The Position Velocity free essay sample

The Position, Velocity, and Acceleration Vectors 1. A motorist drives south at 20. 0 m/s for 3. 00 min, then turns west and travels at 25. 0 m/s for 2. 00 min, and finally travels northwest at 30. 0 m/s for 1. 00 min. For this 6. 00-min trip, find (a) the total vector displacement, (b) the average speed, and (c) the average velocity. Let the positive x axis point east. 2. A golf ball is hit off a tee at the edge of a cliff. Its x and y coordinates as functions of time are given by the following expressions: = (18. 0 m/s)t and y = (4. 00 m/s)t – (4. 90 m/s2)t2 (a) Write a vector expression for the ball’s position as a function of time, using the unit vectors [pic] and [pic]. By taking derivatives, obtain expressions for (b) the velocity vector v as a function of time and (c) the acceleration vector a as a function of time. Next use unit-vector notation to write expressions for (d) the position, (e) the velocity, and (f) the acceleration of the golf ball, all at t = 3. We will write a custom essay sample on The Position Velocity or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page 00 s. 3. When the Sun is directly overhead, a hawk dives toward the ground with a constant velocity of 5. 00 m/s at 60. ( below the horizontal. Calculate the speed of her shadow on the level ground. 4. The coordinates of an object moving in the xy plane vary with time according to the equations x = –(5. 00 m) sin(wt) and y = (4. 00 m) – (5. 00 m)cos(wt), where w is a constant and t is in seconds. (a) Determine the components of velocity and components of acceleration at t = 0. (b) Write expressions for the position vector, the velocity vector, and the acceleration vector at any time t gt; 0. (c) Describe the path of the object in an xy plot. Section 4. 2 Two-Dimensional Motion with Constant Acceleration 5. At t = 0, a particle moving in the xy plane with constant acceleration has a velocity of [pic] and is at the origin. At t = 3. 00 s, the particles velocity is [pic]. Find (a) the acceleration of the particle and (b) its coordinates at any time t. 6. The vector position of a particle varies in time according to the expression [pic]. (a) Find expressions for the velocity and acceleration as functions of time. (b) Determine the particles position and velocity at t = 1. 00 s. 7. A fish swimming in a horizontal plane has velocity [pic] at a point in the ocean where the position relative to a certain rock is [pic]. After the fish swims with constant acceleration for 20. 0 s, its velocity is [pic]. (a) What are the components of the acceleration? (b) What is the direction of the acceleration with respect to unit vector [pic]? (c) If the fish maintains constant acceleration, where is it at t = 25. 0 s, and in what direction is it moving? 8. A particle initially located at the origin has an acceleration of [pic]and an initial velocity of [pic]. Find (a) the vector position and velocity at any time t and (b) the coordinates and speed of the particle at t = 2. 00 s. 9. It is not possible to see very small objects, such as viruses, using an ordinary light microscope. An electron microscope can view such objects using an electron beam instead of a light beam. Electron microscopy has proved invaluable for investigations of viruses, cell membranes and subcellular structures, bacterial surfaces, visual receptors, chloroplasts, and the contractile properties of muscles. The â€Å"lenses† of an electron microscope consist of electric and magnetic fields that control the electron beam. As an example of the manipulation of an electron beam, consider an lectron traveling away from the origin along the x axis in the xy plane with initial velocity [pic]. As it passes through the region x = 0 to x = d, the electron experiences acceleration [pic] , where ax and ay are constants. For the case vi = 1. 80 ( 107 m/s, ax = 8. 00 ( 1014 m/s2 and ay = 1. 60 ( 1015m/s2, determine at x = d = 0. 0100 m (a) the position of the electron, (b) the velocity of the elect ron, (c) the speed of the electron, and (d) the direction of travel of the electron (i. e. , the angle between its velocity and the x axis). Section 4. 3 Projectile Motion Note: Ignore air resistance in all problems and take g = 9. 80 m/s2 at the Earth’s surface. 10. To start an avalanche on a mountain slope, an artillery shell is fired with an initial velocity of 300 m/s at 55. 0 ° above the horizontal. It explodes on the mountainside 42. 0 s after firing. What are the x and y coordinates of the shell where it explodes, relative to its firing point? 11. In a local bar, a customer slides an empty beer mug down the counter for a refill. The bartender is momentarily distracted and does not see the mug, which slides off the counter and strikes the floor 1. 40 m from the base of the counter. If the height of the counter is 0. 860 m, (a) with what velocity did the mug leave the counter, and (b) what was the direction of the mugs velocity just before it hit the floor? 12. In a local bar, a customer slides an empty beer mug down the counter for a refill. The bartender is momentarily distracted and does not see the mug, which slides off the counter and strikes the floor at distance d from the base of the counter. The height of the counter is h. (a) With what velocity did the mug leave the counter, and (b) what was the direction of the mugs velocity just before it hit the floor? 3. One strategy in a snowball fight is to throw a snowball at a high angle over level ground. While your opponent is watching the first one, a second snowball is thrown at a low angle timed to arrive before or at the same time as the first one. Assume both snowballs are thrown with a speed of 25. 0 m/s. The first one is thrown at an angle of 70. 0( with respect to the horizontal. (a) At what angle should the second snowball be thrown to arrive at the same point as the first? (b) How many seconds later should the second snowball be thrown after the first to arrive at the same time? 4. An astronaut on a strange planet finds that she can jump a maximum horizontal distance of 15. 0 m if her initial speed is 3. 00 m/s. What is the free-fall acceleration on the planet? 15. A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection? 16. A rock is thrown upward from the level ground in such a way that the maximum height of its flight is equal to its horizontal range d. (a) At what angle [pic] is the rock thrown? (b) What If? Would your answer to part (a) be different on a different planet? c) What is the range dmax the rock can attain if it is launched a t the same speed but at the optimal angle for maximum range? 17. A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8. 00 m/s at an angle of 20. 0 ° below the horizontal. It strikes the ground 3. 00 s later. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point 10. 0 m below the level of launching? 18. The small archerfish (length 20 to 5 cm) lives in brackish waters of southeast Asia from India to the Philippines. This aptly named creature captures its prey by shooting a stream of water drops at an insect, either flying or at rest. The bug falls into the water and the fish gobbles it up. The archerfish has high accuracy at distances of 1. 2 m to 1. 5 m, and it sometimes makes hits at distances up to 3. 5 m. A groove in the roof of its mouth, along with a curled tongue, forms a tube that enables the fish to impart high velocity to the water in its mouth when it suddenly closes its gill flaps. Suppose the archerfish shoots at a target 2. 0 m away, at an angle of 30. 0o above the horizontal. With what velocity must the water stream be launched if it is not to drop more than 3. 00 cm vertically on its path to the target? 19. A place-kicker must kick a football from a point 36. 0 m (about 40 yards) from the goal, and half the crowd hopes the ball will clear the crossbar, which is 3. 05 m high. When kicked, the ball leaves the ground with a speed of 20. 0 m/s at an angle of 53. 0 ° to the horizont al. (a) By how much does the ball clear or fall short of clearing the crossbar? (b) Does the ball approach the crossbar while still rising or while falling? 0. A firefighter, a distance d from a burning building, directs a stream of water from a fire hose at angle [pic]i above the horizontal as in Figure P4. 20. If the initial speed of the stream is vi, at what height h does the water strike the building? [pic] Figure P4. 20 21. A playground is on the flat roof of a city school, 6. 00 m above the street below. The vertical wall of the building is 7. 00 m high, to form a meter-high railing around the playground. A ball has fallen to the street below, and a passerby returns it by launching it at an angle of 53. 0( above the horizontal at a point 24. meters from the base of the building wall. The ball takes 2. 20 s to reach a point vertically above the wall. (a) Find the speed at which the ball was launched. (b) Find the vertical distance by which the ball clears the wall. (c) Find the distance from the wall to the point on the roof where the ball lands. 22. A dive bomber has a velocity of 280 m/s at an angle [pic] below the horizontal. When t he altitude of the aircraft is 2. 15 km, it releases a bomb, which subsequently hits a target on the ground. The magnitude of the displacement from the point of release of the bomb to the target is 3. 25 km. Find the angle [pic]. 23. A soccer player kicks a rock horizontally off a 40. 0-m high cliff into a pool of water. If the player hears the sound of the splash 3. 00 s later, what was the initial speed given to the rock? Assume the speed of sound in air to be 343 m/s. 24. A basketball star covers 2. 80 m horizontally in a jump to dunk the ball. His motion through space can be modeled precisely as that of a particle at his center of mass, which we will define in Chapter 9. His center of mass is at elevation 1. 02 m when he leaves the floor. It reaches a maximum height of 1. 85 m above the floor, and is at elevation 0. 00 m when he touches down again. Determine (a) his time of flight (his â€Å"hang time†), (b) his horizontal and (c) vertical velocity components at the instant of takeoff, and (d) his takeoff angle. (e) For comparison, determine the hang time of a whitetail deer making a jump with center-of-mass elevations yi = 1. 20 m, ymax = 2. 50 m, yf = 0. 700 m. 25. An archer shoots an arrow with a velocity of 45. 0 m/s at an angle of 50. 0o with the horizontal. An assistant standing on the level ground 150 m downrange from the launch point throws an apple straight up with the minimum initial speed necessary to meet the path of the arrow. a) What is the initial speed of the apple? (b) At what time after the arrow launch should the apple be thrown so that the arrow hits the apple? 26. A fireworks rocket explodes at height h, the peak of its vertical trajectory. It throws out burning fragments in all directions, but all at the same speed v. Pellets of solidified metal fall to the ground without air resistance. Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal. Section 4. 4 Uniform Circular Motion Note: Problems 8, 10, 12, and 16 in Chapter 6 can also be assigned with this section. 27. The athlete shown in Figure P4. 27 rotates a 1. 00-kg discus along a circular path of radius 1. 06 m. The maximum speed of the discus is 20. 0 m/s. Determine the magnitude of the maximum radial acceleration of the discus. [pic] Figure P4. 27 28. From information on the endsheets of this book, compute the radial acceleration of a point on the surface of the Earth at the equator, due to the rotation of the Earth about its axis. 29. A tire 0. 500 m in radius rotates at a constant rate of 200 rev/min. Find the speed and acceleration of a small stone lodged in the tread of the tire (on its outer edge). 30. As their booster rockets separate, Space Shuttle astronauts typically feel accelerations up to 3g, where g = 9. 80 m/s2. In their training, astronauts ride in a device where they experience such an acceleration as a centripetal acceleration. Specifically, the astronaut is fastened securely at the end of a mechanical arm that then turns at constant speed in a horizontal circle. Determine the rotation rate, in revolutions per second, required to give an astronaut a centripetal acceleration of 3. 00g while in circular motion with radius 9. 45 m. 31. Young David who slew Goliath experimented with slings before tackling the giant. He found that he could revolve a sling of length 0. 600 m at the rate of 8. 00 rev/s. If he increased the length to 0. 900 m, he could revolve the sling only 6. 00 times per second. (a) Which rate of rotation gives the greater speed for the stone at the end of the sling? (b) What is the centripetal acceleration of the stone at 8. 00 rev/s? (c) What is the centripetal acceleration at 6. 00 rev/s? 32. The astronaut orbiting the Earth in Figure P4. 32 is preparing to dock with a Westar VI satellite. The satellite is in a circular orbit 600 km above the Earths surface, where the free-fall acceleration is 8. 1 m/s2. Take the radius of the Earth as 6 400 km. Determine the speed of the satellite and the time interval required to complete one orbit around the Earth. [pic] Figure P4. 32 Section 4. 5 Tangential and Radial Acceleration 33. A train slows down as it rounds a sharp horizontal turn, slowing from 90. 0 km/h to 50. 0 km/h in the 15. 0 s that it takes to round the bend. The radius of the curve is 150 m. Compute the acceleration at the moment the train speed reaches 50. 0 km/h. Assume it continues to slow down at this time at the same rate. 34. An automobile whose speed is increasing at a rate of 0. 00 m/s2 travels along a circular road of radius 20. 0 m. When the instantaneous speed of the automobile is 4. 00 m/s, find (a) the tangential acceleration component, (b) the centripetal acceleration component, and (c) the magnitude and direction of the total acceleration. 35. Figure P4. 35 represents the total acceleration of a particle moving clockwise in a circle of radius 2. 50 m at a certain of time. At this instant, find (a) the radial acceleration, (b) the speed of the particle, and (c) its tangential acceleration. [pic] Figure P4. 35 36. A ball swings in a vertical circle at the end of a rope 1. 0 m long. When the ball is 36. 9( past the lowest point on its way up, its total acceleration is [pic]. At that instant, (a) sketch a vector diagram showin g the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball. 37. A race car starts from rest on a circular track. The car increases its speed at a constant rate at as it goes once around the track. Find the angle that the total acceleration of the car makes with the radius connecting the center of the track and the car at the moment the car completes the circle. Section 4. 6 Relative Velocity and Relative Acceleration 38. Heather in her Corvette accelerates at the rate of [pic]m/s2, while Jill in her Jaguar accelerates at [pic]m/s2. They both start from rest at the origin of an xy coordinate system. After 5. 00 s, (a) what is Heathers speed with respect to Jill, (b) how far apart are they, and (c) what is Heathers acceleration relative to Jill? 39. A car travels due east with a speed of 50. 0 km/h. Raindrops are falling at a constant speed vertically with respect to the Earth. The traces of the rain on the side windows of the car make an angle of 60.  ° with the vertical. Find the velocity of the rain with respect to (a) the car and (b) the Earth. 40. How long does it take an automobile traveling in the left lane at 60. 0 km/h to pull alongside a car traveling in the right lane at 40. 0 km/h if the cars front bumpers are initially 100 m apart? 41. A river has a steady speed of 0. 500 m/s. A student swims upstream a distance of 1. 00 km and swims back to the starting point. If the student can swim at a speed of 1. 20 m/s in still water, how long does the trip take? Compare this with the time the trip would take if the water were still. 42. The pilot of an airplane notes that the compass indicates a heading due west. The airplanes speed relative to the air is 150 km/h. If there is a wind of 30. 0 km/h toward the north, find the velocity of the airplane relative to the ground. 43. Two swimmers, Alan and Beth, start together at the same point on the bank of a wide stream that flows with a speed v. Both move at the same speed c (c gt; v), relative to the water. Alan swims downstream a distance L and then upstream the same distance. Beth swims so that her motion relative to the Earth is perpendicular to the banks of the stream. She swims the distance L and then back the same distance, so that both swimmers return to the starting point. Which swimmer returns first? (Note: First guess the answer. ) 44. A bolt drops from the ceiling of a train car that is accelerating northward at a rate of 2. 50 m/s2. What is the acceleration of the bolt relative to (a) the train car? (b) the Earth? 45. A science student is riding on a flatcar of a train traveling along a straight horizontal track at a constant speed of 10. 0 m/s. The student throws a ball into the air along a path that he judges to make an initial angle of 60.  ° with the horizontal and to be in line with the track. The students professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise? 46. A Coast Guard cutter detects an unidentified ship at a distance of 20. 0 km in the direction 15. 0( east of north. The ship is traveling at 26. 0 km/h on a course at 40. 0( east of north. The Coast Guard w ishes to send a speedboat to intercept the vessel and investigate it. If the speedboat travels 50. 0 km/h, in what direction should it head? Express the direction as a compass bearing with respect to due north. Additional Problems 47. The â€Å"Vomit Comet. † In zero-gravity astronaut training and equipment testing, NASA flies a KC135A aircraft along a parabolic flight path. As shown in Figure P4. 47, the aircraft climbs from 24 000 ft to 31 000 ft, where it enters the zero-g parabola with a velocity of 143 m/s nose-high at 45. 0o and exits with velocity 143 m/s at 45. 0o nose-low. During this portion of the flight the aircraft and objects inside its padded cabin are in free fall—they have gone ballistic. The aircraft then pulls out of the dive with an upward acceleration of 0. 800 g, moving in a vertical circle with radius 4. 13 km. During this portion of the flight, occupants of the plane perceive an acceleration of 1. 8 g’s. ) What are the aircraft (a) speed and (b) altitude at the top of the maneuver? (c) What is the time spent in zero gravity? (d) What is the speed of the aircraft at the bottom of the flight path? [pic] [pic] Figure P4. 47 48. As some molten metal splashes, one droplet flies off to the east with initial velocity vi at angle [pic]i above the horizontal, and another droplet to the west with the same speed at the same angle above the horizontal, as in Figure P4. 48. In terms of vi and [pic]i, find the distance between them as a function of time. pic] Figure P4. 48 49. A ball on the end of a string is whirled around in a horizontal circle of radius 0. 300 m. The plane of the circle is 1. 20 m above the ground. The string breaks and the ball lands 2. 00 m (horizontally) away from the point on the ground directly beneath the balls location when the string breaks. Find the radial acceleration of the ball during its circular motion. 50. A projectile is fired up an incline (incline angle [pic]) with an initial speed vi at an angle [pic]i with respect to the horizontal ([pic]i gt; [pic]), as shown in Figure P4. 50. a) Show that the projectile travels a distance d up the incline, where [pic] (b) For what value of [pic]i is d a maximum, and what is that maximum value? [pic] Figure P4. 50 51. Barry Bonds hits a home run so that the baseball just clears the top row of bleachers, 21. 0 m high, located 130 m from home plate. The ball is hit at an angle of 35. 0 ° to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time at which the ball reaches the cheap seats, and (c) the velocity components and the speed of the ball when it passes over the top row. Assume the ball is hit at a height of 1. 00 m above the ground. 52. An astronaut on the surface of the Moon fires a cannon to launch an experiment package, which leaves the barrel moving horizontally. (a) What must be the muzzle speed of the probe so that it travels completely around the Moon and returns to its original location? (b) How long does this trip around the Moon take? Assume that the free-fall acceleration on the Moon is one-sixth that on the Earth. 53. A pendulum with a cord of length r = 1. 00 m swings in a vertical plane (Fig. P4. 3). When the pendulum is in the two horizontal positions [pic] = 90. 0 ° and [pic] = 270 °, its speed is 5. 00 m/s. (a) Find the magnitude of the radial acceleration and tangential acceleration for these positions. (b) Draw vector diagrams to determine the direction of the total acceleration for these two positions. (c) Calculate the magnitude and direction of the total acceleration. [pic] Figure P4. 53 54. A basketball player who is 2. 00 m tall is standing on the floor 10. 0 m from the basket, as in Figure P4. 54. If he shoots the ball at a 40. ( angle with the horizontal, at what initial speed must he throw so that it goes through the hoop without striking the backboard? The basket height is 3. 05 m. [pic] Figure P4. 54 55. When baseball players throw the ball in from the outfield, they usually allow it to take one bounce before it reaches the infield, on the theory that the ball arrives sooner that way. Suppose that the angle at which a bounced ball leaves the ground is the same as the angle at which the outfielder threw it, as in Figure P4. 55, but that the ball’s speed after the bounce is one half of what it was before the bounce. a) Assuming the ball is always thrown with the same initial speed, at what angle ( should the fielder throw the ball to make it go the same distance D with one bounce (blue path) as a ball thrown upward at 45. 0 ° with no bounce (green path)? (b) Determine the ratio of the ti mes for the one-bounce and no-bounce throws. [pic] Figure P4. 55 56. A boy can throw a ball a maximum horizontal distance of R on a level field. How far can he throw the same ball vertically upward? Assume that his muscles give the ball the same speed in each case. 57. A stone at the end of a sling is whirled in a vertical circle of radius 1. 0 m at a constant speed v0 = 1. 50 m/s as in Figure P4. 57. The center of the sling is 1. 50 m above the ground. What is the range of the stone if it is released when the sling is inclined at 30. 0 ° with the horizontal (a) at A? (b) at B? What is the acceleration of the stone (c) just before it is released at A? (d) just after it is released at A? [pic] Figure P4. 57 58. A quarterback throws a football straight toward a receiver with an initial speed of 20. 0 m/s, at an angle of 30. 0 ° above the horizontal. At that instant, the receiver is 20. 0 m from the quarterback. In what direction and with what constant speed should the receiver run in order to catch the football at the level at which it was thrown? 59. Your grandfather is copilot of a bomber, flying horizontally over level terrain, with a speed of 275 m/s relative to the ground, at an altitude of 3 000 m. (a) The bombardier releases one bomb. How far will it travel horizontally between its release and its impact on the ground? Neglect the effects of air resistance. (b) Firing from the people on the ground suddenly incapacitates the bombardier before he can call, â€Å"Bombs away! Consequently, the pilot maintains the plane’s original course, altitude, and speed through a storm of flak. Where will the plane be when the bomb hits the ground? (c) The plane has a telescopic bomb sight set so that the bomb hits the target seen in the sight at the time of release. At what angle from the vertical was the bomb sight set? 60. A high-powered rifle fires a bullet with a muzzle speed of 1. 00 km /s. The gun is pointed horizontally at a large bull’s eye target—a set of concentric rings—that is 200 m away. (a) How far below the extended axis of the rifle barrel does a bullet hit the target? The rifle is equipped with a telescopic sight. It is â€Å"sighted in† by adjusting the axis of the telescope so that it points precisely at the location where the bullet hits the target at 200 m. (b) Find the angle between the telescope axis and the rifle barrel axis. When shooting at a target at a distance other than 200 m, the marksman uses the telescopic sight, placing its crosshairs to â€Å"aim high† or â€Å"aim low† to compensate for the different range. Should she aim high or low, and approximately how far from the bull’s eye, when the target is at a distance of (c) 50. 0 m, (d) 150 m, or (e) 250 m? Note: The trajectory of the bullet is everywhere so nearly horizontal that it is a good approximation to model the bullet as fired horizontally in each case. What if the target is uphill or downhill? (f) Suppose the target is 200 m away, but the sight line to the target is above the horizontal by 30(. Should the marksman aim high, low, or right on? (g) Suppose the target is downhill by 30(. Should the marksman aim high, low, or right on? Explain your answers. 61. A hawk is flying horizontally at 10. 0 m/s in a straight line, 200 m above the ground. A mouse it has been carrying struggles free from its grasp. The hawk continues on its path at the same speed for 2. 00 seconds before attempting to retrieve its prey. To accomplish the retrieval, it dives in a straight line at constant speed and recaptures the mouse 3. 00 m above the ground. (a) Assuming no air resistance, find the diving speed of the hawk. (b) What angle did the hawk make with the horizontal during its descent? (c) For how long did the mouse enjoy free fall? 62. A person standing at the top of a hemispherical rock of radius R kicks a ball (initially at rest on the top of the rock) to give it horizontal velocity vi as in Figure P4. 62. a) What must be its minimum initial speed if the ball is never to hit the rock after it is kicked? (b) With this initial speed, how far from the base of the rock does the ball hit the ground? [pic] Figure P4. 62 63. A car is parked on a steep incline overlooking the ocean, where the incline makes an angle of 37. 0 ° below the horizontal. The negligent driver leaves the car in neutral, and the parking brakes are defective. Starting from rest at t = 0, the car rolls down the incline with a constant acceleration of 4. 00 m/s2, traveling 50. 0 m to the edge of a vertical cliff. The cliff is 30. 0 m above the ocean. Find (a) the speed of the car when it reaches the edge of the cliff and the time at which it arrives there, (b) the velocity of the car when it lands in the ocean, (c) the total time interval that the car is in motion, and (d) the position of the car when it lands in the ocean, relative to the base of the cliff. 64. A truck loaded with cannonball watermelons stops suddenly to avoid running over the edge of a washed-out bridge (Fig. P4. 64). The quick stop causes a number of melons to fly off the truck. One melon rolls over the edge with an initial speed vi = 10. 0 m/s in the horizontal direction. A cross-section of the bank has the shape of the bottom half of a parabola with its vertex at the edge of the road, and with the equation y2 = 16x, where x and y are measured in meters. What are the x and y coordinates of the melon when it splatters on the bank? [pic] Figure P4. 64 65. The determined coyote is out once more in pursuit of the elusive roadrunner. The coyote wears a pair of Acme jet-powered roller skates, which provide a constant horizontal acceleration of 15. 0 m/s2 (Fig. P4. 65). The coyote starts at rest 70. 0 m from the brink of a cliff at the instant the roadrunner zips past him in the direction of the cliff. a) If the roadrunner moves with constant speed, determine the minimum speed he must have in order to reach the cliff before the coyote. At the edge of the cliff, the roadrunner escapes by making a sudden turn, while the coyote continues straight ahead. His skates remain horizontal and continue to operate while he is in flight, so that the coyote’s accel eration is [pic]. (b) If the cliff is 100 m above the flat floor of a canyon, determine where the coyote lands in the canyon. (c) Determine the components of the coyotes impact velocity. [pic] Figure P4. 65 66. Do not hurt yourself; do not strike your hand against anything. Within these limitations, describe what you do to give your hand a large acceleration. Compute an order-of-magnitude estimate of this acceleration, stating the quantities you measure or estimate and their values. 67. A skier leaves the ramp of a ski jump with a velocity of 10. 0 m/s, 15. 0 ° above the horizontal, as in Figure P4. 67. The slope is inclined at 50. 0 °, and air resistance is negligible. Find (a) the distance from the ramp to where the jumper lands and (b) the velocity components just before the landing. How do you think the results might be affected if air resistance were included? Note that jumpers lean forward in the shape of an airfoil, with their hands at their sides, to increase their distance. Why does this work? ) [pic] Figure P4. 67 68. In a television picture tube (a cathode ray tube) electrons are emitted with velocity vi from a source at the origin of coordinates. The initial velocities of di fferent electrons make different angles [pic] with the x axis. As they move a distance D along the x axis, the electrons are acted on by a constant electric field, giving each a constant acceleration a in the x direction. At x = D the electrons pass through a circular aperture, oriented perpendicular to the x axis. At the aperture, the velocity imparted to the electrons by the electric field is much larger than vi in magnitude. Show that velocities of the electrons going through the aperture radiate from a certain point on the x axis, which is not the origin. Determine the location of this point. This point is called a virtual source, and it is important in determining where the electron beam hits the screen of the tube. 69. A fisherman sets out upstream from Metaline Falls on the Pend Oreille River in northwestern Washington State. His small boat, powered by an outboard motor, travels at a constant speed v in still water. The water flows at a lower constant speed vw. He has traveled upstream for 2. 00 km when his ice chest falls out of the boat. He notices that the chest is missing only after he has gone upstream for another 15. 0 minutes. At that point he turns around and heads back downstream, all the time traveling at the same speed relative to the water. He catches up with the floating ice chest just as it is about to go over the falls at his starting point. How fast is the river flowing? Solve this problem in two ways. a) First, use the Earth as a reference frame. With respect to the Earth, the boat travels upstream at speed v – vw and downstream at v + vw. (b) A second much simpler and more elegant solution is obtained by using the water as the reference frame. This approach has important applications in many more complicated problems; examples are calculating the motion of rockets and satellites a nd analyzing the scattering of subatomic particles from massive targets. 70. The water in a river flows uniformly at a constant speed of 2. 50 m/s between parallel banks 80. 0 m apart. You are to deliver a package directly across the river, but you can swim only at 1. 50 m/s. (a) If you choose to minimize the time you spend in the water, in what direction should you head? (b) How far downstream will you be carried? (c) What If? If you choose to minimize the distance downstream that the river carries you, in what direction should you head? (d) How far downstream will you be carried? 71. An enemy ship is on the east side of a mountain island, as shown in Figure P4. 71. The enemy ship has maneuvered to within 2 500 m of the 1 800-m-high mountain peak and can shoot projectiles with an initial speed of 250 m/s. If the western shoreline is horizontally 300 m from the peak, what are the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship? [pic] 72. In the What If? section of Example 4. 7, it was claimed that the maximum range of a ski jumper occurs for a launch angle [pic] given by [pic] where [pic] is the angle that the hill makes with the horizontal in Figure 4. 16. Prove this claim by deriving the equation above. Figure P4. 71  © Copyright 2004 Thomson. All rights reserved.