Solved: Starting from rest, two skaters push off against

A baseball (m 0.14 kg) has an initial velocity of 0 38 m/s as it approaches a bat. We have chosen the direction of approach as the negative direction. The bat applies an average force that is much larger than the weight of the ball, and the ball departs from the bat with a final velocity of f 58 m/s. (a) Determine the impulse applied to the ball by the bat. (b) Assuming that the time of contact is t 1.6 s, find the average force exerted on the ball by the bat. R

During a storm, rain comes straight down with a velocity of 0 15 m/s and hits the roof of a car perpendicularly (see Figure 7.4). The mass of rain per second that strikes the car roof is 0.060 kg/s. Assuming that the rain comes to rest upon striking the car ( f 0 m/s), find the average force exerted by the rain on the roof.

In Example 2 rain is falling on the roof of a car and exerts a force on it. Instead of rain, suppose hail is falling. The hail comes straight down at a mass rate of m/t 0.060 kg/s and an initial velocity of 0 15 m /s and strikes the roof perpendicularly, just as the rain does in Example 2. However, unlike rain, hail usually does not come to rest after striking a surface. Instead, the hailstones bounce off the roof of the car. If hail fell instead of rain, would the force on the roof be (a) smaller than, (b) equal to, or (c) greater than that calculated in Example 2? Re

Two identical automobiles have the same speed, one traveling east and one traveling west. Do these cars have the same momentum?

In Times Square in New York City, people celebrate on New Years Eve. Some just stand around, but many move about randomly. Consider a group consisting of all of these people. Approximately, what is the total linear momentum of this group at any given instant?

Two objects have the same momentum. Do the velocities of these objects necessarily have (a) the same directions and (b) the same magnitudes?

a) Can a single object have a kinetic energy but no momentum? (b) Can a group of two or more objects have a total kinetic energy that is not zero but a total momentum that is zero? 5. Suppose you are standing on the edge of a dock and jump straight down. If you land on sand your stopping time is much shorter than if you land on water. Using the impulsemomentum theorem as a guide, determine which one of the following statements is correct. (a) In bringing you to a halt, the sand exerts a greater impulse on you than does the water. (b) In bringing you to a halt, the sand and the water exert the same impulse on you, but the sand exerts a greater average force. (c) In bringing you to a halt, the sand and the water exert the same impulse on you, but the sand exerts a smaller average force.

An airplane is flying horizontally with a constant momentum during a time interval t. (a) Is there a net impulse acting on the plane during this time? Use the impulsemomentum theorem to guide your thinking. (b) In the horizontal direction, both the thrust generated by the engines and air resistance act on the plane. Considering your answer to part (a), how is the impulse of the thrust related (in magnitude and direction) to the impulse of the force due to the air resistance?

Imagine two balls colliding on a billiard table that is friction-free. Using the momentumconservation principle as a guide, decide which statement is correct: (a) The total momentum of the system that contains only one of the two balls is the same before and after the collision. (b) The total momentum of the system that contains both of the two balls is the same before and after the collision.

A freight train is being assembled in a switching yard, and Figure 7.8 shows two boxcars. Car 1 has a mass of m1 65 103 kg and moves at a velocity of v01 0.80 m/s. Car 2, with a mass of m2 92 103 kg and a velocity of v02 1.3 m/s, overtakes car 1 and couples to it. Neglecting friction, find the common velocity vf of the cars after they become coupled. Exam

Starting from rest, two skaters push off against each other on smooth level ice, where friction is negligible. As Figure 7.9a shows, one is a woman (m1 54 kg), and one is a man (m2 88 kg). Part b of the drawing shows that the woman moves away with a velocity of vf1 2.5 m/s. Find the recoil velocity vf2 of the man. Re

An object slides along the surface of the earth and slows down because of kinetic friction. If the object alone is considered as the system, the kinetic frictional force must be identified as an external force that, according to Equation 7.4, decreases the momentum of the system. (a) If both the object and the earth are considered to be the system, is the force of kinetic friction still an external force? (b) Can the frictional force change the total linear momentum of the two-body system?

A satellite explodes in outer space, far from any other body, sending thousands of pieces in all directions. Is the linear momentum of the satellite before the explosion less than, equal to, or greater than the total linear momentum of all the pieces after the explosion?

On a distant asteroid, a large catapult is used to throw chunks of stone into space. Could such a device be used as a propulsion system to move the asteroid closer to the earth?

A canoe with two people aboard is coasting with an initial momentum of 110 kg  m/s. Then, one of the people (person 1) dives off the back of the canoe. During this time, the net average external force acting on the system (the canoe and the two people) is zero. The table lists four possibilities for the final momentum of person 1 and the final momentum of person 2 plus the canoe, immediately after person 1 dives off. Only one possibility could be correct. Which one is it?

You are a passenger on a jetliner that is flying at a constant velocity. You get up from your seat and walk toward the front of the plane. Because of this action, your forward momentum increases. Does the forward momentum of the plane itself decrease, remain the same, or increase?

An ice boat is coasting on a frozen lake. Friction between the ice and the boat is negligible, and so is air resistance. Nothing is propelling the boat. From a bridge someone jumps straight down into the boat, which continues to coast straight ahead. (a) Does the total horizontal momentum of the boat plus the jumper change? (b) Does the speed of the boat itself increase, decrease, or remain the same?

Concept Simulation 7.1 at www.wiley.com/college/cutnell reviews the concepts that are pertinent in this question. In movies, Superman hovers in midair, grabs a villain by the neck, and throws him forward. Superman, however, remains stationary. This is not possible, because it violates which one or more of the following: (a) The law of conservation of energy (b) Newtons second law (c) Newtons third law (d) The principle of conservation of linear momentum

The energy released by the exploding gunpowder in a cannon propels the cannonball forward. Simultaneously, the cannon recoils. The mass of the cannonball is less than that of the cannon. Which has the greater kinetic energy, the launched cannonball or the recoiling cannon? Assume that momentum conservation applies.

Figure 7.10 illustrates an elastic head-on collision between two balls. One ball has a mass of m1 0.250 kg and an initial velocity of 5.00 m/s. The other has a mass of m2 0.800 kg and is initially at rest. No external forces act on the balls. What are the velocities of the balls after the collision?

A ballistic pendulum can be used to measure the speed of a projectile, such as a bullet. The ballistic pendulum shown in Figure 7.12a consists of a stationary 2.50-kg block of wood suspended by a wire of negligible mass. A 0.0100-kg bullet is fired into the block, and the block (with the bullet in it) swings to a maximum height of 0.650 m above the initial position (see part b of the drawing). Find the speed with which the bullet is fired, assuming that air resistance is negligible.

Two balls collide in a one-dimensional elastic collision. The two balls constitute a system, and the net external force acting on them is zero. The table shows four possible sets of values for the initial and final momenta of the two balls, as well as their initial and final kinetic energies. Only one set of values could be correct. Which set is it?

In an elastic collision, is the kinetic energy of each object the same before and after the collision?

Concept Simulation 7.2 at www.wiley.com/college/cutnell illustrates the concepts that are involved in this question. Also review Multiple-Concept Example 7. Suppose two objects collide head on, as in Example 7, where initially object 1 (mass m1) is moving and object 2 (mass m2) is stationary. Now assume that they have the same mass, so m1 m2. Which one of the following statements is true? (a) Both objects have the same velocity (magnitude and direction) after the collision. (b) Object 1 rebounds with one-half its initial speed, while object 2 moves to the right, as in Figure 7.10, with one-half the speed that object 1 had before the collision. (c) Object 1 stops completely, while object 2 acquires the same velocity (magnitude and direction) that object 1 had before the collision. Co

For the situation in Figure 7.13a, use momentum conservation to determine the magnitude and direction of the final velocity of ball 1 after the collision.

Would you expect the center of mass of a baseball bat to be located halfway between the ends of the bat, nearer the lighter end, or nearer the heavier end?

A sunbather is lying on a floating raft that is stationary. She then gets up and walks to one end of the raft. Consider the sunbather and raft as an isolated system. (a) What is the velocity of the center of mass of this system while she is walking? (b) Does the raft itself move while she is walking? If so, what is the direction of the rafts velocity relative to that of the sunbather?

Water, dripping at a constant rate from a faucet, falls to the ground. At any instant there are many drops in the air between the faucet and the ground. Where does the center of mass of the drops lie relative to the halfway point between the faucet and the ground? (a) Above it (b) Below it (c) Exactly at the halfway point

Two joggers, Jim and Tom, are both running at a speed of 4.00 m/s. Jim has a mass of 90.0 kg, and Tom has a mass of 55.0 kg. Find the kinetic energy and momentum of the two-jogger system when (a) Jim and Tom are both running due north (Figure 7.17a) and (b) Jim is running due north and Tom is running due south (Figure 7.17b).

Does the total kinetic energy have a smaller value in case (a) or (b), or is it the same in both cases?

Does the total momentum have a smaller magnitude in case (a) or (b), or is it the same for each?

The following table gives mass and speed data for the two objects in Figure 7.18. Find the magnitude of the momentum and the kinetic energy for each object.

Is it possible for two objects to have different speeds when their momenta have the same magnitude?

Two identical cars are traveling at the same speed. One is heading due east and the other due north, as the drawing shows. Which statement is true regarding the kinetic energies and momenta of the cars? (a) They both have the same kinetic energies and the same momenta. (b) They have the same kinetic energies, but different momenta. (c) They have different kinetic energies, but the same momenta. (d) They have different kinetic energies and different momenta.

Six runners have the mass (in multiples of speed (in multiples of v0), and direction of travel that are indicated in the table. Which two runners have identical momenta? (a) B and C (b) A and C (c) C and D (d) A and E (e) D and F

A particle is moving along the axis, and the graph shows its momentum p as a function of time t. Use the impulsemomentum theorem to rank (largest to smallest) the three regions according to the magnitude of the impulse applied to the particle. (a) A, B, C (b) A, C, B (c) A and C (a tie), B (d) C, A, B (e) B, A, C

A particle moves along the axis, and the graph shows its momentum p as a function of time t. In each of the four regions a force, which may or may not be nearly zero, is applied to the particle. In which region is the magnitude of the force largest and in which region is it smallest? (a) B largest, D smallest (b) C largest, B smallest (c) A largest, D smallest (d) C largest, A smallest (e) A largest, C smallest

As the text discusses, the conservation of linear momentum is applicable only when the system of objects is an isolated system. Which of the systems listed below are isolated systems? 1. A ball is dropped from the top of a building. The system is the ball. 2. A ball is dropped from the top of a building. The system is the ball and the earth.3. A billiard ball collides with a stationary billiard ball on a frictionless pool table. The system is the moving ball. 4. A car slides to a halt in an emergency. The system is the car. 5. A space probe is moving in deep space where gravitational and other forces are negligible. The system is the space probe. (a) Only 2 and 5 are isolated systems. (b) Only 1 and 3 are isolated systems. (c) Only 3 and 5 are isolated systems. (d) Only 4 and 5 are isolated systems. (e) Only 5 is an isolated system.

Two objects are involved in a completely inelastic one-dimensional collision. The net external force acting on them is zero. The table lists four possible sets of the initial and final momenta and kinetic energies of the two objects. Which is the only set that could occur?

Object 1 is moving along the x axis with an initial momentum of where the  sign indicates that it is moving to the right. As the drawing shows, object 1 collides with a second object that is initially at rest. The collision is not head-on, so the objects move off in different directions after the collision. The net external force acting on the two-object system is zero. After the collision, object 1 has a momentum whose y component is 5 kg m/s. What is the y component of the momentum of object 2 after the collision? (a) 0 kg m/s (b) 16 kg m/s (c) 5 kg m/s (d) 16 kg m/s (e) The y component of the momentum of object 2 cannot be determined.

The drawing shows three particles that are moving with different velocities. Two of the particles have mass m, and the third has a mass 2m. At the instant shown, the center of mass (cm) of the three particles is at the coordinate origin. What is the velocity vcm (magnitude and direction) of the center of mass?

A 46-kg skater is standing still in front of a wall. By pushing against the wall she propels herself backward with a velocity of 1.2 m/s. Her hands are in contact with the wall for 0.80 s. Ignore friction and wind resistance. Find the magnitude and direction of the average force she exerts on the wall (which has the same magnitude as, but opposite direction to, the force that the wall applies to her).

A model rocket is constructed with a motor that can provide a total impulse of 29.0 N s. The mass of the rocket is 0.175 kg. What is the speed that this rocket achieves when launched from rest? Neglect the effects of gravity and air resistance.

Before starting this problem, review Conceptual Example 3. Suppose that the hail described there bounces off the roof of the car with a velocity of 15 m/s. Ignoring the weight of the hailstones, calculate the force exerted by the hail on the roof. Compare your answer to that obtained in Example 2 for the rain, and verify that your answer is consistent with the conclusion reached in Conceptual Example 3.

In a performance test, each of two cars takes 9.0 s to accelerate from rest to 27 m/s. Car A has a mass of 1400 kg, and car B has a mass of 1900 kg. Find the net average force that acts on each car during the test.

A volleyball is spiked so that its incoming velocity of 4.0 m/s is changed to an outgoing velocity of 21 m/s. The mass of the volleyball is 0.35 kg. What impulse does the player apply to the ball?

Two arrows are fired horizontally with the same speed of 30.0 m/s. Each arrow has a mass of 0.100 kg. One is fired due east and the other due south. Find the magnitude and direction of the total momentum of this two-arrow system. Specify the direction with respect to due east.

Refer to Conceptual Example 3 as an aid in understanding this problem. A hockey goalie is standing on ice. Another player fires a puck (m 0.17 kg) at the goalie with a velocity of 65 m/s. (a) If the goalie catches the puck with his glove in a time of 5.0 103 s, what is the average force (magnitude and direction) exerted on the goalie by the puck? (b) Instead of catching the puck, the goalie slaps it with his stick and returns the puck straight back to the player with a velocity of 65 m/s. The puck and stick are in contact for a time of 5.0 103 s. Now what is the average force exerted on the goalie by the puck? Verify that your answers to parts (a) and (b) are consistent with the conclusion of Conceptual Example 3.

When jumping straight down, you can be seriously injured if you land stiff-legged. One way to avoid injury is to bend your knees upon landing to reduce the force of the impact. A 75-kg man just before contact with the ground has a speed of 6.4 m/s. (a) In a stiff-legged landing he comes to a halt in 2.0 ms. Find the average net force that acts on him during this time. (b) When he bends his knees, he comes to a halt in 0.10 s. Find the average net force now. (c) During the landing, the force of the ground on the man points upward, while the force due to gravity points downward. The average net force acting on the man includes both of these forces. Taking into account the directions of the forces, find the force of the ground on the man in parts (a) and (b).

A space probe is traveling in outer space with a momentum that has a magnitude of 7.5 107 kg m/s. A retrorocket is fired to slow down the probe. It applies a force to the probe that has a magnitude of 2.0 106 N and a direction opposite to the probes motion. It fires for a period of 12 s. Determine the momentum of the probe after the retrorocket ceases to fire.

A stream of water strikes a stationary turbine blade horizontally, as the drawing illustrates. The incident water stream has a velocity of 16.0 m/s, while the exiting water stream has a velocity of 16.0 m/s. The mass of water per second that strikes the blade is 30.0 kg/s. Find the magnitude of the average force exerted on the water by the blade.

A student (m 63 kg) falls freely from rest and strikes the ground. During the collision with the ground, he comes to rest in a time of 0.040 s. The average force exerted on him by the ground is 18 000 N, where the upward direction is taken to be the positive direction. From what height did the student fall? Assume that the only force acting on him during the collision is that due to the ground.

A golf ball strikes a hard, smooth floor at an angle of 30.0 and, as the drawing shows, rebounds at the same angle. The mass of the ball is 0.047 kg, and its speed is 45 m/s just before and after striking the floor. What is the magnitude of the impulse applied to the golf ball by the floor? (Hint: Note that only the vertical component of the balls momentum changes during impact with the floor, and ignore the weight of the ball.)

An 85-kg jogger is heading due east at a speed of 2.0 m/s. A 55-kg jogger is heading 32 north of east at a speed of 3.0 m/s. Find the magnitude and direction of the sum of the momenta of the two joggers.

A basketball (m 0.60 kg) is dropped from rest. Just before striking the floor, the ball has a momentum whose magnitude is 3.1 kg m/s. At what height was the basketball dropped?

A dump truck is being filled with sand. The sand falls straight downward from rest from a height of 2.00 m above the truck bed, and the mass of sand that hits the truck per second is 55.0 kg/s. The truck is parked on the platform of a weight scale. By how much does the scale reading exceed the weight of the truck and sand?

In a science fiction novel two enemies, Bonzo and Ender, are fighting in outer space. From stationary positions they push against each other. Bonzo flies off with a velocity of 1.5 m/s, while Ender recoils with a velocity of 2.5 m/s. (a) Without doing any calculations, decide which person has the greater mass. Give your reasoning. (b) Determine the ratio mBonzo /mEnder of the masses of these two enemies.

A 2.3-kg cart is rolling across a frictionless, horizontal track toward a 1.5-kg cart that is held initially at rest. The carts are loaded with strong magnets that cause them to attract one another. Thus, the speed of each cart increases. At a certain instant before the carts collide, the first carts velocity is 4.5 m/s, and the second carts velocity is 1.9 m/s. (a) What is the total momentum of the system of the two carts at this instant? (b) What was the velocity of the first cart when the second cart was still at rest? 1

As the drawing illustrates, two disks with masses and are moving horizontally to the right at a speed of v0. They are on an air-hockey table, which supports them with an essentially frictionless cushion of air. They move as a unit, with a compressed spring between them, which has a negligible mass. Then the spring is released and allowed to push the disks outward. Consider the situation where disk 1 comes to a momentary halt shortly after the spring is released. Assuming that m1  1.2 kg, m2  2.4 kg, and v0  5.0 m/s, find the velocity of disk 2 at that moment.

A lumberjack (mass 98 kg) is standing at rest on one end of a floating log (mass  230 kg) that is also at rest. The lumberjack runs to the other end of the log, attaining a velocity of 3.6 m/s relative to the shore, and then hops onto an identical floating log that is initially at rest. Neglect any friction and resistance between the logs and the water. (a) What is the velocity of the first log just before the lumberjack jumps off? (b) Determine the velocity of the second log if the lumberjack comes to rest on it.

An astronaut in his space suit and with a propulsion unit (empty of its gas propellant) strapped to his back has a mass of 146 kg. The astronaut begins a space walk at rest, with a completely filled propulsion unit. During the space walk, the unit ejects some gas with a velocity of 32 m/s. As a result, the astronaut recoils with a velocity of 0.39 m/s. After the gas is ejected, the mass of the astronaut (now wearing a partially empty propulsion unit) is 165 kg. What percentage of the gas was ejected from the completely filled propulsion unit? 2

A two-stage rocket moves in space at a constant velocity of 4900 m/s. The two stages are then separated by a small explosive charge placed between them. Immediately after the explosion the velocity of the 1200-kg upper stage is 5700 m/s in the same direction as before the explosion. What is the velocity (magnitude and direction) of the 2400-kg lower stage after the explosion?

A 40.0-kg boy, riding a 2.50-kg skateboard at a velocity of 5.30 m/s across a level sidewalk, jumps forward to leap over a wall. Just after leaving contact with the board, the boys velocity relative to the sidewalk is 6.00 m/s, 9.50 above the horizontal. Ignore any friction between the skateboard and the sidewalk. What is the skateboards velocity relative to the sidewalk at this instant? Be sure to include the correct algebraic sign with your answer.

The lead female character in the movie Diamonds Are Forever is standing at the edge of an offshore oil rig. As she fires a gun, she is driven back over the edge and into the sea. Suppose the mass of a bullet is 0.010 kg and its velocity is 720 m/s. Her mass (including the gun) is 51 kg. (a) What recoil velocity does she acquire in response to a single shot from a stationary position, assuming that no external force keeps her in place? (b) Under the same assumption, what would be her recoil velocity if, instead, she shoots a blank cartridge that ejects a mass of 5.0  104 kg at a velocity of 720 m/s? *

A 0.015-kg bullet is fired straight up at a falling wooden block that has a mass of 1.8 kg. The bullet has a speed of 810 m/s when it strikes the block. The block originally was dropped from rest from the top of a building and has been falling for a time t when the collision with the bullet occurs. As a result of the collision, the block (with the bullet in it) reverses direction, rises, and comes to a momentary halt at the top of the building. Find the time t.

By accident, a large plate is dropped and breaks into three pieces. The pieces fly apart parallel to the floor. As the plate falls, its momentum has only a vertical component and no component parallel to the floor. After the collision, the component of the total momentum parallel to the floor must remain zero, since the net external force acting on the plate has no component parallel to the floor. Using the data shown in the drawing, find the masses of pieces 1 and 2.

Adolf and Ed are wearing harnesses and are hanging at rest from the ceiling by means of ropes attached to them. Face to face, they push off against one another. Adolf has a mass of 120 kg, and Ed has a mass of 78 kg. Following the push, Adolf swings upward to a height of 0.65 m above his starting point. To what height above his own starting point does Ed rise?

A cannon of mass 5.80  103 kg is rigidly bolted to the earth so it can recoil only by a negligible amount. The cannon fires an 85.0-kg shell horizontally with an initial velocity of 551 m/s. Suppose the cannon is then unbolted from the earth, and no external force hinders its recoil. What would be the velocity of an identical shell fired by this loose cannon? (Hint: In both cases assume that the burning gunpowder imparts the same kinetic energy to the system.)

After sliding down a snow-covered hill on an inner tube, Ashley is coasting across a level snowfield at a constant velocity of 2.7 m/s. Miranda runs after her at a velocity of 4.5 m/s and hops on the inner tube. How fast do the two slide across the snow together on the inner tube? Ashleys mass is 71 kg and Mirandas is 58 kg. Ignore the mass of the inner tube and any friction between the inner tube and the snow. 2

Consult Multiple-Concept Example 8 for background pertinent to this problem. A 2.50-g bullet, traveling at a speed of 425 m/s, strikes the wooden block of a ballistic pendulum, such as that in Figure 7.12. The block has a mass of 215 g. (a) Find the speed of the bulletblock combination immediately after the collision. (b) How high does the combination rise above its initial position?

One object is at rest, and another is moving. The two collide in a one-dimensional, completely inelastic collision. In other words, they stick together after the collision and move off with a common velocity. Momentum is conserved. The speed of the object that is moving initially is 25 m/s. The masses of the two objects are 3.0 and 8.0 kg. Determine the final speed of the two-object system after the collision for the case when the large-mass object is the one moving initially and the case when the small-mass object is the one moving initially.

Batman (mass  91 kg) jumps straight down from a bridge into a boat (mass  510 kg) in which a criminal is fleeing. The velocity of the boat is initially 11 m/s. What is the velocity of the boat after Batman lands in it?

A car (mass  1100 kg) is traveling at 32 m/s when it collides headon with a sport utility vehicle (mass  2500 kg) traveling in the opposite direction. In the collision, the two vehicles come to a halt. At what speed was the sport utility vehicle traveling?

5.00-kg ball, moving to the right at a velocity of 2.00 m/s on a frictionless table, collides head-on with a stationary 7.50-kg ball.Find the final velocities of the balls if the collision is (a) elastic and (b) completely inelastic.

Multiple-Concept Example 9 reviews the concepts that play roles in this problem. The drawing shows a collision between two pucks on an air-hockey table. Puck A has a mass of 0.025 kg and is moving along the x axis with a velocity of 5.5 m/s. It makes a collision with puck B, which has a mass of 0.050 kg and is initially at rest. The collision is not head-on. After the collision, the two pucks fly apart with the angles shown in the drawing. Find the final speeds of (a) puck A and (b) puck B.

A projectile (mass  0.20 kg) is fired at and embeds itself in a stationary target (mass  2.50 kg). With what percentage of the projectiles incident kinetic energy does the target (with the projectile in it) fly off after being struck?

Object A is moving due east, while object B is moving due north. They collide and stick together in a completely inelastic collision. Momentum is conserved. Object A has a mass of mA  17.0 kg and an initial velocity of 0A , due east. Object B, however, has a mass of and an initial velocity of 0B , due north. Find the magnitude and direction of the total momentum of the two-object system after the collision.

Multiple-Concept Example 7 deals with some of the concepts that are used to solve this problem. A cue ball (mass  0.165 kg) is at rest on a frictionless pool table. The ball is hit dead center by a pool stick, which applies an impulse of 1.50 N s to the ball. The ball then slides along the table and makes an elastic head-on collision with a second ball of equal mass that is initially at rest. Find the velocity of the second ball just after it is struck.

A ball is attached to one end of a wire, the other end being fastened to the ceiling. The wire is held horizontal, and the ball is released from rest (see the drawing). It swings downward and strikes a block initially at rest on a horizontal frictionless surface. Air resistance is negligible, and the collision is elastic. The masses of the ball and block are, respectively, 1.60 kg and 2.40 kg, and the length of the wire is 1.20 m. Find the velocity (magnitude and direction) of the ball (a) just before the collision, and (b) just after the collision.

A girl is skipping stones across a lake. One of the stones accidentally ricochets off a toy boat that is initially at rest in the water (see the drawing). The 0.072-kg stone strikes the boat at a velocity of 13 m/s, 15 below due east, and ricochets off at a velocity of 11 m/s, 12 above due east. After being struck by the stone, the boats velocity is 2.1 m/s, due east. What is the mass of the boat? Assume the water offers no resistance to the boats motion.

A mine car (mass  440 kg) rolls at a speed of 0.50 m/s on a horizontal track, as the drawing shows. A 150-kg chunk of coal has a speed of 0.80 m/s when it leaves the chute. Determine the speed of the carcoal system after the coal has come to rest in the car.

A 50.0-kg skater is traveling due east at a speed of 3.00 m/s. A 70.0-kg skater is moving due south at a speed of 7.00 m/s. They collide and hold on to each other after the collision, managing to move off at an angle south of east, with a speed of vf. Find (a) the angle and (b) the speed vf, assuming that friction can be ignored. *

A 4.00-g bullet is moving horizontally with a velocity of , where the sign indicates that it is moving to the right (see part a of the drawing). The bullet is approaching two blocks resting on a horizontal frictionless surface. Air resistance is negligible. The bullet passes completely through the first block (an inelastic collision) and embeds itself in the second one, as indicated in part b. Note that both blocks are moving after the collision with the bullet. The mass of the first block is 1150 g, and its velocity is 0.550 m/s after the bullet passes through it. The mass of the second block is 1530 g. (a) What is the velocity of the second block after the bullet embeds itself? (b) Find the ratio of the total kinetic energy after the collisions to that before the collisions. *

An electron collides elastically with a stationary hydrogen atom. The mass of the hydrogen atom is 1837 times that of the electron. Assume that all motion, before and after the collision, occurs along the same straight line. What is the ratio of the kinetic energy of the hydrogen atom after the collision to that of the electron before the collision?

A 60.0-kg person, running horizontally with a velocity of 3.80 m/s, jumps onto a 12.0-kg sled that is initially at rest. (a) Ignoring the effects of friction during the collision, find the velocity of the sled and person as they move away. (b) The sled and person coast 30.0 m on level snow before coming to rest. What is the coefficient of kinetic friction between the sled and the snow? *

Starting with an initial speed of 5.00 m/s at a height of 0.300 m, a 1.50-kg ball swings downward and strikes a 4.60-kg ball that is at rest, as the drawing shows. (a) Using the principle of conservation of mechanical energy, find the speed of the 1.50-kg ball just before impact. (b) Assuming that the collision is elastic, find the velocities (magnitude and direction) of both balls just after the collision. (c) How high does each ball swing after the collision, ignoring air resistance?

Multiple-Concept Example 7 outlines the general approach to problems like this one. Two identical balls are traveling toward each other with velocities of 4.0 and 7.0 m/s, and they experience an elastic head-on collision. Obtain the velocities (magnitude and direction) of each ball after the collision. *

A ball is dropped from rest from the top of a 6.10-m-tall building, falls straight downward, collides inelastically with the ground, and bounces back. The ball loses 10.0% of its kinetic energy every time it collides with the ground. How many bounces can the ball make and still reach a windowsill that is 2.44 m above the ground?

Two particles are moving along the x axis. Particle 1 has a mass m1 and a velocity v1  4.6 m/s. Particle 2 has a mass m2 and a velocity v2  6.1 m/s. The velocity of the center of mass of these two particles is zero. In other words, the center of mass of the particles remains stationary, even though each particle is moving. Find the ratio m1/m2 of the masses of the particles. 4

Consider the two moving boxcars in Example 5. Determine the velocity of their center of mass (a) before and (b) after the collision. (c) Should your answer in part (b) be less than, greater than, or equal to the common velocity vf of the two coupled cars after the collision? Justify your answer.

Johns mass is 86 kg, and Barbaras is 55 kg. He is standing on the x axis at xJ 9.0 m, while she is standing on the x axis at xB  2.0 m. They switch positions. How far and in which direction does their center of mass move as a result of the switch?

Two stars in a binary system orbit around their center of mass.The centers of the two stars are apart. The larger of the two stars has a mass of and its center is from the systems center of mass. What is the mass of the smaller star?

The drawing shows a sulfur dioxide molecule. It consists of two oxygen atoms and a sulfur atom. A sulfur atom is twice as massive as an oxygen atom. Using this information and the data provided in the drawing, find (a) the x coordinate and (b) the y coordinate of the center of mass of the sulfur dioxide molecule. Express your answers in nanometers (1 nm  109 m).

Two friends, Al and Jo, have a combined mass of 168 kg. At an ice skating rink they stand close together on skates, at rest and facing each other, with a compressed spring between them. The spring is kept from pushing them apart because they are holding each other. When they release their arms, Al moves off in one direction at a speed of 0.90 m/s, while Jo moves off in the opposite direction at a speed of 1.2 m/s. Assuming that friction is negligible, find Als mass.

Multiple-Concept Example 7 presents a model for solving problems such as this one. A 1055-kg van, stopped at a traffic light, is hit directly in the rear by a 715-kg car traveling with a velocity of 2.25 m/s. Assume that the transmission of the van is in neutral, the brakes are not being applied, and the collision is elastic. What are the final velocities of (a) the car and (b) the van?

A baseball (m  149 g) approaches a bat horizontally at a speed of 40.2 m/s (90 mi/h) and is hit straight back at a speed of 45.6 m/s (102 mi/h). If the ball is in contact with the bat for a time of 1.10 ms, what is the average force exerted on the ball by the bat? Neglect the weight of the ball, since it is so much less than the force of the bat. Choose the direction of the incoming ball as the positive direction.

One average force has a magnitude that is three times as large as that of another average force . Both forces produce the same impulse. The average force acts for a time interval of 3.2 ms. For what time interval does the average force act? F

For tests using a ballistocardiograph, a patient lies on a horizontal platform that is supported on jets of air. Because of the air jets, the friction impeding the horizontal motion of the platform is negligible. Each time the heart beats, blood is pushed out of the heart in a direction that is nearly parallel to the platform. Since momentum must be conserved, the body and the platform recoil, and this recoil can be detected to provide information about the heart. For each beat, suppose that 0.050 kg of blood is pushed out of the heart with a velocity of 0.25 m/s and that the mass of the patient and platform is 85 kg. Assuming that the patient does not slip with respect to the platform, and that the patient and platform start from rest, determine the recoil velocity.

The carbon monoxide molecule (CO) consists of a carbon atom and an oxygen atom separated by a distance of 1.13  1010 m. The mass mC of the carbon atom is 0.750 times the mass mO of the oxygen atom, or mC  0.750 mO. Determine the location of the center of mass of this molecule relative to the carbon atom.

A wagon is rolling forward on level ground. Friction is negligible. The person sitting in the wagon is holding a rock. The total mass of the wagon, rider, and rock is 95.0 kg. The mass of the rock is 0.300 kg. Initially the wagon is rolling forward at a speed of 0.500 m/s. Then the person throws the rock with a speed of 16.0 m/s. Both speeds are relative to the ground. Find the speed of the wagon after the rock is thrown directly forward in one case and directly backward in another.

See Multiple-Concept Example 9 to review the approach used in problems of this type. Three guns are aimed at the center of a circle, and each fires a bullet simultaneously. The directions in which they fire are 120 apart. Two of the bullets have the same mass of 4.50  103 kg and the same speed of 324 m/s. The other bullet has an unknown mass and a speed of 575 m/s. The bullets collide at the center and mash into a stationary lump. What is the unknown mass? A

A fireworks rocket is moving at a speed of 45.0 m/s. The rocket suddenly breaks into two pieces of equal mass, which fly off with velocities 1 and 2, as shown in the drawing. What are the magnitudes of (a) 1 and (b) 2?

Two ice skaters have masses m1 and m2 and are initially stationary. Their skates are identical. They push against one another, as in Figure 7.9, and move in opposite directions with different speeds. While they are pushing against each other, any kinetic frictional forces acting on their skates can be ignored. However, once the skaters separate, kinetic frictional forces eventually bring them to a halt. As they glide to a halt, the magnitudes of their accelerations are equal, and skater 1 glides twice as far as skater 2. What is the ratio m1/m2 of their masses?

The drawing shows a human figure in a sitting position. For purposes of this problem, there are three parts to the figure, and the center of mass of each one is shown in the drawing. These parts are: (1) the torso, neck, and head (total mass 41 kg) with a center of mass located on the y axis at a point 0.39 m above the origin, (2) the upper legs (mass 17 kg) with a center of mass located on the x axis at a point 0.17 m to the right of the origin, and (3) the lower legs and feet (total mass 9.9 kg) with a center of mass located 0.43 m to the right of and 0.26 m below the origin. Find the x and y coordinates of the center of mass of the human figure. Note that the mass of the arms and hands (approximately 12% of the whole-body mass) has been ignored to simplify the drawing. *

Two people are standing on a 2.0-m-long stationary platform, one at each end. The platform floats parallel to the ground on a cushion of air, like a hovercraft. One person throws a 6.0-kg ball to the other, who catches it. The ball travels nearly horizontally. Excluding the ball, the total mass of the platform and people is 118 kg. Because of the throw, this 118-kg mass recoils. How far does it move before coming to rest again?

A golf ball bounces down a flight of steel stairs, striking several steps on the way down, but never hitting the edge of a step. The ball starts at the top step with a vertical velocity component of zero. If all the collisions with the stairs are elastic, and if the vertical height of the staircase is 3.00 m, determine the bounce height when the ball reaches the bottom of the stairs. Neglect air resistance.