Answers. (d) What is the angle between the positive direction of the x axis and a line tangent to the particle's path at t = 2.00 s?

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1 1 A particle moving in a plane has a position vector r = 2t i + 8t 3 j + k in SI units. By finding the acceleration, prove that the particle does not move in a parabola. 6 The position r of a particle moving in an xy plane is given by r = (2.00t t)i + ( t 4 ) with r in meters and t in seconds. (a) In unit-vector notation, calculate r for t = 2.00 s. (b) In unit-vector notation, calculate v for t = 2.00 s. (c) In unit-vector notation, calculate a for t = 2.00 s. (d) What is the angle between the positive direction of the x axis and a line tangent to the particle's path at t = 2.00 s? 2 At time to the velocity of an object is given in feet per second, by v 0 = 125 i + 25 j. At 3.0 s later the velocity is v = 100 i - 75 j. What was the average acceleration of the object during this time interval? (a) (6.00 m)ˆi - (106 m)ˆj; (b) (19.0 m/s)ˆi - (224m/s)ˆj; (c) (24.0m/s 2 )ˆi - (336m/s 2 )ˆj; (d) i-33.3j, ft/s 2 or ft/s 2 Here are four descriptions of the position (in meters) of a puck as it moves in an xy plane: (1) x = -3t 2 + 4t 2 and y = 6t 2-4t (2) x -3t 3-4t and y = -5t (3) r = 2t 2 i - (4t + 3)j (4) r = (4t 2-2t)i + 3j 7 A room has dimensions 3.00 m (height) x 3.70 m x 4.30 m. A fly starting at one corner flies around, ending up at the diagonally opposite corner. (a) What is the magnitude of its displacement? (b) Could the length of its path be less than this magnitude? (c) Could the length of its path be greater than this magnitude? (d) Could the length of its path be equal to this magnitude? (e) Choose a suitable coordinate system and express the components of the displacement vector in that system in unitvector notation. (f) If the fly walks rather than flies, what is the length of the shortest path it can take? (Hint.This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.) For each description, determine whether the x and y components of the puck's acceleration are constant and whether the acceleration a is constant. (1) and (3) a x and a y are both constant, so a is constant; (2) and (4) a y is constant but a x is not, so a is not (a) 6.42 m; (b) no; (c) yes; (d) yes; (e) a possible answer: (4.30 m)ˆi + (3.70 m)ˆj + (3.00 m) ˆk; (f) 7.96 m 4 A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn. The coordinates of the rabbit's position as functions of time t are given by x = -0.31t t + 28 and y = 0.22t 2-9.1t + 30, with t in seconds and x and y in meters. (a) At t = 15 s, what is the rabbit's position vector y in unit vector notation and in magnitude-angle notation? (b) Graph the rabbit's path for t = 0 to t = 25 s. 8 The rock shown below is acted on by one or more forces. Draw the free body diagram for the rock at the top of a parabolic trajectory. 5 If the position of a hobo's marble is given by r = (4t 3-2t) i + 3j with r in meters and t in seconds, what must be the units of the coefficients 4, -2, and 3? ANSWERS 4 m/s 2, -2 m/s, 3 m

2 9 A 3.00 kg ball is dropped from the roof of a building m high. While the ball is falling to Earth, a horizontal wind exerts a constant force of 12.0 N on it. (a) Draw the free body diagram. (b) How long does it take the ball to hit the ground? (c) How far from the building does the ball hit the ground? (d) What is its speed and angle when it hits the ground? (e) With what force and angle will the ball hit the ground? 12 A bank in downtown Boston is robbed (see the map ). To elude police, the robbers escape by helicopter, making three successive flights described by the following displacements: 32 km, 45 0 south of east; 53 km, 26 0 north of west; 26 km, 18 0 east of south. At the end of the third flight they are captured. In what town are they apprehended? (Give your answere in unit vector notation) ANSWERS Good review problem of projectile motion (a) (b) 72.0 m (c) 6.00 s (d) 63.6 m/s (e) N A 3.00 kg ball is dropped from the roof of a building m high. While the ball is falling to Earth, a horizontal wind exerts a constant force of 12.0 N on it. (a) Draw the free body diagram. (b) How long does it take the ball to hit the ground? (c) How far from the building does the ball hit the ground? (d) What is its speed and angle when it hits the ground? (e) With what force and angle will the ball hit the ground? Walpole (where the state prison is located) ANSWERS Good review problem of projectile motion (a) (b) 72.0 m (c) 6.00 s (d) 63.6 m/s (e) N The figure shows the path taken by a skunk foraging for trash food, from initial point 1. The skunk took the same time T to go from each labeled point to the next along its path. Rank points a, b, and c according to the magnitude of the average velocity of the skunk to reach them from initial point i, greatest first. 11 A 3.00 kg ball is dropped from the roof of a building m high. While the ball is falling to Earth, a horizontal wind exerts a constant force of 12.0 N on it. (a) Draw the free body diagram. (b) How long does it take the ball to hit the ground? (c) How far from the building does the ball hit the ground? (d) What is its speed and angle when it hits the ground? (e) With what force and angle will the ball hit the ground? a and c tie, then b ANSWERS Good review problem of projectile motion (a) (b) 72.0 m (c) 6.00 s (d) 63.6 m/s (e) N

3 14 In the figure, a radar station detects an airplane approaching directly from the east. At first observation, the airplane is at distance d 1 = 360 m from the station and at angle θ 1 = 40 0 above the horizon. The airplane is tracked through an angular change θ = in the vertical east-west plane; its distance is then d 2 = 790 m. (a) Find the magnitude of the airplanels displacement during this period. (b) Find the direction of the airplanels displacement during this period. 16 Rock faults are ruptures along which opposite faces of rock have slid past each other. In the figure, points A and B coincided before the rock in the foreground slid down to the right. The net displacement AB is along the plane of the fault. The horizontal component of AB is the strike-slip AC. The component of AB that is directed down the plane of the fault is the dip-slip AD. (a) What is the magnitude of the net displacement AB if the strike-slip is 22.0 m and the dip-slip is 17.0 m? (b) If the plane of the fault is inclined at angle φ = to the horizontal, what is the vertical component of AB? 15 (a) 1030 m; (b) west A wheel with a radius of 45.0 cm rolls without slipping along a horizontal floor (see the figure). At time t 1, the dot P painted on the rim of the wheel is at the point of contact between the wheel and the floor. At a later time t 2, the wheel has rolled through one-half of a revolution. (a) What is the magnitude (relative to the floor) of the displacement of P? (b) What is the angle (relative to the floor) of the displacement of P? 17 (a) 27.8 m; (b) 13.4 m Answer the following questions: (a) If a wily bat flies from xyz coordinates (-2 m, 4 m, -3 m) to coordinates (6 m, -2 m, -3 m), what is its displacement r in unit-vector notation? (b) Is r parallel to one of the three coordinate planes? If so, which plane? (a) (8 m)ˆi - (6 m)ˆj; (b) yes, the xy plane (no z component) (a) 168 cm; (b) 32.5

4 18 The figure gives the path of a squirrel moving about on level ground, from point A (at time t = 0), to points B (at t = 5.00 min), C (at t = 10.0 min), and finally D (at t = 15.0 min). Consider the average velocities of the squirrel from point A to each of the other three points. (a) Of those three average velocities, what is the magnitude of the one with the least magnitude? (b) Of those three average velocities, what is the angle of the one with the least magnitude? What are the (c) Of those three average velocities, what is the magnitude of the one with the greatest magnitude? (d) Of those three average velocities, what is the angle of the one with the greatest magnitude? 20 The New Hampshire State Police use aircraft to enforce highway speed limits. Suppose that one of these airplanes has a speed of 135 mi./h in still air. It is flying straight north so that it is at all times directly above a north-south highway. A ground observer tells the pilot by radio that a 70-mi./h wind is blowing; but neglects to give him the wind direction. The pilot observes that in spite of the wind he can travel 135 mi. along the highway in 1 h. In other words, his ground speed is the same as if there were no wind. (a) What is the direction of the wind? (b) What is the heading of the plane, that is, the angle between the axis along its length and the highway? (a) 75 degrees East of South (b) 30 degrees East of North 21 A light plane attains an airspeed of 500 km/h The pilot sets out for a destination 800 km to the north but discovers that the plane must be headed east of north to fly there directly. The plane arrives in 2.00 h. What was the wind velocity vector? 19 a) 0.83 cm/s; (b) 0; (c) 0.11 m/s; (d) -63 A World War II battleship steams due east at 24 km/h. A submarine 4.0 km away fires a torpedo that has a speed of 50 km/hr. The bearing of the ship as seen from the submarine is 20 0 east of north (a) In what direction should the torpedo be fired to hit the ship. (b) What will be the running time for the torpedo to reach the battleship? km/h, 22 0 south of west A plane flies 483 km east from city A to city B in 45.0 min and then 966 km south from city B to city C in 1.50 h. (a) For the total trip, what is the magnitude of the plane's displacement? (b) For the total trip, what is the direction of the plane's displacement? (c) For the total trip, what is the magnitude of the plane's displacement? (d) For the total trip, what is the direction of its average velocity? (e) For the total trip, what is the its average speed? (a) km; (b) 26.6 east of south; (c) 480km/h; (d) 26.6 east of south; (e) 644km/h (a) degrees (b) 6.55 min hr

5 23 In the figure particle A moves along the line y = 30 in with a constant velocity V of magnitude 3.0 m/s and parallel to the x axis. At the instant particle A passes the y axis, particle B leaves the origin with zero initial speed and constant accleration d of magnitude 0.40 m/s 2. What angle θ between a and the position. - direction of the y axis would result in a collision? 26 The pitcher in a slow-pitch softball game releases the ball at a point 3.0 ft above ground level. A stroboscopic plot of the position of the ball is shown in the figure, where the readings are 0.25 s apart and the ball is released at t = 0. (a) What is the initial speed of the ball? (b) What is the speed of the ball at the instant it reaches its maximum height above ground level? (c) What is that maximum height? 60 (a) 38 ft/s; (b) 32 ft/s; (c) 9.3 ft 24 The position r of a particle moving in the xy plane is given by r = 2ti + 2 sin[(π/4 rad/s)t]j, where r is in meters and t is in seconds. (a) Calculate the x and y components of the particle's position at t = 0, 1.0, 2.0, 3.0, and 4.0 s and sketch the particle's path in the xy plane for the interval 0 < = t < = 4.0 s. (b) Calculate the components of the particle's velocity at t = 1.0, 2.0, and 3.0 s. (c) Show that the velocity is tangent to the path of the particle and in the direction the particle is moving at each time by drawing the velocity vectors on the plot of the particle's path in part (a). (d) Calculate the components of the particle's acceleration at t = 1.0, 2.0, and 3.0 s. 27 A boat can travel at a speed of 8 km/h in still water on a lake. In the flowing water of a stream, it can move at 8 km/h relative to the water in the stream. The stream speed is 3 km/h. (a) how fast can the boat move past a tree on the shore in traveling upstream? (b) how fast can the boat move past a tree on the shore in traveling downstream? Answer (a) 5 km/h. (b) 11 km/h. 25 (a) 0, 0; 2.0 m, 1.4 m; 4.0 m, 2.0 m; 6.0 m, 1.4 m; 8.0 m, 0; (b) 2.0 m/s, 1.1 m/s; 2.0 m/s, 0; 2.0 m/s, -1.1 m/s; (c) 0, m/s 2 ; 0, -1.2 m/s 2 ; 0, -.87 m/s 2 You are to ride a jet-cycle over a lake, starting from rest at point i. First, moving at 30 0 north of due east: 28 A river is 0.76 km wide. The banks are straight and parallel. The current is 5.0 km/h and is parallel to the banks. A boat has a maximum speed of 3 km/h in still water. The pilot of the boat wishes to travel on a straight line from A to B, where AB is perpendicular to the banks. The pilot should do which of the following? a. head directly across the river. b. head 68 upstream from the line AB. c. head 22 upstream from the line AB. d. give up. The trip from A to Bis not possible with this boat. e. do none of these. 1. Increase your speed at m/s 2 for 6.00 s. 2. With whatever speed you then have, move for 8.00 s. 3. Then slow at m/s 2 for 6.00 s. Immediately next, moving due west: 4. Increase your speed at m/s 2 for 5.00 s. 5. With whatever speed you then have, move for 10.0 s. 6. Then slow at m/s 2 until you stop. In magnitue angle notation, what is your average velocity for the trip from point i? give up. The trip from A to Bis not possible with this boat m/s at 3.1 west of north

6 29 A woman can row a boat 4.0 mi./hr in still water (a) If she is crossing a river where the current is 2.0 mi./h, in what direction will her boat be headed if she wants to reach a point directly opposite from her starting point? (b) If the river is 4.0 mi. wide, how long will it take her to go straight across the river? (c) How long will it take her to row 2.0 mi. down the river and then back up to her starting point? (d) How long will it take her to row 2.0 mi. up the river and back to her starting point? (e) In what direction should she head the boat if she wants to cross in the shortest possible time? 32 A boat can travel 2.70 m/s in still water. (a) If the boat points its prow directly across a stream whose current is m/s, what is the velocity (magnitude and direction) of the boat relative to the shore? (b) What will be the position of the boat, relative to its point of origin, after 3.0 s? (a) 30 degrees (b) 1.15 hr. 69 min. (c) 80 in (d) 80 in (e) 1 hr straight across the river 30 A rowboat is pointing perpendicular to the bank of a river. A rower can propel the boat with a speed of 3.0 m/s with respect to the water. The river has a current of 4.0 m/s (a) Construct a diagram in which the two velocities are represented as vectors. (b) Find the vector which represents the boat's velocity with respect to the shore. (c) At what angle is this vector inclined to the direction in which the boat is pointing? (d) What is the boat's speed with respect to the launch point? (e) If the river is 100 m wide, determine how far downstream of the launch point the rowboat is when it reaches the opposite bank. 33 A 200-m-wide river has a uniform flow speed of 1.1 m/s through a jungle and toward the east. An explorer wishes to leave a small clearing on the south bank and cross the river in a powerboat that moves at a constant speed of 4.0 m/s with respect to the water. There is a clearing on the north bank 82 m upstream from a point directly opposite the clearing on the south bank. (a) In what direction must the boat be pointed in order to travel in a straight line and land in the clearing on the north bank? (b) How long will the boat take to cross the river and land in the clearing? 31 (a) (b) 5 m/s (c) 53.1 degrees (d) 5 m/s (e) 133 m Huck Finn walks at a speed of 1.0 m/s across his raft (that is, he walks perpendicular to the raft's motion relative to the shore). The raft is traveling down the Mississippi River at a speed of 2.5 m/s relative to the river bank (see the figure to the left). What is the velocity (speed and direction) of Huck relative to the river bank? 34 a) 37 west of north; (b) 62.6 s A boat is traveling upstream in the positive direction of an x axis at 14 km/h with respect to the water of a river. The water is flowing at 9.0 km/h with respect to the ground. (a) What is the magnitude of the boat's velocity with respect to the ground? (b) What is the direction of the boat's velocity with respect to the ground? A child on the boat walks from front to rear at 6.0 km/h with respect to the boat. (c) What is the magnitude of the child's velocity with respect to the ground? (d) What is the direction of the child's velocity with respect to the ground? (a) 5 km/h; (b) positive x; (c) 1 km/h; (d) negative x

7 35 A woman who can row a boat at 6.4 km/h in still water faces a long, straight river with a width of 6.4 km and a current of 3.2 km/h. Let i point directly across the river and j point directly downstream. (a) If she rows in a straight line to a point directly opposite her starting position, at what angle to i must she point the boat? (b) If she rows in a straight line to a point directly opposite her starting position, how long will she take? (c) How long will she take if, instead, she rows 3.2 km down the river and then back to her starting point? (d) How long if she rows 3.2 km up the river and then back to her starting point? (e) At what angle to i should she point the boat if she wants to cross the river in the shortest possible time? (f) How long is that shortest time? The initial path of a rocket is 30 above the horizontal, and the horizontal component of the rocket's velocity is 326 m/s. The initial vertical component of the rocket's velocity is 188 m/s A plane, diving at an angle of 53 0 with the vertical, releases a projectile at an altitude of 730 m. The projectile hits the ground 5.0 s after being released. (a) What is the speed of the plane? (b) How far did the projectile travel horizontally during its flight? (c) What were the horizontal and vertical components of its velocity just before striking the ground? (a) -30 ; (b) 69 min; (c) 80 min; (d) 80 min; (e) 0; (f) 60 min (a) 202 m/s (b) 806 m (c) Horizontal 161 m/s Vertical 171 m/s 36 A 200-m-wide river flows due east at a uniform speed of 2.0 m/s. A boat with a speed of 8.0 m/s relative to the water leaves the south bank pointed in a direction 30 0 west of north. (a) What is the magnitude of the boat's velocity relative to the ground? (b) What is the direction of the boat's velocity relative to the ground? (c) How long does the boat take to cross the river? 41 An anti-tank gun is located on the edge of a plateau that is 60 m above the surrounding plain. The gun crew sights an enemy tank station on the plain at a horizontal distance of 2.2 km from the gun. At the same moment, the tank crew sees the gun and starts to move directly away from it with an acceleration of 0.90 m/s 2. If the anti-tank gun fires a shell with a muzzle speed of 240 m/s at an elevation angle of 10 degrees above the horizontal, how long should the gun crew wait before firing if they are to hit the tank? (a) 7.2 m/s; (b) 16 west of north; (c) 29 s (not 28 s) 37 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? 5.60 seconds 38 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 > 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.)

8 42 In the figure, a rescue plane flies at 198 km/h (= 55.0 m/s) and a constant elevation of 500 m toward a point directly over a boating accident victim struggling in the water very close to the victim. (a) What should be the angle θ of the pilot's line of sight to the victim when the release is made? (b) As the capsule reaches the water, what is its velocity v in unit-vector notation? (c) As the capsule reaches the water, what is its velocity v in magnitude-angle notation? 44 In the figure, a cream tangerine is thrown up past windows 1, 2, and 3, which are identical in size and regularly spaced vertically. (a) Rank those three windows according to the time the cream tangerine takes to pass them during the passage, greatest first. (b) Rank those three windows according to the average speed of the cream tangerine during the passage, greatest first. The cream tangerine then moves down past windows 4, 5, and 6, which are identical in size and irregularly spaced horizontally. (c) Rank those three windows according to the time the cream tangerine takes to pass them the greatest first. (d) Rank those three windows according to the average speed of the cream tangerine during the passage, greatest first. 43 The figure illustrates the ramps for the current world-record motorcycle jump, set by Jason Renie in The ramps were H = 3.00 m high, angled at θ R = , and separated by distance D = 77.0 m. Assuming that he landed halfway down the landing ramp and that the slowing effects of the air were negligible, calculate the speed at which he left the launch ramp. (a) 3, 2, 1; (b) 1, 2, 3; (c) all tie; (d) 6, 5, 4 45 Two seconds after being projected from ground level, a projectile is displaced 40 m horizontally and 53 m vertically above its launch point. (a) What is the horizontal components of the initial velocity of the projectile? (b) What is the vertical components of the initial velocity of the projectile? (c) At the instant the projectile achieves its maximum height above ground level, how far is it displaced horizontally from the launch point? (a) 20 m/s; (b) 36 m/s; (c) 74 m

9 46 A track meet is held on a planet in a distant solar system. A shot-putter releases a shot at a point 2.0 m above ground level. A stroboscopic plot of the position of the shot is shown in the figure, where the readings are 0.50 s apart and the shot is released at time t = 0. (a) What is the initial velocity of the shot in unit-vector notation? (b) What is the magnitude of the free-fall acceleration on the planet? (c) How long after it is released does the shot reach the ground? (d) If an identical throw of the shot is made on the surface of Earth, how long after it is released does it reach the ground? 49 An anti-tank gun is located on the edge of a plateau that is 60 m above the surrounding plain. The gun crew sights an enemy tank station on the plain at a horizontal distance of 2.2 km from the gun. At the same moment, the tank crew sees the gun and starts to move directly away from it with an acceleration of 0.90 m/s 2. If the anti-tank gun fires a shell with a muzzle speed of 240 m/s at an elevation angle of 10 degrees above the horizontal, how long should the gun crew wait before firing if they are to hit the tank? 5.60 seconds (a) (10 m/s)ˆi + (10 m/s)ˆj; (b) 8.0 m/s 2 ; (c) 2.7 s; (d) 2.2 s 50 A rocket is launched from rest and moves in a straight line at 70 degrees above the horizontal with an observed acceleration of 46 m/s 2. After 30 s of powered flight, the engines shut off and the rocket follows a parabolic path back to earth. (a) Find the time of flight from launching to impact. (b) What is the maximum altitude reached? (c) What is the distance from launch pad to impact point? 47 In a detective story, a body is found 4.6 m from the base of a building and 24 m below an open window. (a) Assuming the victim left that window horizontally, what was the body's initial speed? (b) Would you guess the death to be accidental? Explain your answer. (a) 2.1 m/s; (b) not accidental because horizontal launch speed is about 20% of world class sprint speed 48 A baseball is hit at ground level. The ball reaches its maximum height above ground level 3.0 s after being hit. Then 2.5 s after reaching its maximum height, the ball barely clears a fence that is 97.5 m from where it was hit. Assume the ground is level. (a) What maximum height above ground level is reached by the ball? (b) How high is the fence? (c) How far beyond the fence does the ball strike the ground? (a) 309 s (b) 105 km (c) 139 km (a) 44 m; (b) 13 m; (c) 8.9 m

10 51 The figure shows a pirate ship 560 m from a fort defending the harbor entrance of an island. A defense cannon, located at sea level, fires balls at initial speed v 0 = 82 m/s. (a) At what angle θ 0 from the horizontal must a ball be fired to hit the ship? (b) How far should the pirate ship be from the cannon if it is to be beyond the maximum range of the cannonballs? 54 Long flights at midlatitudes in the northern hemisphere encounter the jet stream, an eastward airflow that can affect a plane's speed relative to Earth's surface. If a pilot maintains a certain speed relative to the air (the plane's airspeed), the speed relative to the surface (the plane's ground speed) is more when the flight is in the direction of the jet stream and less when the flight is opposite the jet stream. Suppose a round-trip flight is scheduled between two cities separated by 4000 km, with the outgoing flight in the direction of the jet stream and the return flight opposite it. The airline computer advises an airspeed of 1000 km/h, for which the difference in flight times for the outgoing and return flights is 70.0 min. What jet-stream speed is the computer using? 143 km/h 55 A bolt drops from the ceiling of a train car that is accelerating northward at a rate of 2.50 m/s 2. (a) What is the acceleration of the bolt relative to the train car? (b) What is the acceleration of the bolt relative to the Earth? (a) (b) 686 m 52 An enemy ship is on the east side of a mountain island, as shown in the figure. The enemy ship has maneuvered to within m of the 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? 56 An elevator is moving upward at a constant speed of 4 m/s. A light bulb falls out of a socket in the ceiling of the elevator. A man in the building watching the cage sees the bulb rise for (4/9.8) s and then fall for (4/9.8) s; at t = (4/9.8) s the bulb appears to be at rest to the man. Compute the velocity of the bulb at t = (4/9.8) s from the view of an observer in the elevator. -4 m/s j 57 A suspicious-looking man runs as fast as he can along a moving sidewalk from one end to the other, taking 2.50s. Then security agents appear, and the man runs as fast as he can back along the sidewalk to his starting point, taking 10.0 s. What is the ratio of the man's running speed to the sidewalk's speed? 53 An elevator is moving upward at a constant speed of 4 m/s. A light bulb falls out of a socket in the ceiling of the elevator. A man in the building watching the cage sees the bulb rise for (4/9.8) s and then fall for (4/9.8) s; at t = (4/9.8) s the bulb appears to be at rest to the man. Compute the velocity of the bulb at t = (4/9.8) s from the view of an observer in the elevator. 5/3

11 58 In the figure a ball is shot directly upward from the ground with an initial speed of v 0 = 7.00 m/s. Simultaneously, a construction elevator cab begins to move upward from the ground with a constant speed of v, = 3.00 m/s. (a) What maximum height does the ball reach relative to the ground? (b) What maximum height does the ball reach relative to the cab floor? (c) At what rate does the speed of the ball change relative to the ground? (d) At what rate does the speed of the ball change relative to the cab floor? 61 A cameraman on a pickup truck is traveling westward at 20 km/h while he videotapes a cheetah that is moving westward 30 km/h faster than the truck. Suddenly, the cheetah stops, turns, and then runs at 45 km/h eastward, as measured by a suddenly nervous crew member who stands alongside the cheetah's path. The change in the animal's velocity takes 2.0 s. (a) What is the magnitude of the animal's acceleration according to the cameraman? (b) What is the direction of the animal's acceleration according to the cameraman? (c) What is the magnitude according to the nervous crew member? (d) What is the direction according to the nervous crew member? (a) 13 m/s 2 (b) eastward (c) 13 m/s 2 (d) eastward (a) 2.5 m (b) 0.82 m (c) 9.8 m/s 2 (d) 9.8 m/s 2 62 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? 59 (Optional) Long flights at midlatitudes in the northern hemisphere encounter the jet stream, an eastward airflow that can affect a plane's speed relative to Earth's surface. If a pilot maintains a certain speed relative to the air (the plane's airspeed), the speed relative to the surface (the plane's ground speed) is more when the flight is in the direction of the jet stream and less when the flight is opposite the jet stream. Suppose a round-trip flight is scheduled between two cities separated by 4000 km, with the outgoing flight in the direction of the jet stream and the return flight opposite it. The airline computer advises an airspeed of 1000 km/h, for which the difference in flight times for the outgoing and return flights is 70.0 min. What jet-stream speed is the computer using? s A river has a steady speed of 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. 64 A bolt drops from the ceiling of a train car that is accelerating northward at a rate of 2.50 m/s km/h (a) What is the acceleration of the bolt relative to the train car? 60 An airport terminal has a moving sidewalk to speed passengers through a long corridor. Larry does not use the moving sidewalk; he takes 150 s to walk through the corridor. Curly, who simply stands on the moving sidewalk, covers the same distance in 70 s. Moe boards the sidewalk and walks along it. How long does Moe take to move through the corridor? Assume that Larry and Moe walk at the same speed. 48 s (b) What is the acceleration of the bolt relative to the Earth?

12 65 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 v w. 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 and analyzing the scattering of subatomic particles from massive targets. 68 Two highways intersect, as shown in the figure to the left. At the instant a police car P is 800 m from the intersection and moving at 80 km/hr. Motorist M is 600 m from the intersection moving at 60 km/hr. (a) In unit-vector notation, what is the velocity of the motorist with respect to the police car? (b) For the instant shown in the figure, how does the direction of the velocity found in a compare to the line of sight between the two cars? (c) If the cars maintain their velocities, do the answers to a and b change as the cars move nearer the intersection? 66 A plane is traveling eastward at an airspeed of 500 km/h. But a 90 km/h wind is blowing southward. What are the direction and speed of the plane relative to the ground? Answer (a) (80i - 60j) km/h (b) v happens to be along the line of sight (c) The answers do not change 508 km/h at 10.2 south of east. 67 An ocean liner is traveling at 18 km/hr. A passenger on deck walks toward the rear of the ship at a rate of 4.0 m/s. After walking 30 meters he turns right and walks at the save rate to the rail, which is 12 meters from his turning point. (a) What is his velocity relative to the water surface while walking to the rear? (b) What is his velocity relative to the water surface while walking toward the rail? (c) What was his total displacement frm the starting point (relative to the water surface)? 69 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 60.0 with the vertical. (a) Find the velocity of the rain with respect to the car (b) Find the velocity of the rain with respect to the Earth.

13 70 Two highways intersect as shown in the figure. At the instant shown, a police car P is distance d P = 800 m from the intersection and moving at speed v p = 80 km/h. Motorist M is distance d M = 600 m from the intersection and moving at speed v M = 60 km/h. (a) In unit-vector notation, what is the velocity of the motorist with respect to the police car? (b) For the instant shown in the figure, what is the angle between the velocity found in (a) and the line of sight between the two cars? (c) If the cars maintain their velocities, do the answers to (a) and (b) change as the cars move nearer the intersection? 72 An ocean liner is traveling at 15 km/hr. A passenger walks toward the rear of the ship at a rate of 4.0 m/s. After walking 30 meters he turns right and walks at the same rate to the rail, which is 12 meters from his turning point. (a) What is his velocity relative to the water surface while walking to the rear? (b) What is his relative velocity to the water surface while walking toward the rail? (c) What was his total displacement from the starting point (relative to the water surface)? (a) (80 km/h)ˆi - (60 km/h)ˆj (b) 800 i j (c) unchanged 71 A car is traveling down a highway at 25 m/s. Just as the car crosses a perpendicular intersecting crossroads, the passenger throws out a pop can at a 45 degree angle of elevation in a plane perpendicular to the motion of the car. The initial speed of the can relative to the car is 10 m/s. It is released at a height of 1.2 m above the road. (a) Write the initial velocity of the pop can (relative to the road) in terms of the unit vectors i, j, and k. 73 In a large department store, a shopper is standing on the "up" escalator, which is traveling at an angle of 40 0 above the horizontal and at a speed of 0.75 m/s. He passes his daughter, who is standing on the identical, adjacent "down" escalator. (see the figure) Find the velocity of the shopper relative to his daughter in unit (b) What is the final position of the can relative to an observer in the car. Give the answer in terms of the unit vectors i, j, and k? (0.96j) m/s

14 74 Curtain of death. A large metallic asteroid strikes Earth and quickly digs a crater into the rocky material below ground level by launching rocks upward and outward. The following table gives five pairs of launch speeds and angles (from the horizontal) for such rocks, based on a model of crater formation. (Other rocks, with intermediate speeds and angles, are also launched.) Suppose that you are at x = 20 km when the asteroid strikes the ground at time t = 0 and position x = 0 (see figure). (a) At t = 20 s, what are the x and y coordinates of the rocks headed in your direction from launches A through E? (b) Plot these coordinates and then sketch a curve through the points to include rocks with intermediate launch speeds and angles. The curve should give you an idea of what you would see as you look up into the approaching rocks and what dinosaurs must have seen during asteroid strikes long ago. 76 In the figure, suppose that Barbra s velocity relative to Alex is a constant v BA = 52 km/h and car P is moving in the negative direction of the x axis. (a) If Alex measures a constant velocity v PA = -78 km/h for car P, what velocity v PB will Barbara measure? (b) If car P brakes to a stop relative to Alex (and thus relative to the ground) in time t = 10 s at constant acceleration, what is its acceleration a PA relative to Alex? (c) What is the acceleration a PB of car P relative to Barbara during the braking? (a) A: 10.1 km, km B: 12.1 km, 1.51 km C: 14.3 km, 2.68 km D: 16.4 km, 3.99 km E: 18.5k m, 5.53 km (b) the rocks form a curtain that curves upward and away from you 77 In the figure a plane moves due east while the pilot points the plane somewhat south of east, toward a steady wind that blows to the northeast. The plane has velocity V PW relative to the wind, with an airspeed (speed relative to the wind) of 215 km/h, directed at angle θ south of east. The wind has velocity V WG relative to the ground, with a speed of 65.0 km/h, directed 20.0 o east of north. Suppose the pilot turns the plane to point it due east without changing the airspeed. (a) Does the magnitude of the v PG,x increase, decrease, or remain the same? (b) Does the magnitude of the V PGy increase, decrease, or remain the same? (c) Does the magnitude of thev PG increase, decrease, or remain the same? (You can answer these questions without computation.) 75 The table here gives velocities(km/h) for Barbara and car P of the figure for three situations. For each, what is the missing value and how is the distance between Barbara and car P changing? (a) (c) increase (a) 0, distance not changing (b) +70 km/h, distance increasing (c) +80 km/h, distance decreasing

15 78 A rugby player runs with the ball directly toward his opponent's goal, along the positive direction of an x axis. He can legally pass the ball to a teammate as long as the ball's velocity relative to the field does not have a positive x component. Suppose the player runs at speed 4.0 m/s relative to the field while he passes the ball with velocity V BP relative to himself. If V BP has magnitude 6.0 m/s, what is the smallest angle it can have for the pass to be legal? In the overhead view of the figure, Jeeps P and B race along straight lines, across flat terrain, and past stationary border guard A. Relative to the guard, B travels at a constant speed of 20.0 m/s, at the angle θ 2 = Relative to the guard, P has accelerated from rest at a constant rate of m/s 2 at the angle θ 1 = At a certain time during the acceleration, P has a speed of 40.0 m/s. (a) At that time, what is the magnitude f the velocity of P relative to B? (b) At that time, what is the direction of the velocity of P relative to B? (c) At that time, what is themagnitude of the acceleration of P relative to B? (d) At that time, what is thedirection of the acceleration of P relative to B? 79 Snow is falling vertically at a constant speed of 8.0 m/s. At what angle from the vertical do the snowflakes appear to be falling as viewed by the driver of a car traveling on a straight, level road with a speed of 50 km/h? Two ships, A and B, leave port at the same time. Ship A travels northwest at 24 knots, and ship B travels at 28 knots in a direction 40 0 west of south. (1 knot = 1 nautical mile per hour; see Appendix D.) (a) What is the magnitude of the velocity of ship A relative to B? (b) What is the direction of the velocity of ship A relative to B? (c) After what time will the ships be 160 nautical miles apart? (d) What will be the bearing of B (the direction of B's position) relative to A at that time? (a) 24.8 m/s (b) north of east (c) 0.40 m/s 2 (d) north of east (a) 38 knots (b) east of north (c) 4.2h (d) west of south 83 Ship A is located 4.0 km north and 2.5 km east of ship B. Ship A has a velocity of 22 km/h toward the south, and ship B has a velocity of 40 km/h in a direction 37 0 north of east. (a) What is the velocity of A relative to B in unit-vector notation with i toward the east? (b) Write an expression (in terms of i and j) for the position of A relative to B as a function of t, where t = 0 when the ships are in the positions described above. Optional 81 A train travels due south at 30 m/s (relative to the ground) in a rain that is blown toward the south by the wind. The path of each raindrop makes an angle of 70 0 with the vertical, as measured by an observer stationary on the ground. An observer on the train, however, sees the drops fall perfectly vertically. Determine the speed of the raindrops relative to the ground. (c) At what time is the separation between the ships least? (d) What is that least separation? - 32 m/s (a) (-32 km/h)ˆi - (46 km/h)ˆj (b) [(2.5 km) - (32 km/h)t] ˆi +[(4.0 km) - (46 km/h)t] ˆj (c) h (d) 0.20 km

16 84 Figure a, a sled moves in the negative x direction at constant speed v s while a ball of ice is shot from the sled with a velocity v 0 = v 0x i + v 0y j relative to the sled. When the ball lands, its horizontal displacement x bg relative to the ground (from its launch position to its landing position) is measured. Figure b gives x bg as a function of v s Assume the ball lands at approximately its launch height. (a) What is the values of v 0x and (b) What is the values of v 0y? The ball's displacement x b, relative to the sled can also be measured. Assume that the sled's velocity is not changed when the ball is shot. (c) What is x b, when v s is 5.0 m/s? (d) What is x b, when v s is 15 m/s? 87 About 4000 car-train collisions occur in the United States each year. Many (perhaps most) occur when a driver spots an oncoming train and then attempts to beat it to a railroad crossing. (This is beyond foolish, and way past stupid as well.) Consider the following situation: One of those 4000 drivers spots a train on a track that is perpendicular to the roadway and judges the train's speed to be equal to his speed of 30.0 m/s. The train and the car are both approaching the rail crossing (see the figure). The driver figures that by accelerating his car at a constant 1.50 m/s 2 toward the crossing, he should reach it first. At the instant of this decision, the distance between car and crossing center and the distance between train and crossing center are both 40.0 m. The length of the car is' 5.0 m, and the width of the train is 3.00 m. (a) Is there a collision? (b) If no, by how much time does the driver miss a collision; if yes, how much more time is needed to barely miss the collision? (c) On the same graph, plot the location of the train and the location of the car, both versus time. (a) 10 m/s (b) 19.6 m/s (c) 40 m (d) 40 m (a) yes (b) 0.16 s The pilot of an aircraft flies due east relative to the ground in a wind blowing 20 km/h toward the south. If the speed of the aircraft in the absence of wind is 70 km/h, what is the speed of the aircraft relative to the ground? 67 km/h A person walks up a stalled 15-m-long escalator in 90 s. When standing on the same escalator, now moving, the person is carried up in 60 s. How much time would it take that person to walk up the moving escalator? Does the answer depend on the length of the escalator? 88 A helicopter is flying in a straight line over a level field at a constant speed of 6.2 m/s and at a constant altitude of 9.5 m. A package is ejected horizontally from the helicopter with an initial velocity of 12 m/s relative to the helicopter and in a direction opposite the helicopter's motion. (a) Find the initial speed of the package relative to the ground. (b) What is the horizontal distance between the helicopter and the package at the instant. the package strikes the ground? (c) What angle does the velocity vector of the package make with the ground at the instant before impact, as seen from the ground? (a) 5.8m/s (b) 17 m (c) s, no 89 Heather in her Corvette accelerates at the rate of (3.00i j) m/s 2, while Jill in her Jaguar accelerates at (1.00 i j) m/s 2. They both start from rest at the origin of an xy coordinate system. After 5.00 s, (a) what is Heather's speed with respect to Jill, (b) how far apart are they (c) what is Heather's acceleration relative to Jill? (a) 10 m/s i - 25 m/s j (b 25 m/s i m/s j (c) 2 m/s i - 5 m/s j

17 90 The pilot of an airplane notes that the compass indicates a heading due west. The airplane's 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. 95 A wooden boxcar is moving along a straight railroad track at speed v 1. A sniper fires a bullet (initial speed v 2 ) at it from a high-powered rifle. The bullet passes through both lengthwise walls of the car, its entrance and exit holes being exactly opposite each other as viewed from within the car. From what direction, relative to the track, is the bullet fired? Assume that the bullet is not deflected upon entering the car, but that its speed decreases by 20%. Take v 1 = 85 km/h and v 2 = 650 m/s. (Why don't you need to know the width of the boxcar?) 91 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 60.0 with the horizontal and to be in line with the track. The student's professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise? 93 0 from the car s direction of motion A car is traveling down a highway at 25 m/s. Just as the car crosses a perpendicular intersecting crossroads, the passenger throws out a pop can at a 45 degree angle of elevation in a plane perpendicular to the motion of the car. The initial speed of the can relative to the car is 10 m/s. It is released at a height of 1.2 m above the road. 92 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 wishes 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. (a) Write the initial velocity of the pop can (relative to the road) in terms of the unit vectors i, j, and k. 93 If a rock is dropped from the top of a sailboat s mast, will it hit the deck at the same point whether the boat is at rest or in motion at constant velocity? (b) What is the final position of the can relative to an observer in the car. Give the answer in terms of the unit vectors i, j, and k? ANSWER yes, neglecting air resistance 94 An ocean liner is traveling at 18 km/hr. A passenger on deck walks toward the rear of the ship at a rate of 4.0 m/s. After walking 30 meters he turns right and walks at the save rate to the rail, which is 12 meters from his turning point. (a) What is his velocity relative to the water surface while walking to the rear? (b) What is his velocity relative to the water surface while walking toward the rail? (c) What was his total displacement frm the starting point (relative to the water surface)? 97 An ocean liner is traveling at 15 km/hr. A passenger walks toward the rear of the ship at a rate of 4.0 m/s. After walking 30 meters he turns right and walks at the same rate to the rail, which is 12 meters from his turning point. (a) What is his velocity relative to the water surface while walking to the rear? (b) What is his relative velocity to the water surface while walking toward the rail? (c) What was his total displacement from the starting point (relative to the water surface)?

18 98 The rock shown below is acted on by one or more forces. Draw the free body diagram for the rock rising in a parabolic trajectory. ANSWERS 99 A 3.00 kg ball is dropped from the roof of a building m high. While the ball is falling to Earth, a horizontal wind exerts a constant force of 12.0 N on it. (a) Draw the free body diagram. (b) How long does it take the ball to hit the ground? (c) How far from the building does the ball hit the ground? (d) What is its speed and angle when it hits the ground? (e) With what force and angle will the ball hit the ground? ANSWERS Good review problem of projectile motion (a) (b) 72.0 m (c) 6.00 s (d) 63.6 m/s (e) N