the second ball if it also returns to its original level in 2.75 s?

Problem 74GP IP A cannon is placed at the bottom of a cliff 61.5 m high. If the cannon is fired straight upward, the cannonball just reaches the top of the cliff, (a) What is the initial speed of the cannonball? (b) Suppose a second cannon is placed at the top of the cliff. This cannon is fired horizontally, giving its cannon bails the same initial speed found in part(a). Show that the range of this cannon is the same as the maximum range of the cannon at the base of the cliff. (Assume the ground at the base of the cliff is level, though the result is valid even if the ground is not level.)

IP To decide who pays for lunch, a passenger on a moving train tosses a coin straight upward with an initial speed of and catches it again when it returns to its initial level. From the point of view of the passenger, then, the coin's initial velocity is \(\mathrm {(4.38\ m/s)\hat{y}}\). The train's velocity relative to the ground is \(\mathrm {(12.1\ m/s)\hat{x}}\). (a) What is the minimum speed of the coin relative to the ground during its flight? At what point in the coin's flight does this minimum speed occur? Explain. (b) Find the initial speed and direction of the coin as seen by an observer on the ground. (c) Use the expression for \(y_\mathrm{max}\) derived in Example to calculate the maximum height of the , as seen by an observer on the ground. (d) Calculate the maximum height of the coin from the point of view of the passenger, who sees only one-dimensional motion. ________________ Equation Transcription: Text Transcription: (4.38 m/s)hat{y} (12.1 m/s)hat{x} y_{max}

Problem 75GP Shot Put Record The men's world record for the shot put, 23.12 m, was set by Randy Barnes of the United States on May 20,1990. If the shot was launched from 6.00 ft above the ground at an initial angle of 42.0°, what was its initial speed?

Referring to Conceptual Checkpoint 4–3, suppose the two snowballs are thrown from an elevation of 15 m with an initial speed of 12 m/s. What is the speed of each ball when it is 5.0 m above the ground?

Referring to Active Example 4–2, suppose the ball is punted from an initial height of 0.750 m. What is the initial speed of the ball in this case?

Problem 77GP IP A hockey puck just clears the 2.00-m-high boards on its way out of the rink. The base of the boards is 20.2 m from the point where the puck is launched, (a) Given the launch angle of the puck, ?, outline a strategy that you can use to find its initial speed, u0 (b) Use your strategy to find u0 for 0 = 15.0°.

Problem 82GP (a) What is the greatest horizontal distance from which the archerfish can hit the beetle, assuming the same squirt speed and direction as in Problem 81? (b) How much time docs the beetle have to react in this case?

Problem 81GP As discussed in Example 4-7, the archcrfish hunts by dislodging an unsuspecting insect from its resting place with a stream of water expelled from the fish's mouth. Suppose the archerfish squirts water with a speed of 2.15 m/s at an angle of 52.0° above the horizontal, and aims for a beetle on a leaf 3.00 cm above the water's surface, (a) At what horizontal distance from the beetle should the archerfish fire if it is to hit its target in the least time? (b) How much time will the beetle have to react?

Collision Course A useful rule of thumb in boating is that if the heading from your boat to a second boat remains constant, the two boats are on a collision course. Consider the two boats shown in Figure 4-23. At time \(t=0\), boat 1 is at the location and moving in the positive direction; boat 2 is at and moving in the positive direction. The speed of boat 1 is \(v_1\). (a) What speed must boat 2 have if the boats are to collide at the point (b) Assuming boat 2 has the speed found in part (a), calculate the displacement from boat 1 to boat 2 , \(\mathrm {\Delta \vec r=\vec r_2-\vec r_1}\). (c) Use your results from part (b) to show that \((\Delta r)_y/(\Delta r)_x=-Y/X\), independent of time. This shows that \(\mathrm {\Delta \vec r=\vec r_2-\vec r_1}\) maintains a constant direction until the collision, as specified in the rule of thumb. ________________ Equation Transcription: Text Transcription: t=0 v_1 Delta vec{r}=vec{r}_2-vec{r}_1 (Delta vec{r})_y/(Delta r)_x=-Y/X Delta vec{r}=vec{r}_2-vec{r}_1 vec{r}_2 vec{r}_1 Delta vec{r}

Problem 83GP Find the launch angle for which the range and maximum height of a projectile are the same.

Problem 84GP A mountain climber jumps a crevasse of width W by leaping horizontally with speed u0.(a) if the height difference between the two sides of the crevasse is h, what is the minimum value of u0 for the climber to land safely on the other side? (b) In this case, what is the cumber's direction of motion on landing?

Problem 85G Prove that the landing speed of a projectile is independent of launch angle for a given height of launch.

Maximum Height and Range Prove that the maximum height of a projectile, \(H\), divided by the range of the projectile, \(R\), satisfies the relation \(H / R=\frac{1}{4} \tan \theta\). Equation Transcription: Text Transcription: H R H/R=1/4tan theta

Problem 87GP Landing on a Different Level A projectile fired from y = 0 with initial speed u0 e:\04-02-2016\chapter 4\1403\9781111788452\exercises and initial angle ? lands on a different level, y = h. Show that the time of flight of the projectile is where T0 is the time of flight for h = 0 and H is the maximum height of the projectile.

Problem 88GP A mountain climber jumps a crevasse by leaping horizontally with speed u0. If the climber's direction of motion on landing is ? below the horizontal, what is the height difference h between the two sides of the crevasse?

IP Referring to Problem 73, suppose the initial velocity of the coin tossed by the passenger is \(\mathrm {\vec{v}=(-2.25\ m/s)\hat{x}+(4.38\ m/s)\hat{y}}\). The train's velocity relative to the ground is still \(\mathrm {(12.1\ m/s)\hat{x}}\). (a) What is the minimum speed of the coin relative to the ground during its flight? At what point in the coin's flight does this minimum speed occur? Explain. (b) Find the initial speed and direction of the coin as seen by an observer on the ground. (c) Use the expression for \(y_{\mathrm {max}}\) derived in Example 4-7 to calculate the maximum height of the coin, as seen by an observer on the ground. (d) Repeat part (c) from the point of view of the passenger. Verify that both observers calculate the same maximum height. ________________ Equation Transcription: Text Transcription: vec{v}=(-2.25 m/s)hat{x}+(4.38 m/s)hat{y} (12.1 m/s)hat{x} y_{max}

Projectiles: Coming or Going? Most projectiles continually move farther from the origin during their flight, but this is not the case if the launch angle is greater than \(\cos ^{-1}\left(\frac{1}{3}\right)=70.5^{\circ}\). For example, the projectile shown in Figure 4-24 has a launch angle of \(75.0^{\circ}\) and an initial speed of 10.1 m/s. During the portion of its motion shown in red, it is moving closer to the origin it is moving away on the blue portions. Calculate the distance from the origin to the projectile (a) at the start of the red portion, (b) at the end of the red portion, and (c) just before the projectile lands. Notice that the distance for part (b) is the smallest of the three. Equation Transcription: Text Transcription: cos^-1(1/3)=70.5degree 75degree

Landing Rovers on Mars When the twin Mars exploration rovers, Spirit and Opportunity, set down on the surface of the red planet in January of 2004, their method of landing was both unique and elaborate. After initial braking with retro rockets, the rovers began their long descent through the thin Martian atmosphere on a parachute until they reached an altitude of about 16.7 m. At that point a system of four air bags with six lobes each were inflated, addi- tional retro rocket blasts brought the craft to a virtual standstill, and the rovers detached from their parachutes. After a period of free fall to the surface, with an acceleration of \(3.72\ \mathrm {m/s^2}\), the rovers bounced about a dozen times before coming to rest. They then deflated their air bags, righted themselves, and began to explore the surface. Figure 4–25 shows a rover with its surrounding cushion of air bags making its first contact with the Martian surface. After a typical first bounce the upward velocity of a rover would be 9.92 m/s at an angle of \(75.0^\circ\) above the horizontal. Assume this is the case for the problems that follow. What is the maximum height of a rover between its first and second bounces? A. 2.58 m B. 4.68 m C. 12.3 m D. 148 m Equation Transcription: Text Transcription: 3.72 m/s^2 75.0^o vec{r}(t)

Referring to Example 4–5 (a) At what launch angle greater than \(54.0^\circ\) does the golf ball just barely miss the top of the tree in front of the green? Assume the ball has an initial speed of 13.5 m/s, and that the tree is 3.00 m high and is a horizontal distance of 14.0 m from the launch point. (b) Where does the ball land in the case described in part (a)? (c) At what launch angle less than \(54.0^\circ\) does the golf ball just barely miss the top of the tree in front of the green? (d) Where does the ball land in the case described in part (c)? ________________ Equation Transcription: Text Transcription: 54.0^o 54.0^o

Problem 92PP How much time elapses between the first and second bounces? A. 1.38 s B. 2.58 s C. 5.15 s D. 5.33 s

Landing Rovers on Mars When the twin Mars exploration rovers, Spirit and Opportunity, set down on the surface of the red planet in January of 2004, their method of landing was both unique and elaborate. After initial braking with retro rockets, the rovers began their long descent through the thin Martian atmosphere on a parachute until they reached an altitude of about 16.7 m. At that point a system of four air bags with six lobes each were inflated, addi- tional retro rocket blasts brought the craft to a virtual standstill, and the rovers detached from their parachutes. After a period of free fall to the surface, with an acceleration of \(3.72\ \mathrm {m/s^2}\), the rovers bounced about a dozen times before coming to rest. They then deflated their air bags, righted themselves, and began to explore the surface. Figure 4–25 shows a rover with its surrounding cushion of air bags making its first contact with the Martian surface. After a typical first bounce the upward velocity of a rover would be 9.92 m/s at an angle of \(75.0^\circ\) above the horizontal. Assume this is the case for the problems that follow. How far does a rover travel in the horizontal direction between its first and second bounces? A. 13.2 m B. 49.4 m C. 51.1 m D. 98.7 m Equation Transcription: Text Transcription: 3.72 m/s^2 75.0^o vec{r}(t)

Landing Rovers on Mars When the twin Mars exploration rovers, Spirit and Opportunity, set down on the surface of the red planet in January of 2004, their method of landing was both unique and elaborate. After initial braking with retro rockets, the rovers began their long descent through the thin Martian atmosphere on a parachute until they reached an altitude of about 16.7 m. At that point a system of four air bags with six lobes each were inflated, additional retro rocket blasts brought the craft to a virtual standstill, and the rovers detached from their parachutes. After a period of free fall to the surface, with an acceleration of \(3.72\ \mathrm {m/s^2}\), the rovers bounced about a dozen times before coming to rest. They then deflated their air bags, righted themselves, and began to explore the surface. Figure 4–25 shows a rover with its surrounding cushion of air bags making its first contact with the Martian surface. After a typical first bounce the upward velocity of a rover would be 9.92 m/s at an angle of \(75.0^\circ\) above the horizontal. Assume this is the case for the problems that follow. What is the average velocity of a rover between its first and second bounces? A. 0 B. 2.57 m/s in the x direction C. 9.92 m/s at \(75.0^\circ\) above the x axis D. 9.58 m/s in the y direction Equation Transcription: Text Transcription: 3.72 m/s^2 75.0^o vec{r}(t) 75.0^o

Problem 97IP Referring to Example 4-6 Suppose the ball is dropped at the horizontal distance of 5.50 m, but from a new height of 5.00 m. The dolphin jumps with the same speed of 12.0 m/s. (a) What launch angle must the dolphin have if it is to catch the ball? (b) At what height does the dolphin catch the ball in this case? (c) What is the minimum initial speed the dolphin must have to catch the ball before it hits the water?

Problem 96IP Referring to Example 4-5 Suppose that the golf ball is launched with a speed of 15.0 m/s at an angle of 57.5° above the horizontal, and that it lands on a green 3.50 m above the level where it was struck, (a) What horizontal distance does the ball cover during its flight? (b) What increase in initial speed would be needed to increase the horizontal distance in part (a) by 7.50 m? Assume everything else remains the same.