Reference no: EM132636918

PROBLEMS

Components of Velocity and Acceleration†

The Velocity and Acceleration Vectors

Question 1. A sailboat tacking against the wind moves as follows: 3.2 km at 45 east of north, 4.5 km at 50 west of north, and 2.6 km at 45 east of north. The entire motion takes 1 h

8. The components of the position of a body as a function of time are given by: x  5t  4t2
y  3t 2  2t3 z0
15 min.

(a) What is the total displacement for this motion?

(b) What is the average velocity for this motion?

(c) What is the speed if it is assumed to be constant?

Question 2. In one-half year, the Earth moves halfway around its orbit, a circle of radius 1.50  10 11 m centered on the Sun. What is the average speed, and what is the magnitude of the average velocity for this time interval?

Question 3. The fastest bird is the spine-tailed swift, which reaches speeds of 171 km/h. Suppose that you wish to shoot such a bird with a .22-caliber rifle that fires a bullet with a speed of 366 m/s. If you fire at the instant when the bird is 30 m directly overhead, how many meters ahead of the bird must you aim the rifle? Ignore gravity in this problem.

Question 4. An automobile with a drunken driver at the wheel travels round and round a traffic circle at 30 km/h. The automobile takes 80 s to go once around the circle. At t  0, the automobile is at the east of the traffic circle; at t  20 s it is at the north; at t  40 s it is at the west; etc. What are the components of the velocity of the automobile at t  0, t  10 s, t  20 s, t  30 s, and t  40 s? The x axis points eastward and the y axis points northward.

Question 5. Suppose that a particle moving in three dimensions has a position vector r  (4  2t)i  (3  5t  4t 2 )j  (2  2t  3t2 )k

where distance is measured in meters and time in seconds.

(a) Find the instantaneous velocity vector.

(b) Find the instantaneous acceleration vector. What are the magnitude and the direction of the acceleration?

Question 6. A particle is moving in the x-y plane; the components of its position are x  A cos bt y  A sin bt where A and b are constants.

(a) What are the components of the instantaneous velocity vector? The instantaneous acceleration vector?

(b) What is the magnitude of the instantaneous velocity? The instantaneous acceleration?

Question 7. For the motion of the cruise missile described in Example 3, calculate the displacement of the missile relative to the aircraft at t  2.0 s and at t  3.0 s. What are the magnitude and the direction of the displacement vector at each of these times?

Question 8. where x and y are in meters and t is in seconds. What is the velocity vector as a function of time? What is the acceleration vector as a function of time? What is the speed at t ¾ 2.0 s?

Question 9. An airplane traveling at a constant speed of 300 km/h flies 30¾ north of east for 0.50 h and then flies 30¾ west of south for 1.00 h. What is the average velocity vector for the entire flight? What is the average acceleration vector for the entire flight?

Question 10. The components of the position vector of a particle moving in the x-y plane are x  A cos bt y  Bt where A, b, and B are constants. What are the components of the instantaneous velocity vector? The instantaneous acceleration vector? What is the speed of the particle?

Question 11. As an aircraft approaches landing, the components of its position are given by x  90t y  500  15t where x and y are in meters and t is in seconds. What is the velocity vector of the aircraft during this descent? What is the value of its speed during the descent? What angle does the velocity vector make with the horizontal?

Question 12. Two football players are initially 20 m apart. The first player (a receiver) runs perpendicularly to the initial line joining the two players at a constant speed of 7.0 m/s. After two seconds, the second player (the quarterback) throws the ball at a horizontal speed of 15 m/s (ignore any vertical motion). In what horizontal direction should the quarterback aim so that the ball reaches the same spot the receiver will be? At what time will the ball be caught?

Motion with Constant Acceleration

The Motion of Projectiles†

Question 13. Suppose that the acceleration vector of a particle moving in the x-y plane is a 3i 2j where the acceleration is measured in m/s2 . The vector and the position vector are zero at t ¾ 0.

velocity

(a) What is the velocity vector of this particle as a function of time?

(b) What is the position vector as a function of time?

Question 14. The fastest recorded speed of a baseball thrown by a pitcher is

162.3 km/h (100.9 mi/h), achieved by Nolan Ryan in 1974 at

CHAPTER 4

Motion in Two and Three Dimensions

Anaheim Stadium. If the baseball leaves the pitcher's hand with a horizontal velocity of this magnitude, how far will the ball have fallen vertically by the time it has traveled 20 m horizontally?

Question 15. At Acapulco, professional divers jump from a 36-m-high cliff into the sea (compare Example 9 in Chapter 2). At the base of the cliff, a rocky ledge sticks out for a horizontal distance of

6.4 m. With what minimum horizontal velocity must the divers jump off if they are to clear this ledge?

Question 16. Consider the bomb dropped from the bomber described in Example 6.

(a) What are the final horizontal and vertical components of the velocity of the bomb when it strikes the surface of the sea?

(b) What is the final speed of the bomb? Compare this with the initial speed of the bomb.

Question 17. A stunt driver wants to make his car jump over 10 cars parked side by side below a horizontal ramp (Fig. 4.29). With what minimum speed must he drive off the ramp? The vertical height of the ramp is 2.0 m, and the horizontal distance he must clear is 24 m.

Question 18. A particle has an initial position vector r ¾ 0 and an initial velocity v 0 ¾ 3i ¾ 2j (where distance is measured in meters and velocity in meters per second). The particle moves with a constant acceleration a ¾ i ¾ 4j (measured in m/s2 ). At what time does the particle reach a maximum y coordinate? What is the position vector of the particle at that time?

Question 19. According to a reliable report, in 1795 a member of the Turkish embassy in England shot an arrow to a distance of 441 m. According to a less reliable report, a few years later the Turkish Sultan Selim shot an arrow to 889 m. In each of these cases calculate what must have been the minimum initial speed of the arrow.

Question 20. A golfer claims that a golf ball launched with an elevation angle of 12¾ can reach a horizontal range of 250 m. Ignoring air friction, what would the initial speed of such a golf ball have to be? What maximum height would it reach?

Question 21. According to the Guinness Book of World Records, during a catastrophic explosion in Halifax on December 6, 1917, William Becker was thrown through the air for some 1500 m and was found, still alive, in a tree. Assume that Becker left the ground and returned to the ground (ignore the height of the tree) at an angle of 45¾. With what speed did he leave the ground? How high did he rise? How long did he stay in flight?

Question 22. In a circus act at the Ringling Bros. and Barnum & Bailey Circus, a "human cannonball" was fired from a large cannon with a muzzle speed of 87 km/h. Assume that the firing angle was 45¾ from the horizontal. How many seconds did the human cannonball take to reach maximum height? How high did he rise? How far from the cannon did he land?

Question 23. The world record for the javelin throw by a woman established in 1976 by Ruth Fuchs in Berlin was 69.11 m (226 ft 9 in.). If Fuchs had thrown her javelin with the same initial velocity in Buenos Aires rather than in Berlin, how much farther would it have gone? The acceleration of gravity is 9.8128 m/s 2 in Berlin and 9.7967 m/s 2 in Buenos Aires. Pretend that air resistance plays no role in this problem.

Question 24. The motion of an ICBM can be regarded as the motion of a projectile, because along the greatest part of its trajectory the missile is in free fall, outside of the atmosphere. Suppose that the missile is to strike a target 1000 km away. What minimum speed must the missile have at the beginning of its trajectory? What maximum height does it reach when launched with this minimum speed? How long does it take to reach its target? For these calculations assume that g
¾ 9.8 m/s 2 everywhere along the trajectory and ignore the (short) portions of the trajectory inside the atmosphere.

Question 25. The natives of the South American Andes throw stones by means of slings which they whirl (see Fig. 4.30). They can accurately throw a 0.20-kg stone to a distance of 50 m.

(a) What is the minimum speed with which the stone must leave the sling to reach this distance?

(b) Just before the release, the stone is being whirled around a circle of radius 1.0 m with the speed calculated in part

Question 26. A gunner wants to fire a gun at a target at a horizontal distance of 12500 m from his position.

(a) If his gun fires with a muzzle speed of 700 m/s and if g ¾ 9.81 m/s2 , what is the correct elevation angle? Pretend that there is no air resistance.

(b) If the gunner mistakenly assumes g ¾ 9.80 m/s2 , by how many meters will he miss the target?

(a). How many revolutions per second does the stone make? v0

2.0 m 24 m
FIGURE 4.29 A stunt.

FIGURE 4.30 Whirling a stone before slinging. Problems 123

Question 27. The nozzle of a fire hose ejects 280 liters of water per minute at a speed of 26 m/s. How far away will the stream of water land if the nozzle is aimed at an angle of 35¾ with the horizontal? How many liters of water are in the air at any given instant?

Question 28. According to an ancient Greek source, a stone-throwing machine on one occasion achieved a range of 730 m. If this is true, what must have been the minimum initial speed of the stone as it was ejected from the engine? When thrown with this speed, how long would the stone have taken to reach its target?

Question 29. For what launch angle will the height and range of a projectile be equal?

Question 30. A juggler tosses and catches balls at waist level; the balls are tossed at launch angles of 60¾. If a ball attains a height 60 cm above waist level, how long is a ball in the air?

Question 31. At t ¾ 0, a small particle begins at the origin with initial velocity components v 0x ¾ ¾10 m/s and v 0y ¾ 25 m/s. Throughout its motion, the particle experiences an acceleration a ¾ (2.0i ¾ 4.5j) m/s2 . Find the speed of the particle at t ¾ 3.0 s. Find the position vector of the particle at t
¾ 3.0 s.

Question 32. A baseball is popped up, remaining aloft for 6.0 s before being caught at a horizontal distance of 75 m from the starting point. What was the launch angle?

Question 33. An errant speeding bus launches from an unfinished highway ramp angled 10¾ upward. To complete the jump across a horizontal roadway gap of 15 m, what minimum initial speed must the bus have?

Question 34. A child rolls a ball horizontally off the edge of a table. For what initial speed will the ball strike the floor a horizontal distance away from the table edge equal to the table height? In that case, what is the velocity of the ball just before it hits the floor?

Question 35. A boy stands at the edge of a cliff and launches a rock upward at an angle of 45.0¾. The rock comes back down to the elevation where it was released 2.25 s later, then continues until it is seen to splash into the lake below 4.00 s after release. How far below the point of release is the lake surface? What horizontal distance from the point of release is the splash?

Question 36. A rock is thrown from a bridge at an upward launch angle of 30¾ with an initial speed of 25 m/s. The bridge is 30 m above the river. How much time elapses before the rock hits the water?

Question 37. A hockey player 25 m from the goal hits the hockey puck

toward the goal, imparting a launch speed of 65 m/s at a launch angle of 10¾. If the goal is 1.5 m high, does the shot score? At what vertical height does the puck pass the goal? How long does the puck take to reach the goal?

Question 38. (a) A golfer wants to drive a ball to a distance of 240 m. If he launches the ball with an elevation angle of 14.0¾, what is the appropriate initial speed? Ignore air resistance.

(b) If the speed is too great by 0.6 m/s, how much farther will the ball travel when launched at the same angle?

(c) If the elevation angle is 0.5¾ larger than 14.0¾, how much farther will the ball travel if launched with the speed calculated in part (a)?

Question 39. Show that for a projectile launched with an elevation angle of 45¾, the maximum height reached is one-quarter of the range.

Question 40. During a famous jump in Richmond, Virginia, in 1903, the

horse Heatherbloom with its rider jumped over an obstacle 8 ft 8 in. high while covering a horizontal distance of 37 ft. At what angle and with what speed did the horse leave the ground? Make the (somewhat doubtful) assumption that the motion of the horse is particle motion.

Question 41. With what elevation angle must you launch a projectile if its range is to equal twice its maximum height?

Question 42. In a baseball game, the batter hits the ball and launches it

upward at an angle of 52¾ with a speed of 38 m/s. At the same instant, the center fielder starts to run toward the (expected) point of impact of the ball from a distance of 45 m. If he runs at
8.0 m/s, can he reach the point of impact before the ball?

Question 43. The gun of a coastal battery is emplaced on a hill 50 m above the water level. It fires a shot with a muzzle speed of 600 m/s at a ship at a horizontal distance of 12000 m. What elevation angle must the gun have if the shot is to hit the ship? Pretend there is no air resistance.

Question 44. In a flying ski jump, the skier acquires a speed of 110 km/h by racing down a steep hill and then lifts off into the air from a horizontal ramp. Beyond this ramp, the ground slopes downward at an angle of 45¾.

(a) Assuming that the skier is in a free-fall motion after he leaves the ramp, at what distance down the slope will he land?

(b) In actual jumps, skiers reach distances of up to 165 m.

Why does this not agree with the result you obtained in part (a)?

Question 45. Olympic target archers shoot arrows at a bull's-eye 12 cm

across from a distance of 90.00 m. If the initial speed of the arrow is 70.00 m/s, what must be the elevation angle? If the archer misaims the arrow by 0.03¾ in the vertical direction, will it hit the bull's-eye? If the archer misaims the arrow by 0.03¾ in the horizontal direction, will it hit the bull's-eye? Assume that the height of the bull's-eye above the ground is the same as the initial arrow height of the bow and ignore air resistance.

Question 46. The muzzle speed for a Lee-Enfield rifle is 630 m/s. Suppose you fire this rifle at a target 700 m away and at the same level as the rifle.

(a) In order to hit the target, you must aim the barrel at a point above the target. How many meters above the target must you aim? Pretend there is no air resistance.

(b) What will be the maximum height that the bullet reaches along its trajectory?

(c) How long does the bullet take to reach the target?

Question 47. In artillery, it is standard practice to fire a sequence of trial

shots at a target before commencing to fire "for effect." The artillerist first fires a shot short of the target, then a shot beyond the target, and then makes the necessary adjustment in elevation so that the third shot is exactly on target. Suppose that the first shot fired from a gun aimed with an elevation 124

CHAPTER 4

Motion in Two and Three Dimensions

angle of 7¾20¾ lands 180 m short of the target; the second shot fired with an elevation of 7¾35¾ lands 120 m beyond the target. What is the correct elevation angle to hit the target?

where ¾ is the angle of the slope and the other symbols have their usual meaning. For what value of ¾ is this range a maximum?

Question 48. A hay-baling machine throws each finished bundle of hay

2.5 m up in the air so it can land on a trailer waiting 5.0 m behind the machine. What must be the speed with which the bundles are launched? What must be the angle of launch?

Question 49. Consider the trajectories for projectiles with the same launch speed, but different elevation angles. If you launch a large number of such projectiles simultaneously, will any of them ever collide while in flight? Explain carefully.

Question 50. Suppose that at the top of its parabolic trajectory a projectile has a horizontal speed v0x . The segment at the top of the parabola can be approximated by a circle, called the osculating circle (Fig. 4.31). What is the radius of this circle? (Hint: The projectile is instantaneously in uniform circular motion at the top of the parabola.)

FIGURE 4.31 The osculating circle.

Question 51. A battleship steaming at 45 km/h fires a gun at right angles to the longitudinal axis of the ship. The elevation angle of the gun is 30¾, and the muzzle velocity of the shot is 720 m/s; the gravitational acceleration is 9.8 m/s2 . What is the range of this shot in the reference frame of the ground? Pretend that there is no air resistance.

Question 52. The maximum speed with which you can throw a stone is

about 25 m/s (a professional baseball pitcher can do much better than this). Can you hit a window 50 m away and 13 m up from the point where the stone leaves your hand? What is the maximum height of a window you can hit at this distance?

Question 53. A gun standing on sloping ground (see Fig. 4.32) fires up the slope. Show that the slant range of the gun (measured along the slope) is

2v 0 2 cos 2 u l ¾ (tan u ¾ tan ¾) g cos ¾

l θ a
FIGURE 4.32 Projectile motion up a slope.

Question 54. Two football players are initially 15 m apart. The first player (a receiver) runs perpendicular to the line joining the two players at a constant speed of 8.0 m/s. After two seconds, the second player (the quarterback) throws the ball with a horizontal component velocity of 20 m/s. In what horizontal direction and with what vertical launch angle should the

quarterback throw so that the ball reaches the same spot the receiver will be? At what time will the ball be caught?

Question 55. When a tractor leaves a muddy field and drives on the high-

way, clumps of mud will sometimes come off the rear wheels and be launched into the air (see Fig. 4.33). In terms of the speed u of the tractor and the radius R of the wheel, find the maximum possible height that a clump of dirt can reach. In your calculation be careful to take into account both the initial velocity of the clump and the initial height at which it comes off the wheel. Evaluate numerically for u ¾ 30 km/h and R¾0.80 m. (Hint: Solve this problem in the reference frame of the tractor.)

u R
FIGURE 4.33 Tractor wheel flinging mud.

Question 56. A gun on the shore (at sea level) fires a shot at a ship which is heading directly toward the gun at a speed of 40 km/h. At the instant of firing, the distance to the ship is 15000 m. The muzzle speed of the shot is 700 m/s. Pretend that there is no air resistance.

(a) What is the required elevation angle for the gun? Assume g ¾ 9.8 m/s2 .
(b) What is the time interval between firing and impact?

Question 57. A ship is steaming at 30 km/h on a course parallel to a straight shore at a distance of 17000 m. A gun emplaced on the shore (at sea level) fires a shot with a muzzle speed of 700 m/s when the ship is at the point of closest approach. If the shot is to hit the ship, what must be the elevation angle of the gun? How far ahead of the ship must the gun be aimed? Give the answer to the latter question both in meters and in minutes of arc. Pretend that there is no air resistance. (Hint: Solve this problem by the following method of successive approximations.
First calculate the time of flight of the shot, neglecting the motion of the ship; then calculate how far the ship moves in this time; and then calculate the elevation angle and the aiming angle required to hit the ship at this new position.) Problems

125

4.5 Uniform Circular Motion

Question 58. An audio compact disk (CD) player is rotating at an angular velocity of 32.5 radians per second when playing a track at a radius of 4.0 cm. What is the linear speed at that radius? What is the rotation rate in revolutions per minute?

Question 59. In science fiction movies, large, ring-shaped space stations rotate so that astronauts experience an acceleration, which feels the same as gravity. If the station is 200 m in radius, how many revolutions per minute are required to provide an acceleration of 9.81 m/s2 ?

Question 60. When drilling metals, excess heat is avoided by staying below a recommended linear cutting speed. A 3.0-mm-diameter hole and a 25-mm-diameter hole need to be drilled. At what maximum number of revolutions per minute can the drill bit rotate so that a point on its perimeter does not exceed the material's linear cutting speed limit of 3.0 m/s?

Question 61. The Space Shuttle orbits the Earth on a circle of radius 6500 km every 87 minutes. What is the centripetal acceleration of the Space Shuttle in this orbit?

Question 62. A mechanical pitcher hurls baseballs for batting practice. The arm of the pitcher is 0.80 m long and is rotating at 45 radians/second at the instant of release. What is the speed of the pitched ball?

Question 63. An ultracentrifuge spins a small test tube in a circle of radius 10 cm at 1000 revolutions per second. What is the centripetal acceleration of the test tube? How many standard g's does this amount to?

Question 64. The blade of a circular saw has a diameter of 20 cm. If this blade rotates at 7000 revolutions per minute (its maximum safe speed), what are the speed and the centripetal acceleration of a point on the rim?

Question 65. At the Fermilab accelerator (one of the world's largest atom smashers), protons are forced to travel in an evacuated tube in a circular orbit of diameter 2.0 km (Fig. 4.34). The protons

FIGURE 4.34 The main accelerator ring at Fermilab.

have a speed nearly equal to the speed of light (99.99995% of the speed of light). What is the centripetal acceleration of these protons? Express your answer in m/s 2 and in standard g's.

Question 66. A phonograph record rotates at 33 3 1 revolutions per minute.

The radius of the record is 15 cm. What is the speed of a point at its rim?

Question 67. The Earth moves around the Sun in a circular path of radius

1.50 ¾ 10 11 m at uniform speed. What is the magnitude of the centripetal acceleration of the Earth toward the Sun?

Question 68. An automobile has wheels of diameter 64 cm. What is the centripetal acceleration of a point on the rim of this wheel when the automobile is traveling at 95 km/h?

Question 69. The Earth rotates about its axis once in one sidereal day of

23 h 56 min. Calculate the centripetal acceleration of a point located on the equator. Calculate the centripetal acceleration of a point located at a latitude of 45¾.

Question 70. When looping the loop, the Blue Angels stunt pilots of the

U.S. Navy fly their jet aircraft along a vertical circle of diameter 1000 m (Fig. 4.35). At the top of the circle, the speed is 350 km/h; at the bottom of the circle, the speed is 620 km/h. What is the centripetal acceleration at the top? At the bottom? In the reference frame of one of these aircraft, what is the acceleration that the pilot feels at the top and at the bottom; i.e., what is the acceleration relative to the aircraft of a small body, such as a coin, released by the pilot?

Question 71. The table inside the book cover lists the radii of the orbits of the planets around the Sun and the time taken to complete an orbit ("period of revolution"). Assume that the planets move along circles at constant speed. Calculate the centripetal acceleration for each of the first three planets (Mercury, Venus, Earth). Verify that the centripetal accelerations are in proportion to the inverses of the squares of the orbital radii.

FIGURE 4.35 Blue Angels looping the loop. 126 CHAPTER 4
Motion in Two and Three Dimensions

4.6 The Relativity of Motion and the Addition of Velocities

Question 72. On a rainy day, a steady wind is blowing at 30 km/h. In the reference frame of the air, the raindrops are falling vertically with a speed of 10 m/s. What are the magnitude and the direction of the velocity of the raindrops in the reference frame of the ground?

Question 73. In an airport, a moving walkway has a speed of 1.5 m/s relative to the ground. What is the speed, relative to the ground, of a passenger running forward on this walkway at 4.0 m/s? What is the speed, relative to the ground, of a passenger running backward on this walkway at 4.0 m/s?

Question 74. On a rainy day, raindrops are falling with a vertical velocity of 10 m/s. If an automobile drives through the rain at 25 m/s, what is the velocity (magnitude and direction) of the raindrops relative to the automobile?

Question 75. A battleship steaming at 13 m/s toward the shore fires a shot in the forward direction. The elevation angle of the gun is 20¾, and the muzzle speed of the shot is 660 m/s. What is the velocity vector of the shot relative to the shore?

Question 76. A wind of 30 m/s is blowing from the west. What will be the speed, relative to the ground, of a sound signal traveling due north? The speed of sound, relative to air, is 330 m/s.

Question 77. On a windy day, a hot-air balloon is ascending at a rate of

1.5 m/s relative to the air. Simultaneously, the air is moving with a horizontal velocity of 12.0 m/s. What is the velocity (magnitude and direction) of the balloon relative to the ground?

Question 78. You can paddle your kayak at a speed of 3.5 km/h relative to the water. If a river is flowing at 2.5 km/h, how far can you paddle downstream in 40 minutes? How long will it take you to paddle back upstream from there?

Question 79. As a train rolls by at 5.00 m/s, you see a cat on one of the flatcars. The cat is walking toward the back of the train at a speed of 0.50 m/s relative to the car. On the cat is a flea which is walking from the cat's neck to its tail at a speed of 0.10 m/s relative to the cat. How fast is the flea moving relative to you?

Question 80. A boat with maximum speed v (relative to the water) is on one shore of a river of width d. The river is flowing at speed V. Traveling in a straight line, how long does it take to get to a point directly opposite? What is the fastest crossing time to any point?

Question 81. Each step on an up escalator is 20 cm high and 30 cm deep.The escalator advances 1.5 step per second. If you also walk up the escalator stairs at a rate of 1.0 step per second, what is your velocity (magnitude and direction) relative to a fixed observer?

Question 82. A villain in a car traveling at 30 m/s fires a projectile along the direction of motion toward the front of the car with a launch speed of 50 m/s relative to the car. A hero standing nearby observes the projectile to travel straight up. What was the launch angle as viewed by the villain? What height does the projectile attain?

Question 83. A blimp is motoring at constant altitude.The airspeed indica-

tor on the blimp shows that its speed relative to the air is 20 km/h, and the compass shows that the heading of the blimp is 10¾ east of north. If the air is moving over the ground with a velocity of 15 km/h due east, what is the velocity (magnitude and direction) of the blimp relative to the ground? For an observer on the ground, what is the angle between the longitudinal axis of the blimp and the direction of motion?

Question 84. A sailboat is moving in a direction 50¾ east of north at a speed of 14 km/h.The wind measured by an instrument aboard the sailboat has an apparent (relative to the sailboat) speed

of 32 km/h coming from an apparent direction of 10¾ east of north. Find the true (relative to ground) speed and direction of the wind.

Question 85. (a) In still air, a high-performance sailplane has a rate of

descent (or sinking rate) of 0.50 m/s at a forward speed (or airspeed) of 60 km/h. Suppose the plane is at an initial altitude of 1500 m. How far can it travel horizontally in still air before it reaches the ground?

(b) Suppose the plane is in a (horizontal) wind of 20 km/h.

With the same initial conditions, how far can it travel in the downwind direction? In the upwind direction?

Question 86. A wind is blowing at 50 km/h from a direction 45¾ west of

north.The pilot of an airplane wishes to fly on a route due north from an airport.The airspeed of the airplane is 250 km/h.

(a) In what direction must the pilot point the nose of the airplane?

(b) What will be the airplane's speed relative to the ground?

Question 87. At the entrance of Ambrose Channel at New York harbor, the tidal current at one time of the day has a velocity of 4.2 km/h in a direction 20¾ south of east. Consider a ship in this current; suppose that the ship has a speed of 16 km/h relative to the water. If the helmsman keeps the bow of the ship aimed due north, what will be the actual velocity (magnitude and direction) of the ship relative to the ground?

Question 88. A white automobile is traveling at a constant speed of

90 km/h on a highway. The driver notices a red automobile 1.0 km behind, traveling in the same direction. Two minutes later, the red automobile passes the white automobile.

(a) What is the average speed of the red automobile relative to the white?

(b) What is the speed of the red automobile relative to the ground?

Question 89. Two automobiles travel at equal speeds in opposite directions on two separate lanes of a highway. The automobiles move at constant speed v 0 on straight parallel tracks separated by a distance h. Find a formula for the rate of change of the distance between the automobiles as a function of time; take the instant of closest approach as t ¾ 0. Plot v vs. t for v 0 ¾ 60 km/h, h ¾ 50 m. Review Problems

127

Question 90. A ferryboat on a river has a speed v relative to the water. The water of the river flows with speed V relative to the ground. The width of the river is d.

(a) Show that the ferryboat takes a time 2d/2v2 travel across the river and back.

¾V2 to
(b) Show that the ferryboat takes a time 2dv/(v 2 ¾ V 2 ) to travel a distance d up the river and back. Which trip takes a shorter time?

Question 91. An AWACS aircraft is flying at high altitude in a wind of

150 km/h from due west. Relative to the air, the heading of the aircraft is due north and its speed is 750 km/h. A radar operator on the aircraft spots an unidentified target approaching from northeast; relative to the AWACS aircraft, the bearing of the target is 45¾ east of north, and its speed is 950 km/h. What is the speed of the unidentified target relative to the ground?

Attachment:- Problems.rar