Solved: Two boats are heading away from shore. Boat 1

In Figure 3.5, the directions to the right and upward are the positive directions. In the x direction, the spacecraft has an initial velocity component of v0x 22 m/s and an acceleration component of ax 24 m/s2 . In the y direction, the analogous quantities are v0y 14 m/s and ay 12 m/s2 . At a time of t 7.0 s, find the x and y components of the spacecrafts displacement. Reasoni

This example also deals with the spacecraft in Figure 3.5. As in Example 1, the x components of the crafts initial velocity and acceleration are v0x 22 m/s and ax 24 m/s2 , respectively. The corresponding y components are v0y 14 m/s and ay 12 m/s2 . At a time of t 7.0 s, find the spacecrafts final velocity (magnitude and direction). Reasonin

Suppose you are driving due east, traveling a distance of 1500 m in 2 minutes. You then turn due north and travel the same distance in the same time. What can be said about the average speeds and the average velocities for the two segments of the trip? (a) The average speeds are the same, and the average velocities are the same. (b) The average speeds are the same, but the average velocities are different. (c) The average speeds are different, but the average velocities are the same.

A power boat, starting from rest, maintains a constant acceleration. After a certain time t, its displacement and velocity are and . At time 2t, what would be its displacement and velocity, assuming the acceleration remains the same? (a) 2 and 2 (b) 2 and 4 (c) 4 and 2 (d) 4 and 4v

Figure 3.7 shows an airplane moving horizontally with a constant velocity of 115 m/s at an altitude of 1050 m. The directions to the right and upward have been chosen as the positive directions. The plane releases a care package that falls to the ground along a curved trajectory. Ignoring air resistance, determine the time required for the package to hit the ground.

Figure 3.7 shows a care package falling from a plane, and Figure 3.8 shows this package as package B. As in Example 3, the directions to the right and upward are chosen as the positive directions, and the plane is moving horizontally with a constant velocity of 115 m/s at an altitude of 1050 m. Ignoring air resistance, find the magnitude v and the directional angle of the final velocity vector that the package has just before it strikes the ground.

Suppose you are driving in a convertible with the top down. The car is moving to the right at a constant velocity. As Figure 3.9 illustrates, you point a rifle straight upward and fire it. In the absence of air resistance, would the bullet land (a) behind you, (b) ahead of you, or (c) in the barrel of the rifle?

A placekicker kicks a football at an angle of 40.0 above the horizontal axis, as Figure 3.10 shows. The initial speed of the ball is v0 22 m/s. Ignore air resistance, and find the maximum height H that the ball attains. Re

For the motion illustrated in Figure 3.10, ignore air resistance and use the data from Example 6 to determine the time of flight between kickoff and landing.

For the motion shown in Figure 3.10 and discussed in Examples 6 and 7, ignore air resistance and calculate the range R of the projectile.

A baseball player hits a home run, and the ball lands in the left-field seats, 7.5 m above the point at which it was hit. It lands with a velocity of 36 m/s at an angle of 28 below the horizontal (see Figure 3.11). The positive directions are upward and to the right in the drawing. Ignoring air resistance, find the magnitude and direction of the initial velocity with which the ball leaves the bat.

From the top of a cliff overlooking a lake, a person throws two stones. The stones have identical initial speeds v0, but stone 1 is thrown downward at an angle below the horizontal, while stone 2 is thrown upward at the same angle above the horizontal, as Figure 3.14 shows. Neglect air resistance and decide which stone, if either, strikes the water with the greater velocity: (a) both stones strike the water with the same velocity, (b) stone 1 strikes with the greater velocity, (c) stone 2 strikes with the greater velocity.

A projectile is fired into the air, and it follows the parabolic path shown in the drawing, landing on the right. There is no air resistance. At any instant, the projectile has a velocity and an acceleration . Which one or more of the drawings could not represent the directions for and at any point on the trajectory?

An object is thrown upward at an angle above the ground, eventually returning to earth. (a) Is there any place along the trajectory where the velocity and acceleration are perpendicular? If so, where? (b) Is there any place where the velocity and acceleration are parallel? If so, where?

Is the acceleration of a projectile equal to zero when the projectile reaches the top of its trajectory?

In baseball, the pitchers mound is raised to compensate for the fact that the ball falls downward as it travels from the pitcher toward the batter. If baseball were played on the moon, would the pitchers mound have to be (a) higher than, (b) lower than, or (c) the same height as it is on earth?

A tennis ball is hit upward into the air and moves along an arc. Neglecting air resistance, where along the arc is the speed of the ball (a) a minimum and (b) a maximum?

A wrench is accidentally dropped from the top of the mast on a sailboat. Air resistance is negligible. Will the wrench hit at the same place on the deck whether the sailboat is at rest or moving with a constant velocity?

A rifle, at a height H above the ground, fires a bullet parallel to the ground. At the same instant and at the same height, a second bullet is dropped from rest. In the absence of air resistance, which bullet, if either, strikes the ground first?

A stone is thrown horizontally from the top of a cliff and eventually hits the ground below. A second stone is dropped from rest from the same cliff, falls through the same height, and also hits the ground below. Ignore air resistance. Is each of the following quantities different or the same in the two cases? (a) Displacement (b) Speed just before impact with the ground (c) Time of flight

A leopard springs upward at a 45 angle and then falls back to the ground. Air resistance is negligible. Does the leopard, at any point on its trajectory, ever have a speed that is one-half its initial value?

Two balls are launched upward from the same spot at different angles with respect to the ground. Both balls rise to the same maximum height. Ball A, however, follows a trajectory that has a greater range than that of ball B. Ignoring air resistance, decide which ball, if either, has the greater launch speed.

The engine of a boat drives it across a river that is 1800 m wide. The velocity BW of the boat relative to the water is 4.0 m/s, directed perpendicular to the current, as in Figure 3.17. The velocity WS of the water relative to the shore is 2.0 m/s. (a) What is the velocity BS of the boat relative to the shore? (b) How long does it take for the boat to cross the river?

Figure 3.18a shows two cars approaching an intersection along perpendicular roads. The cars have the following velocities: velocity of relative to the 25.0 m/s, eastward velocity of relative to the 15.8 m/s, northward Find the magnitude and direction of AB, where velocity of as measured by a passenger in Reas

Three cars, A, B, and C, are moving along a straight section of a highway. The velocity of A relative to B is AB, the velocity of A relative to C is AC, and the velocity of C relative to B is vCB. Fill in the missing velocities in the table.

On a riverboat cruise, a plastic bottle is accidentally dropped overboard. A passenger on the boat estimates that the boat pulls ahead of the bottle by 5 meters each second. Is it possible to conclude that the magnitude of the velocity of the boat with respect to the shore is 5 m/s?

A plane takes off at St. Louis, flies straight to Denver, and then returns the same way. The plane flies at the same speed with respect to the ground during the entire flight, and there are no head winds or tail winds. Since the earth revolves around its axis once a day, you might expect that the times for the outbound trip and the return trip differ, depending on whether the plane flies against the earths rotation or with it. Is this true, or are the two times the same?

A child is playing on the floor of a recreational vehicle (RV) as it moves along the highway at a constant velocity. He has a toy cannon, which shoots a marble at a fixed angle and speed with respect to the floor. The cannon can be aimed toward the front or the rear of the RV. Is the range toward the front the same as, less than, or greater than the range toward the rear? Answer this question (a) from the childs point of view and (b) from the point of view of an observer standing still on the ground.

Three swimmers can swim equally fast relative to the water. They have a race to see who can swim across a river in the least time. Swimmer A swims perpendicular to the current and lands on the far shore downstream, because the current has swept him in that direction. Swimmer B swims upstream at an angle to the current, choosing the angle so that he lands on the far shore directly opposite the starting point. Swimmer C swims downstream at an angle to the current in an attempt to take advantage of the current. Who crosses the river in the least time?

In a circus act, Biff the clown is fired from a cannon at an initial velocity 0 directed at an angle above the horizontal, as Figure 3.20 shows. Simultaneously, two other clowns are also launched. Bongo is launched horizontally on roller skates at a speed of 4.6 m/s. He rolls along the ground while Biff flies through the air. When Biff returns to the ground, he lands side by side with his roller-skating friend, who is gliding by just at the instant of landing. The third clown, Bingo, however, is fired straight upward at a speed of 10.0 m/s and reaches the same maximum height at the same instant as Biff. Ignore air resistance, and assume that the roller skates are unimpeded by friction. Find the speed v0 and the angle for Biff.

Is Bongos left-to-right motion the same as or different from the horizontal part of Biffs motion along his trajectory?

Is Biffs motion in the horizontal direction determined by the initial velocity 0, just its horizontal component v0x, or just its vertical component v0y?

Is Bingos up-and-down motion the same as or different from the vertical part of Biffs motion along his trajectory?

Is Biffs initial motion in the vertical direction determined by the initial velocity 0, just its horizontal component v0x, or just its vertical component v0y?

A projectile is launched from and returns to ground level, as Figure 3.21 shows. Air resistance is absent. The horizontal range of the projectile is measured to be R 175 m, and the horizontal component of the launch velocity is v0x 25 m/s. Find the vertical component v0y of the launch velocity.

What is the final value of the horizontal component vx of the projectiles velocity?

Can the time be determined for the horizontal part of the motion?

Is the time for the horizontal part of the motion the same as the time for the vertical part of the motion?

For the vertical part of the motion, what is the displacement of the projectile?

Two trees have perfectly straight trunks and are both growing perpendicular to the flat horizontal ground beneath them. The sides of the trunks that face each other are separated by 1.3 m. A frisky squirrel makes three jumps in rapid succession. First, he leaps from the foot of one tree to a spot that is 1.0 m above the ground on the other tree. Then, he jumps back to the first tree, landing on it at a spot that is 1.7 m above the ground. Finally, he leaps back to the other tree, now landing at a spot that is 2.5 m above the ground. What is the magnitude of the squirrels displacement?

A meteoroid is traveling east through the atmosphere at 18.3 km/s while descending at a rate of 11.5 km/s. What is its speed, in km/s?

In a football game a kicker attempts a field goal. The ball remains in contact with the kickers foot for 0.050 s, during which time it experiences an acceleration of 340 m/s2 . The ball is launched at an angle of 51 above the ground. Determine the horizontal and vertical components of the launch velocity

A baseball player hits a triple and ends up on third base. A baseball diamond is a square, each side of length 27.4 m, with home plate and the three bases on the four corners. What is the magnitude of the players displacement?

In diving to a depth of 750 m, an elephant seal also moves 460 m due east of his starting point. What is the magnitude of the seals displacement?

A mountain-climbing expedition establishes two intermediate camps, labeled A and B in the drawing, above the base camp. What is the magnitude r of the displacement between camp A and camp B? 3200 m 4900 m B

A radar antenna is tracking a satellite orbiting the earth. At a certain time, the radar screen shows the satellite to be 162 km away. The radar antenna is pointing upward at an angle of 62.3 from the ground. Find the x and y components (in km) of the position vector of the satellite, relative to the antenna.

In a mall, a shopper rides up an escalator between floors. At the top of the escalator, the shopper turns right and walks 9.00 m to a store. The magnitude of the shoppers displacement from the bottom of the escalator to the store is 16.0 m. The vertical distance between the floors is 6.00 m. At what angle is the escalator inclined above the horizontal?

A skateboarder, starting from rest, rolls down a 12.0-m ramp. When she arrives at the bottom of the ramp her speed is 7.70 m/s.a) Determine the magnitude of her acceleration, assumed to be constant. (b) If the ramp is inclined at 25.0 with respect to the ground, what is the component of her acceleration that is parallel to the ground?

A bird watcher meanders through the woods, walking 0.50 km due east, 0.75 km due south, and 2.15 km in a direction 35.0 north of west. The time required for this trip is 2.50 h. Determine the magnitude and direction (relative to due west) of the bird watchers (a) displacement and (b) average velocity. Use kilometers and hours for distance and time, respectively.

The earth moves around the sun in a nearly circular orbit of radius 1.50 1011 m. During the three summer months (an elapsed time of 7.89 106 s), the earth moves one-fourth of the distance around the sun. (a) What is the average speed of the earth? (b) What is the magnitude of the average velocity of the earth during this period?

A spacecraft is traveling with a velocity of v0x 5480 m/s along the x direction. Two engines are turned on for a time of 842 s. One engine gives the spacecraft an acceleration in the x direction of ax 1.20 m/s2 , while the other gives it an acceleration in the y direction of ay 8.40 m/s2 . At the end of the firing, find (a) vx and (b) vy. 13

A volleyball is spiked so that it has an initial velocity of 15 m/s directed downward at an angle of 55 below the horizontal. What is the horizontal component of the balls velocity when the opposing player fields the ball?

As a tennis ball is struck, it departs from the racket horizontally with a speed of 28.0 m/s. The ball hits the court at a horizontal distance of 19.6 m from the racket. How far above the court is the tennis ball when it leaves the racket?

A skateboarder shoots off a ramp with a velocity of 6.6 m/s, directed at an angle of 58 above the horizontal. The end of the ramp is 1.2 m above the ground. Let the x axis be parallel to the ground, the y direction be vertically upward, and take as the origin the point on the ground directly below the top of the ramp. (a) How high above the ground is the highest point that the skateboarder reaches? (b) When the skateboarder reaches the highest point, how far is this point horizontally from the end of the ramp?

A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the pucks initial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the pucks initial velocity and acceleration are v0y 2.0 m/s and ay 2.0 m/s2 . Find the magnitude and direction of the pucks velocity at a time of t 0.50 s. Specify the direction relative to the x axis. 17. s

A spider crawling across a table leaps onto a magazine blocking its path. The initial velocity of the spider is 0.870 m/s at an angle of 35.0 above the table, and it lands on the magazine 0.0770 s after leaving the table. Ignore air resistance. How thick is the magazine? Express your answer in millimeters.

A horizontal rifle is fired at a bulls-eye. The muzzle speed of the bullet is 670 m/s. The gun is pointed directly at the center of the bulls-eye, but the bullet strikes the target 0.025 m below the center. What is the horizontal distance between the end of the rifle and the bulls-eye?

A golfer imparts a speed of 30.3 m/s to a ball, and it travels the maximum possible distance before landing on the green. The tee and the green are at the same elevation. (a) How much time does the ball spend in the air? (b) What is the longest hole in one that the golfer can make, if the ball does not roll when it hits the green?

A golfer hits a shot to a green that is elevated 3.0 m above the point where the ball is struck. The ball leaves the club at a speed of 14.0 m/s at an angle of 40.0 above the horizontal. It rises to its maximum height and then falls down to the green. Ignoring air resistance, find the speed of the ball just before it lands

In the aerials competition in skiing, the competitors speed down a ramp that slopes sharply upward at the end. The sharp upward slope launches them into the air, where they perform acrobatic maneuvers. The end of a launch ramp is directed 63 above the horizontal. With this launch angle, a skier attains a height of 13 m above the end of the ramp. What is the skiers launch speed?

A space vehicle is coasting at a constant velocity of 21.0 m/s in the y direction relative to a space station. The pilot of the vehicle fires a RCS (reaction control system) thruster, which causes it to accelerate at 0.320 m/s2 in the x direction. After 45.0 s, the pilot shuts off the RCS thruster. After the RCS thruster is turned off, find (a) the magnitude and (b) the direction of the vehicles velocity relative to the space station. Express the direction as an angle measured from the y direction. 2

As preparation for this problem, review Conceptual Example 10. The drawing shows two planes each about to drop an empty fuel tank. At the moment of release each plane has the same speed of 135 m/s, and each tank is at the same height of 2.00 km above the ground. Although the speeds are the same, the velocities are different at the instant of release, because one plane is flying at an angle of 15.0 above the horizontal and the other is flying at an angle of 15.0 below the horizontal. Find the magnitude and direction of the velocity with which the fuel tank hits the ground if it is from (a) plane A and (b) plane B. In each part, give the directional angles with respect to the horizontal.

A criminal is escaping across a rooftop and runs off the roof 15 15 Fuel tank Plane A Plane B horizontally at a speed of 5.3 m/s, hoping to land on the roof of an adjacent building. Air resistance is negligible. The horizontal distance between the two buildings is D, and the roof of the adjacent building is 2.0 m below the jumping-off point. Find the maximum value for D.

On a spacecraft, two engines are turned on for 684 s at a moment when the velocity of the craft has x and y components of v0x 4370 m/s and v0y 6280 m/s. While the engines are firing, the craft undergoes a displacement that has components of x 4.11 106 m and y 6.07 106 m. Find the x and y components of the crafts acceleration. 26

In the absence of air resistance, a projectile is launched from and returns to ground level. It follows a trajectory similar to that shown in Figure 3.10 and has a range of 23 m. Suppose the launch speed is doubled, and the projectile is fired at the same angle above the ground. What is the new range?

A fire hose ejects a stream of water at an angle of 35.0 above the horizontal. The water leaves the nozzle with a speed of 25.0 m/s. Assuming that the water behaves like a projectile, how far from a building should the fire hose be located to hit the highest possible fire?

Baseball player A bunts the ball by hitting it in such a way that it acquires an initial velocity of 1.9 m/s parallel to the ground. Upon contact with the bat the ball is 1.2 m above the ground. Player B wishes to duplicate this bunt, in so far as he also wants to give the ball a velocity parallel to the ground and have his ball travel the same horizontal distance as player As ball does. However, player B hits the ball when it is 1.5 m above the ground. What is the magnitude of the initial velocity that player Bs ball must be given?

A major-league pitcher can throw a baseball in excess of 41.0 m/s. If a ball is thrown horizontally at this speed, how much will it drop by the time it reaches a catcher who is 17.0 m away from the point of release?

A quarterback claims that he can throw the football a horizontal distance of 183 m (200 yd). Furthermore, he claims that he can do this by launching the ball at the relatively low angle of 30.0 above the horizontal. To evaluate this claim, determine the speed with which this quarterback must throw the ball. Assume that the ball is launched and caught at the same vertical level and that air resistance can be ignored. For comparison, a baseball pitcher who can accurately throw a fastball at 45 m/s (100 mph) would be considered exceptional

An eagle is flying horizontally at 6.0 m/s with a fish in its claws. It accidentally drops the fish. (a) How much time passes before the fishs speed doubles? (b) How much additional time would be required for the fishs speed to double again?

The perspective provided by Multiple-Concept Example 9 is useful here. The highest barrier that a projectile can clear is 13.5 m, when the projectile is launched at an angle of 15.0 above the horizontal. What is the projectiles launch speed?

Consult Multiple-Concept Example 4 for background before beginning this problem. Suppose the water at the top of Niagara Falls has a horizontal speed of 2.7 m/s just before it cascades over the edge of the falls. At what vertical distance below the edge does the velocity vector of the water point downward at a 75 angle below the horizontal?

On a distant planet, golf is just as popular as it is on earth. A golfer tees off and drives the ball 3.5 times as far as he would have on earth, given the same initial velocities on both planets. The ball is launched at a speed of 45 m/s at an angle of 29 above the horizontal. When the ball lands, it is at the same level as the tee. On the distant planet, what are (a) the maximum height and (b) the range of the ball?

A rocket is fired at a speed of 75.0 m/s from ground level, at an angle of 60.0 above the horizontal. The rocket is fired toward an 11.0-m-high wall, which is located 27.0 m away. The rocket attains its launch speed in a negligibly short period of time, after which its engines shut down and the rocket coasts. By how much does the rocket clear the top of the wall?

A rifle is used to shoot twice at a target, using identical cartridges. The first time, the rifle is aimed parallel to the ground and directly at the center of the bulls-eye. The bullet strikes the target at a distance of HA below the center, however. The second time, the rifle is similarly aimed, but from twice the distance from the target. This time the bullet strikes the target at a distance of HB below the center. Find the ratio HB/HA.

An airplane with a speed of 97.5 m/s is climbing upward at an angle of 50.0 with respect to the horizontal. When the planes altitude is 732 m, the pilot releases a package. (a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth. (b) Relative to the ground, determine the angle of the velocity vector of the package just before impact.

Multiple-Concept Example 4 deals with a situation similar to that presented here. A marble is thrown horizontally with a speed of 15 m/s from the top of a building. When it strikes the ground, the marble has a velocity that makes an angle of 65 with the horizontal. From what height above the ground was the marble thrown?

Review Conceptual Example 5 before beginning this problem. You are traveling in a convertible with the top down. The car is moving at a constant velocity of 25 m/s, due east along flat ground. You throw a tomato straight upward at a speed of 11 m/s. How far has the car moved when you get a chance to catch the tomato?

See Multiple-Concept Example 9 for the basic idea behind problems such as this. A diver springs upward from a diving board. At the instant she contacts the water, her speed is 8.90 m/s, and her body is extended at an angle of 75.0 with respect to the horizontal surface of the water. At this instant her vertical displacement is 3.00 m, where downward is the negative direction. Determine her initial velocity, both magnitude and direction.

A soccer player kicks the ball toward a goal that is 16.8 m in front of him. The ball leaves his foot at a speed of 16.0 m/s and an angle of 28.0 above the ground. Find the speed of the ball when the goalie catches it in front of the net.

In the javelin throw at a track-and-field event, the javelin is launched at a speed of 29 m/s at an angle of 36 above the horizontal. As the javelin travels upward, its velocity points above the horizontal at an angle that decreases as time passes. How much time is required for the angle to be reduced from 36 at launch to 18? *

An airplane is flying with a velocity of 240 m/s at an angle of 30.0 with the horizontal, as the drawing shows. When the altitude of the plane is 2.4 km, a flare is released from the plane. The flare hits the target on the ground. What is the angle ? T

A child operating a radio-controlled model car on a dock accidentally steers it off the edge. The cars displacement 1.1 s after leaving the dock has a magnitude of 7.0 m. What is the cars speed at the instant it drives off the edge of the dock?

After leaving the end of a ski ramp, a ski jumper lands downhill at a point that is displaced 51.0 m horizontally from the end of the ramp. His velocity, just before landing, is 23.0 m/s and points in a direction 43.0 below the horizontal. Neglecting air resistance and any lift he experiences while airborne, find his initial velocity (magnitude and direction) when he left the end of the ramp. Express the direction as an angle relative to the horizontal.

Stones are thrown horizontally with the same velocity from the tops of two different buildings. One stone lands twice as far from the base of the building from which it was thrown as does the other stone. Find the ratio of the height of the taller building to the height of the shorter building.

The drawing shows an exaggerated view of a rifle that has been sighted in for a 91.4-meter target. If the muzzle speed of the bullet is v0  427 m/s, what are the two possible angles 1 and 2 between the rifle barrel and the horizontal such that the bullet will hit the target? One of these angles is so large that it is never used in target shooting. (Hint: The following trigonometric identity may be useful: 2 sin cos  sin 2.) 91.4

A projectile is launched from ground level at an angle of 12.0 above the horizontal. It returns to ground level. To what value should the launch angle be adjusted, without changing the launch speed, so that the range doubles?

From the top of a tall building, a gun is fired. The bullet leaves the gun at a speed of 340 m/s, parallel to the ground. As the drawing shows, the bullet puts a hole in a window of another building and hits the wall that faces the window. Using the data in the drawing, determine the distances D and H, which locate the point where the gun was fired. Assume that the bullet does not slow down as it passes through the window.

In the annual battle of the dorms, students gather on the roofs of Jackson and Walton dorms to launch water balloons at each other with slingshots. The horizontal distance between the buildings is 35.0 m, and the heights of the Jackson and Walton buildings are, respectively, 15.0 m and 22.0 m. Ignore air resistance. (a) The first balloon launched by the Jackson team hits Walton dorm 2.0 s after launch, striking it halfway between the ground and the roof. Find the direction of the balloons initial velocity. Give your answer as an angle measured above the horizontal. (b) A second balloon launched at the same angle hits the edge of Waltons roof. Find the initial speed of this second balloon.

Two cannons are mounted as shown in the drawing and rigged to fire simultaneously. They are used in a circus act in which two clowns serve as human cannonballs. The clowns are fired toward each other and collide at a height of 1.00 m above the muzzles of the cannons. Clown A is launched at a 75.0 angle, with a speed of 9.00 m/s. The horizontal separation between the clowns as they leave the cannons is 6.00 m. Find the launch speed v0B and the launch angle B (45.0) for clown B. 6.

In a marathon race Chad is out in front, running due north at a speed of 4.00 m/s. John is 95 m behind him, running due north at a speed of 4.50 m/s. How long does it take for John to pass Chad?

A swimmer, capable of swimming at a speed of 1.4 m/s in still water (i.e., the swimmer can swim with a speed of 1.4 m/s relative to the water), starts to swim directly across a 2.8-km-wide river. However, the current is 0.91 m/s, and it carries the swimmer downstream. (a) How long does it take the swimmer to cross the river? (b) How far downstream will the swimmer be upon reaching the other side of the river?

Two friends, Barbara and Neil, are out rollerblading. With respect to the ground, Barbara is skating due south at a speed of 4.0 m/s. Neil is in front of her. With respect to the ground, Neil is skating due west at a speed of 3.2 m/s. Find Neils velocity (magnitude and direction relative to due west), as seen by Barbara.

A police officer is driving due north at a constant speed of 29 m/s relative to the ground when she notices a truck on an eastwest highway ahead of her, driving west at high speed. She finds that the trucks speed relative to her car is 48 m/s (about 110 mph). (a) Sketch the vector triangle that shows how the trucks velocity relative to the ground is related to the police cars velocity relative to the ground and to the trucks velocity relative to the police car. The sketch need not be to scale, but the velocity vectors should be oriented correctly and bear the appropriate labels. (b) What is the trucks speed, relative to the ground?

You are in a hot-air balloon that, relative to the ground, has a velocity of 6.0 m/s in a direction due east. You see a hawk moving directly away from the balloon in a direction due north. The speed of the hawk relative to you is 2.0 m/s. What are the magnitude and direction of the hawks velocity relative to the ground? Express the directional angle relative to due east.

On a pleasure cruise a boat is traveling relative to the water at a speed of 5.0 m/s due south. Relative to the boat, a passenger walks toward the back of the boat at a speed of 1.5 m/s. (a) What are the magnitude and direction of the passengers velocity relative to the water? (b) How long does it take for the passenger to walk a distance of 27 m on the boat? (c) How long does it take for the passenger to cover a distance of 27 m on the water?

Two passenger trains are passing each other on adjacent tracks. Train A is moving east with a speed of 13 m/s, and train B is traveling west with a speed of 28 m/s. (a) What is the velocity (magnitude and direction) of train A as seen by the passengers in train B? (b) What is the velocity (magnitude and direction) of train B as seen by the passengers in train A?

The captain of a plane wishes to proceed due west. The cruising speed of the plane is 245 m/s relative to the air. A weather report indicates that a 38.0-m/s wind is blowing from the south to the north. In what direction, measured with respect to due west, should the pilot head the plane?

A person looking out the window of a stationary train notices that raindrops are falling vertically down at a speed of 5.0 m/s relative to the ground. When the train moves at a constant velocity, the raindrops make an angle of 25 when they move past the window, as the drawing shows. How fast is the train moving?

A ferryboat is traveling in a direction 38.0 north of east with a speed of 5.50 m/s relative to the water. A passenger is walking with a velocity of 2.50 m/s due east relative to the boat. What is the velocity (magnitude and direction) of the passenger with respect to the water? Determine the directional angle relative to due east.

Mario, a hockey player, is skating due south at a speed of 7.0 m/s relative to the ice. A teammate passes the puck to him. The puck has a speed of 11.0 m/s and is moving in a direction of 22 west of south, relative to the ice. What are the magnitude and direction (relative to due south) of the pucks velocity, as observed by Mario?

A jetliner can fly 6.00 hours on a full load of fuel. Without any wind it flies at a speed of 2.40 102 m/s. The plane is to make a roundtrip by heading due west for a certain distance, turning around, and then heading due east for the return trip. During the entire flight, however, the plane encounters a 57.8-m/s wind from the jet stream, which blows from west to east. What is the maximum distance that the plane can travel due west and just be able to return home?

Two boats are heading away from shore. Boat 1 heads due north at a speed of 3.00 m/s relative to the shore. Relative to boat 1, boat 2 is moving 30.0 north of east at a speed of 1.60 m/s. A passenger on boat 2 walks due east across the deck at a speed of 1.20 m/s relative to boat 2. What is the speed of the passenger relative to the shore?

Useful background for this problem can be found in MultipleConcept Example 2. On a spacecraft two engines fire for a time of 565 s. One gives the craft an acceleration in the x direction of ax 5.10 m/s2 , while the other produces an acceleration in the y direction of ay 7.30 m/s2 . At the end of the firing period, the craft has velocity components of vx 3775 m/s and vy 4816 m/s. Find the magnitude and direction of the initial velocity. Express the direction as an angle with respect to the x axis. 67

A dolphin leaps out of the water at an angle of 35 above the horizontal. The horizontal component of the dolphins velocity is 7.7 m/s. Find the magnitude of the vertical component of the velocity.

A hot-air balloon is rising straight up with a speed of 3.0 m/s. A ballast bag is released from rest relative to the balloon at 9.5 m above the ground. How much time elapses before the ballast bag hits the ground?

A golf ball rolls off a horizontal cliff with an initial speed of 11.4 m/s. The ball falls a vertical distance of 15.5 m into a lake below. (a) How much time does the ball spend in the air? (b) What is the speed v of the ball just before it strikes the water?

When chasing a hare along a flat stretch of ground, a greyhound leaps into the air at a speed of 10.0 m/s, at an angle of 31.0 above the horizontal. (a) What is the range of his leap and (b) for how much time is he in the air?

Multiple-Concept Example 4 provides useful background for this problem. A diver runs horizontally with a speed of 1.20 m/s off a platform that is 10.0 m above the water. What is his speed just before striking the water?

A ball is thrown upward at a speed v0 at an angle of 52 above the horizontal. It reaches a maximum height of 7.5 m. How high would this ball go if it were thrown straight upward at speed v0?

A golfer, standing on a fairway, hits a shot to a green that is elevated 5.50 m above the point where she is standing. If the ball leaves her club with a velocity of 46.0 m/s at an angle of 35.0 above the ground, find the time that the ball is in the air before it hits the green.

At some airports there are speed ramps to help passengers get from one place to another. A speed ramp is a moving conveyor belt on which you can either stand or walk. Suppose a speed ramp has a length of 105 m and is moving at a speed of 2.0 m/s relative to the ground. In addition, suppose you can cover this distance in 75 s when walking on the ground. If you walk at the same rate with respect to the speed ramp that you walk on the ground, how long does it take for you to travel the 105 m using the speed ramp?