Consider a system of two particles in the xy plane: m1 5

You are standing on a saucer-shaped sled at rest in the middle of a frictionless ice rink. Your lab partner throws you a heavy Frisbee. You take different actions in successive experimental trials. Rank the following situations according to your final speed from largest to smallest. If your final speed is the same in two cases, give them equal rank. (a) You catch the Frisbee and hold onto it. (b) You catch the Frisbee and throw it back to your partner. (c) You bobble the catch, just touching the Frisbee so that it continues in its original direction more slowly. (d) You catch the Frisbee and throw it so that it moves vertically upward above your head. (e) You catch the Frisbee and set it down so that it remains at rest on the ice.

A boxcar at a rail yard is set into motion at the top of a hump. The car rolls down quietly and without friction onto a straight, level track where it couples with a flatcar of smaller mass, originally at rest, so that the two cars then roll together without friction. Consider the two cars as a system from the moment of release of the boxcar until both are rolling together. Answer the following questions yes or no. (a) Is mechanical energy of the system conserved? (b) Is momentum of the system conserved? Next, consider only the process of the boxcar gaining speed as it rolls down the hump. For the boxcar and the Earth as a system, (c) is mechanical energy conserved? (d) Is momentum conserved? Finally, consider the two cars as a system as the boxcar is slowing down in the coupling process. (e) Is mechanical energy of this system conserved? (f) Is momentum of this system conserved?

A massive tractor is rolling down a country road. In a perfectly inelastic collision, a small sports car runs into the machine from behind. (i) Which vehicle experiences a change in momentum of larger magnitude? (a) The car does. (b) The tractor does. (c) Their momentum changes are the same size. (d) It could be either vehicle. (ii) Which vehicle experiences a larger change in kinetic energy? (a) The car does. (b) The tractor does. (c) Their kinetic energy changes are the same size. (d) It could be either vehicle.

A 2-kg object moving to the right with a speed of 4 m/s makes a head-on, elastic collision with a 1-kg object that is initially at rest. The velocity of the 1-kg object after the collision is (a) greater than 4 m/s, (b) less than 4 m/s, (c) equal to 4 m/s, (d) zero, or (e) impossible to say based on the information provided.

A 5-kg cart moving to the right with a speed of 6 m/s collides with a concrete wall and rebounds with a speed of 2 m/s. What is the change in momentum of the cart? (a) 0 (b) 40 kg ? m/s (c) 240 kg ? m/s (d) 230 kg ? m/s (e) 210 kg ? m/s

A 57.0-g tennis ball is traveling straight at a player at 21.0 m/s. The player volleys the ball straight back at 25.0 m/s. If the ball remains in contact with the racket for 0.060 0 s, what average force acts on the ball? (a) 22.6 N (b) 32.5 N (c) 43.7 N (d) 72.1 N (e) 102 N

The momentum of an object is increased by a factor of 4 in magnitude. By what factor is its kinetic energy changed? (a) 16 (b) 8 (c) 4 (d) 2 (e) 1

The kinetic energy of an object is increased by a factor of 4. By what factor is the magnitude of its momentum changed? (a) 16 (b) 8 (c) 4 (d) 2 (e) 1

If two particles have equal momenta, are their kinetic energies equal? (a) yes, always (b) no, never (c) no, except when their speeds are the same (d) yes, as long as they move along parallel lines

If two particles have equal kinetic energies, are their momenta equal? (a) yes, always (b) no, never (c) yes, as long as their masses are equal (d) yes, if both their masses and directions of motion are the same (e) yes, as long as they move along parallel lines

A 10.0-g bullet is fired into a 200-g block of wood at rest on a horizontal surface. After impact, the block slides 8.00 m before coming to rest. If the coefficient of friction between the block and the surface is 0.400, what is the speed of the bullet before impact? (a) 106 m/s (b) 166 m/s (c) 226 m/s (d) 286 m/s (e) none of those answers is correct

Two particles of different mass start from rest. The same net force acts on both of them as they move over equal distances. How do their final kinetic energies compare? (a) The particle of larger mass has more kinetic energy. (b) The particle of smaller mass has more kinetic energy. (c) The particles have equal kinetic energies. (d) Either particle might have more kinetic energy.

Two particles of different mass start from rest. The same net force acts on both of them as they move over equal distances. How do the magnitudes of their final momenta compare? (a) The particle of larger mass has more momentum. (b) The particle of smaller mass has more momentum. (c) The particles have equal momenta. (d) Either particle might have more momentum.

A basketball is tossed up into the air, falls freely, and bounces from the wooden floor. From the moment after the player releases it until the ball reaches the top of its bounce, what is the smallest system for which momentum is conserved? (a) the ball (b) the ball plus player (c) the ball plus floor (d) the ball plus the Earth (e) momentum is not conserved for any system

A 3-kg object moving to the right on a frictionless, horizontal surface with a speed of 2 m/s collides headon and sticks to a 2-kg object that is initially moving to the left with a speed of 4 m/s. After the collision, which statement is true? (a) The kinetic energy of the system is 20 J. (b) The momentum of the system is 14 kg ? m/s. (c) The kinetic energy of the system is greater than 5 J but less than 20 J. (d) The momentum of the system is 22 kg ? m/s. (e) The momentum of the system is less than the momentum of the system before the collision.

A ball is suspended by a string that is tied to a fixed point above a wooden block standing on end. The ball is pulled back as shown in Figure OQ9.16 and released. In trial A, the ball rebounds elastically from the block. In trial B, two-sided tape causes the ball to stick to the block. In which case is the ball more likely to knock the block over? (a) It is more likely in trial A. (b) It is more likely in trial B. (c) It makes no difference. (d) It could be either case, depending on other factors

A car of mass m traveling at speed v crashes into the rear of a truck of mass 2m that is at rest and in neutral at an intersection. If the collision is perfectly inelastic, what is the speed of the combined car and truck after the collision? (a) v (b) v/2 (c) v/3 (d) 2v (e) None of those answers is correct.

A head-on, elastic collision occurs between two billiard balls of equal mass. If a red ball is traveling to the right with speed v and a blue ball is traveling to the left with speed 3v before the collision, what statement is true concerning their velocities subsequent to the collision? Neglect any effects of spin. (a) The red ball travels to the left with speed v, while the blue ball travels to the right with speed 3v. (b) The red ball travels to the left with speed v, while the blue ball continues to move to the left with a speed 2v. (c) The red ball travels to the left with speed 3v, while the blue ball travels to the right with speed v. (d) Their final velocities cannot be determined because momentum is not conserved in the collision. (e) The velocities cannot be determined without knowing the mass of each ball.

The magnitude of the net force exerted in the x direction on a 2.50-kg particle varies in time as shown in Figure P9.19. Find (a) the impulse of the force over the 5.00-s time interval, (b) the final velocity the particle attains if it is originally at rest, (c) its final velocity if its original velocity is 22.00 i ^ m/s, and (d) the average force exerted on the particle for the time interval between 0 and 5.00 s.

Review. A force platform is a tool used to analyze the performance of athletes by measuring the vertical force the athlete exerts on the ground as a function of time. Starting from rest, a 65.0-kg athlete jumps down onto the platform from a height of 0.600 m. While she is in contact with the platform during the time interval 0 , t , 0.800 s, the force she exerts on it is described by the function F 5 9 200t 2 11 500t 2 where F is in newtons and t is in seconds. (a) What impulse did the athlete receive from the platform? (b) With what speed did she reach the platform? (c) With what speed did she leave it? (d) To what height did she jump upon leaving the platform?

Water falls without splashing at a rate of 0.250 L/s from a height of 2.60 m into a 0.750-kg bucket on a scale. If the bucket is originally empty, what does the scale read in newtons 3.00 s after water starts to accumulate in it?

A 1 200-kg car traveling initially at vCi 5 25.0 m/s in an easterly direction crashes into the back of a 9 000-kg truck moving in the same direction at vTi 5 20.0 m/s (Fig. P9.22). The velocity of the car immediately after the collision is vCf 5 18.0 m/s to the east. (a) What is the velocity of the truck immediately after the collision? (b) What is the change in mechanical energy of the cartruck system in the collision? (c) Account for this change in mechanical energy.

A 10.0-g bullet is fired into a stationary block of wood having mass m 5 5.00 kg. The bullet imbeds into the block. The speed of the bullet-plus-wood combination immediately after the collision is 0.600 m/s. What was the original speed of the bullet?

A car of mass m moving at a speed v1 collides and couples with the back of a truck of mass 2m moving initially in the same direction as the car at a lower speed v2. (a) What is the speed vf of the two vehicles immediately after the collision? (b) What is the change in kinetic energy of the cartruck system in the collision?

A railroad car of mass 2.50 3 104 kg is moving with a speed of 4.00 m/s. It collides and couples with three other coupled railroad cars, each of the same mass as the single car and moving in the same direction with an initial speed of 2.00 m/s. (a) What is the speed of the four cars after the collision? (b) How much mechanical energy is lost in the collision?

Four railroad cars, each of mass 2.50 3 104 kg, are coupled together and coasting along horizontal tracks at speed vi toward the south. A very strong but foolish movie actor, riding on the second car, uncouples the front car and gives it a big push, increasing its speed to 4.00 m/s southward. The remaining three cars continue moving south, now at 2.00 m/s. (a) Find the initial speed of the four cars. (b) By how much did the potential energy within the body of the actor change? (c) State the relationship between the process described here and the process in Problem 25.

A neutron in a nuclear reactor makes an elastic, headon collision with the nucleus of a carbon atom initially at rest. (a) What fraction of the neutrons kinetic energy is transferred to the carbon nucleus? (b) The initial kinetic energy of the neutron is 1.60 3 10213 J. Find its final kinetic energy and the kinetic energy of the carbon nucleus after the collision. (The mass of the carbon nucleus is nearly 12.0 times the mass of the neutron.)

A 7.00-g bullet, when fired from a gun into a 1.00-kg block of wood held in a vise, penetrates the block to a depth of 8.00 cm. This block of wood is next placed on a frictionless horizontal surface, and a second 7.00-g bullet is fired from the gun into the block. To what depth will the bullet penetrate the block in this case?

A tennis ball of mass 57.0 g is held just above a basketball of mass 590 g. With their centers vertically aligned, both balls are released from rest at the same time, to fall through a distance of 1.20 m, as shown in Figure P9.29. (a) Find the magnitude of the downward velocity with which the basketball reaches the ground. (b) Assume that an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down. Next, the two balls meet in an elastic collision. To what height does the tennis ball rebound?

As shown in Figure P9.30, a bullet of mass m and speed v passes completely through a pendulum bob of mass M. The bullet emerges with a speed of v/2. The pendulum bob is suspended by a stiff rod (not a string) of length , and negligible mass. What is the minimum value of v such that the pendulum bob will barely swing through a complete vertical circle?

A 12.0-g wad of sticky clay is hurled horizontally at a 100-g wooden block initially at rest on a horizontal surface. The clay sticks to the block. After impact, the block slides 7.50 m before coming to rest. If the coefficient of friction between the block and the surface is 0.650, what was the speed of the clay immediately before impact?

A wad of sticky clay of mass m is hurled horizontally at a wooden block of mass M initially at rest on a horizontal surface. The clay sticks to the block. After impact, the block slides a distance d before coming to rest. If the coefficient of friction between the block and the surface is m, what was the speed of the clay immediately before impact?

Two blocks are free to slide along the frictionless, wooden track shown in Figure P9.33. The block of mass m1 5 5.00 kg is released from the position shown, at height h 5 5.00 m above the flat part of the track. Protruding from its front end is the north pole of a strong magnet, which repels the north pole of an identical magnet embedded in the back end of the block of mass m2 5 10.0 kg, initially at rest. The two blocks never touch. Calculate the maximum height to which m1 rises after the elastic collision.

(a) Three carts of masses m1 5 4.00 kg, m2 5 10.0 kg, and m3 5 3.00 kg move on a frictionless, horizontal track with speeds of v1 5 5.00 m/s to the right, v2 5 3.00 m/s to the right, and v3 5 4.00 m/s to the left as shown in Figure P9.34. Velcro couplers make the carts stick together after colliding. Find the final velocity of the train of three carts. (b) What If? Does your answer in part (a) require that all the carts collide and stick together at the same moment? What if they collide in a different order?

A 0.300-kg puck, initially at rest on a horizontal, frictionless surface, is struck by a 0.200-kg puck moving initially along the x axis with a speed of 2.00 m/s. After the collision, the 0.200-kg puck has a speed of 1.00 m/s at an angle of u 5 53.0 to the positive x axis (see Figure 9.11). (a) Determine the velocity of the 0.300-kg puck after the collision. (b) Find the fraction of kinetic energy transferred away or transformed to other forms of energy in the collision.

Two automobiles of equal mass approach an intersection. One vehicle is traveling with speed 13.0 m/s toward the east, and the other is traveling north with speed v2i. Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of 55.08 north of east. The speed limit for both roads is 35 mi/h, and the driver of the northward-moving vehicle claims he was within the speed limit when the collision occurred. Is he telling the truth? Explain your reasoning

An object of mass 3.00 kg, moving with an initial velocity of 5.00 i ^ m/s, collides with and sticks to an object of mass 2.00 kg with an initial velocity of 23.00 j ^ m/s. Find the final velocity of the composite object.

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37.08 with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37.08 with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk

A proton, moving with a velocity of vi i ^, collides elastically with another proton that is initially at rest. Assuming that the two protons have equal speeds after the collision, find (a) the speed of each proton after the collision in terms of vi and (b) the direction of the velocity vectors after the collision

A billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.33 m/s at an angle of 30.08 with respect to the original line of motion. Assuming an elastic collision (and ignoring friction and rotational motion), find the struck balls velocity after the collision.

A 90.0-kg fullback running east with a speed of 5.00 m/s is tackled by a 95.0-kg opponent running north with a speed of 3.00 m/s. (a) Explain why the successful tackle constitutes a perfectly inelastic collision. (b) Calculate the velocity of the players immediately after the tackle. (c) Determine the mechanical energy that disappears as a result of the collision. Account for the missing energy.

An unstable atomic nucleus of mass 17.0 3 10227 kg initially at rest disintegrates into three particles. One of the particles, of mass 5.00 3 10227 kg, moves in the y direction with a speed of 6.00 3 106 m/s. Another particle, of mass 8.40 3 10227 kg, moves in the x direction with a speed of 4.00 3 106 m/s. Find (a) the velocity of the third particle and (b) the total kinetic energy increase in the process

The mass of the blue puck in Figure P9.44 is 20.0% greater than the mass of the green puck. Before colliding, the pucks approach each other with momenta of equal magnitudes and opposite directions, and the green puck has an initial speed of 10.0 m/s. Find the speeds the pucks have after the collision if half the kinetic energy of the system becomes internal energy during the collision.

Four objects are situated along the y axis as follows: a 2.00-kg object is at 13.00 m, a 3.00-kg object is at 12.50 m, a 2.50-kg object is at the origin, and a 4.00-kg object is at 20.500 m. Where is the center of mass of these objects?

The mass of the Earth is 5.97 3 1024 kg, and the mass of the Moon is 7.35 3 1022 kg. The distance of separation, measured between their centers, is 3.84 3 108 m. Locate the center of mass of the EarthMoon system as measured from the center of the Earth.

Explorers in the jungle find an ancient monument in the shape of a large isosceles triangle as shown in Figure P9.47. The monument is made from tens of thousands of small stone blocks of density 3 800 kg/m3. The monument is 15.7 m high and 64.8 m wide at its base and is everywhere 3.60 m thick from front to back. Before the monument was built many years ago, all the stone blocks lay on the ground. How much work did laborers do on the blocks to put them in position while building the entire monument? Note: The gravitational potential energy of an objectEarth system is given by Ug 5 MgyCM, where M is the total mass of the object and yCM is the elevation of its center of mass above the chosen reference level.

A uniform piece of sheet metal is shaped as shown in Figure P9.48. Compute the x and y coordinates of the center of mass of the piece.

A rod of length 30.0 cm has linear density (mass per length) given by l 5 50.0 1 20.0x where x is the distance from one end, measured in meters, and l is in grams/meter. (a) What is the mass of the rod? (b) How far from the x 5 0 end is its center of mass?

A water molecule consists of an oxygen atom with two hydrogen atoms bound to it (Fig. P9.50). The angle between the two bonds is 106. If the bonds are 0.100 nm long, where is the center of mass of the molecule?

A 2.00-kg particle has a velocity 12.00 i ^ 2 3.00 j ^ 2 m/s, and a 3.00-kg particle has a velocity 11.00 i ^ 1 6.00 j ^ 2 m/s. Find (a) the velocity of the center of mass and (b) the total momentum of the system.

Consider a system of two particles in the xy plane: m1 5 2.00 kg is at the location r S1 5 11.00i ^ 1 2.00j ^2 m and has a velocity of 13.00i ^ 1 0.500j ^2 m/s; m 2 5 3.00 kg is at r S2 5 124.00i ^ 2 3.00j ^2 m and has velocity 13.00i ^ 2 2.00j ^2 m/s. (a) Plot these particles on a grid or graph paper. Draw their position vectors and show their velocities. (b) Find the position of the center of mass of the system and mark it on the grid. (c) Determine the velocity of the center of mass and also show it on the diagram. (d) What is the total linear momentum of the system?

Romeo (77.0 kg) entertains Juliet (55.0 kg) by playing his guitar from the rear of their boat at rest in still water, 2.70 m away from Juliet, who is in the front of the boat. After the serenade, Juliet carefully moves to the rear of the boat (away from shore) to plant a kiss on Romeos cheek. How far does the 80.0-kg boat move toward the shore it is facing?

The vector position of a 3.50-g particle moving in the xy plane varies in time according to r S1 5 13 i ^ 1 3 j ^2t 1 2 j ^t 2 , where t is in seconds and r S is in centimeters. At the same time, the vector position of a 5.50 g particle varies as r S2 5 3 i ^ 2 2 i ^t 2 2 6 j ^t. At t 5 2.50 s, determine (a) the vector position of the center of mass, (b) the linear momentum of the system, (c) the velocity of the center of mass, (d) the acceleration of the center of mass, and (e) the net force exerted on the two-particle system.

A ball of mass 0.200 kg with a velocity of 1.50 i ^ m/s meets a ball of mass 0.300 kg with a velocity of 20.400 i ^ m/s in a head-on, elastic collision. (a) Find their velocities after the collision. (b) Find the velocity of their center of mass before and after the collision.

For a technology project, a student has built a vehicle, of total mass 6.00 kg, that moves itself. As shown in Figure P9.56, it runs on four light wheels. A reel is attached to one of the axles, and a cord originally wound on the reel goes up over a pulley attached to the vehicle to support an elevated load. After the vehicle is released from rest, the load descends very slowly, unwinding the cord to turn the axle and make the vehicle move forward (to the left in Fig. P9.56). Friction is negligible in the pulley and axle bearings. The wheels do not slip on the floor. The reel has been constructed with a conical shape so that the load descends at a constant low speed while the vehicle moves horizontally across the floor with constant acceleration, reaching a final velocity of 3.00 i ^ m/s. (a) Does the floor impart impulse to the vehicle? If so, how much? (b) Does the floor do work on the vehicle? If so, how much? (c) Does it make sense to say that the final momentum of the vehicle came from the floor? If not, where did it come from? (d) Does it make sense to say that the final kinetic energy of the vehicle came from the floor? If not, where did it come from? (e) Can we say that one particular force causes the forward acceleration of the vehicle? What does cause it?

A particle is suspended from a post on top of a cart by a light string of length L as shown in Figure P9.57a. The cart and particle are initially moving to the right at constant speed vi , with the string vertical. The cart suddenly comes to rest when it runs into and sticks to a bumper as shown in Figure P9.57b. The suspended particle swings through an angle u. (a) Show that the original speed of the cart can be computed from vi 5 !2gL11 2 cos u 2. (b) If the bumper is still exerting a horizontal force on the cart when the hanging particle is at its maximum angle forward from the vertical, at what moment does the bumper stop exerting a horizontal force?

A 60.0-kg person bends his knees and then jumps straight up. After his feet leave the floor, his motion is unaffected by air resistance and his center of mass rises by a maximum of 15.0 cm. Model the floor as completely solid and motionless. (a) Does the floor impart impulse to the person? (b) Does the floor do work on the person? (c) With what momentum does the person leave the floor? (d) Does it make sense to say that this momentum came from the floor? Explain. (e) With what kinetic energy does the person leave the floor? (f) Does it make sense to say that this energy came from the floor? Explain.

Figure P9.59a shows an overhead view of the initial configuration of two pucks of mass m on frictionless ice. The pucks are tied together with a string of length , and negligible mass. At time t 5 0, a constant force of magnitude F begins to pull to the right on the center point of the string. At time t, the moving pucks strike each other and stick together. At this time, the force has moved through a distance d, and the pucks have attained a speed v (Fig. P9.59b). (a) What is v in terms of F, d, ,, and m? (b) How much of the energy transferred into the system by work done by the force has been transformed to internal energy?

A model rocket engine has an average thrust of 5.26 N. It has an initial mass of 25.5 g, which includes fuel mass of 12.7 g. The duration of its burn is 1.90 s. (a) What is the average exhaust speed of the engine? (b) This engine is placed in a rocket body of mass 53.5 g. What is the final velocity of the rocket if it were to be fired from rest in outer space by an astronaut on a spacewalk? Assume the fuel burns at a constant rate.

A garden hose is held as shown in Figure P9.61. The hose is originally full of motionless water. What additional force is necessary to hold the nozzle stationary after the water flow is turned on if the discharge rate is 0.600 kg/s with a speed of 25.0 m/s?

Review. The first stage of a Saturn V space vehicle consumed fuel and oxidizer at the rate of 1.50 3 104 kg/s with an exhaust speed of 2.60 3 103 m/s. (a) Calculate the thrust produced by this engine. (b) Find the acceleration the vehicle had just as it lifted off the launch pad on the Earth, taking the vehicles initial mass as 3.00 3 106 kg.

A rocket for use in deep space is to be capable of boosting a total load (payload plus rocket frame and engine) of 3.00 metric tons to a speed of 10 000 m/s. (a) It has an engine and fuel designed to produce an exhaust speed of 2 000 m/s. How much fuel plus oxidizer is required? (b) If a different fuel and engine design could give an exhaust speed of 5 000 m/s, what amount of fuel and oxidizer would be required for the same task? (c) Noting that the exhaust speed in part (b) is 2.50 times higher than that in part (a), explain why the required fuel mass is not simply smaller by a factor of 2.50

A rocket has total mass Mi 5 360 kg, including Mf 5 330 kg of fuel and oxidizer. In interstellar space, it starts from rest at the position x 5 0, turns on its engine at time t 5 0, and puts out exhaust with relative speed ve 5 1 500 m/s at the constant rate k 5 2.50 kg/s. The fuel will last for a burn time of Tb 5 Mf/k 5 330 kg/(2.5 kg/s) 5 132 s. (a) Show that during the burn the velocity of the rocket as a function of time is given by v 1t2 5 2ve lna1 2 kt Mi b (b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to 132 s. (c) Show that the acceleration of the rocket is a1t2 5 kve Mi 2 kt (d) Graph the acceleration as a function of time. (e) Show that the position of the rocket is x1t2 5 ve a Mi k 2 tb ln a1 2 kt Mi b 1 vet (f) Graph the position during the burn as a function of time.

A ball of mass m is thrown straight up into the air with an initial speed vi . Find the momentum of the ball (a) at its maximum height and (b) halfway to its maximum height.

An amateur skater of mass M is trapped in the middle of an ice rink and is unable to return to the side where there is no ice. Every motion she makes causes her to slip on the ice and remain in the same spot. She decides to try to return to safety by throwing her gloves of mass m in the direction opposite the safe side. (a) She throws the gloves as hard as she can, and they leave her hand with a horizontal velocity v Sgloves. Explain whether or not she moves. If she does move, calculate her velocity v Sgirl relative to the Earth after she throws the gloves. (b) Discuss her motion from the point of view of the forces acting on her.

A 3.00-kg steel ball strikes a wall with a speed of 10.0m/s at an angle of u 5 60.08 with the surface. It bounces off with the same speed and angle (Fig. P9.67). If the ball is in contact with the wall for 0.200 s, what is the average force exerted by the wall on the ball?

(a) Figure P9.68 shows three points in the operation of the ballistic pendulum discussed in Example 9.6 (and shown in Fig. 9.9b). The projectile approaches the pendulum in Figure P9.68a. Figure P9.68b shows the situation just after the projectile is captured in the pendulum. In Figure P9.68c, the pendulum arm has swung upward and come to rest at a height h above its initial position. Prove that the ratio of the kinetic energy of the projectilependulum system immediately after the collision to the kinetic energy immediately before is m1/(m1 1 m2). (b) What is the ratio of the momentum of the system immediately after the collision to the momentum immediately before? (c) A student believes that such a large decrease in mechanical energy must be accompanied by at least a small decrease in momentum. How would you convince this student of the truth?

Review. A 60.0-kg person running at an initial speed of 4.00 m/s jumps onto a 120-kg cart initially at rest (Fig. P9.69). The person slides on the carts top surface and finally comes to rest relative to the cart. The coefficient of kinetic friction between the person and the cart is 0.400. Friction between the cart and ground can be ignored. (a) Find the final velocity of the person and cart relative to the ground. (b) Find the friction force acting on the person while he is sliding across the top surface of the cart. (c) How long does the friction force act on the person? (d) Find the change in momentum of the person and the change in momentum of the cart. (e) Determine the displacement of the person relative to the ground while he is sliding on the cart. (f) Determine the displacement of the cart relative to the ground while the person is sliding. (g) Find the change in kinetic energy of the person. (h) Find the change in kinetic energy of the cart. (i) Explain why the answers to (g) and (h) differ. (What kind of collision is this one, and what accounts for the loss of mechanical energy?)

A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large spring, initially unstretched and with force constant k 5 2.00 3 104 N/m, as shown in Figure P9.70. The cannon fires a 200-kg projectile at a velocity of 125 m/s directed 45.0 above the horizontal. (a) Assuming that the mass of the cannon and its carriage is 5 000 kg, find the recoil speed of the cannon. (b) Determine the maximum extension of the spring. (c) Find the maximum force the spring exerts on the carriage. (d) Consider the system consisting of the cannon, carriage, and projectile. Is the momentum of this system conserved during the firing? Why or why not

A 1.25-kg wooden block rests on a table over a large hole as in Figure P9.71. A 5.00-g bullet with an initial velocity vi is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of 22.0 cm. (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this chapter. (b) Calculate the initial velocity of the bullet from the information provided.

A wooden block of mass M rests on a table over a large hole as in Figure 9.71. A bullet of mass m with an initial velocity of vi is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of h. (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this chapter. (b) Find an expression for the initial velocity of the bullet.

Two particles with masses m and 3m are moving toward each other along the x axis with the same initial speeds vi . The particle with mass m is traveling to the left, and particle with mass 3m is traveling to the right. They undergo a head-on elastic collision, and each rebounds along the same line as it approached. Find the final speeds of the particles

Pursued by ferocious wolves, you are in a sleigh with no horses, gliding without friction across an ice-covered lake. You take an action described by the equations 1270 kg2 17.50 m/s2 i ^ 5 115.0 kg2 12v1f i ^2 1 1255 kg2 1v2f i ^2 v1f 1 v2f 5 8.00 m/s (a) Complete the statement of the problem, giving the data and identifying the unknowns. (b) Find the values of v1f and v2f . (c) Find the amount of energy that has been transformed from potential energy stored in your body to kinetic energy of the system.

Two gliders are set in motion on a horizontal air track. A spring of force constant k is attached to the back end of the second glider. As shown in Figure P9.75, the first glider, of mass m1, moves to the right with speed v1, and the second glider, of mass m2, moves more slowly to the right with speed v2. When m1 collides with the spring attached to m2, the spring compresses by a distance xmax, and the gliders then move apart again. In terms of v1, v2, m1, m2, and k, find (a) the speed v at maximum compression, (b) the maximum compression xmax, and (c) the velocity of each glider after m1 has lost contact with the spring.

Why is the following situation impossible? An astronaut, together with the equipment he carries, has a mass of 150 kg. He is taking a space walk outside his spacecraft, which is drifting through space with a constant velocity. The astronaut accidentally pushes against the spacecraft and begins moving away at 20.0 m/s, relative to the spacecraft, without a tether. To return, he takes equipment off his space suit and throws it in the direction away from the spacecraft. Because of his bulky space suit, he can throw equipment at a maximum speed of 5.00 m/s relative to himself. After throwing enough equipment, he starts moving back to the spacecraft and is able to grab onto it and climb inside

Two blocks of masses m1 5 2.00 kg and m2 5 4.00 kg are released from rest at a height of h 5 5.00 m on a frictionless track as shown in Figure P9.77. When they meet on the level portion of the track, they undergo a head-on, elastic collision. Determine the maximum heights to which m1 and m2 rise on the curved portion of the track after the collision.

Review. A metal cannonball of mass m rests next to a tree at the very edge of a cliff 36.0 m above the surface of the ocean. In an effort to knock the cannonball off the cliff, some children tie one end of a rope around a stone of mass 80.0 kg and the other end to a tree limb just above the cannonball. They tighten the rope so that the stone just clears the ground and hangs next to the cannonball. The children manage to swing the stone back until it is held at rest 1.80 m above the ground. The children release the stone, which then swings down and makes a head-on, elastic collision with the cannonball, projecting it horizontally off the cliff. The cannonball lands in the ocean a horizontal distance R away from its initial position. (a) Find the horizontal component R of the cannonballs displacement as it depends on m. (b) What is the maximum possible value for R, and (c) to what value of m does it correspond? (d) For the stonecannonballEarth system, is mechanical energy conserved throughout the process? Is this principle sufficient to solve the entire problem? Explain. (e) What if? Show that R does not depend on the value of the gravitational acceleration. Is this result remarkable? State how one might make sense of it.

A 0.400-kg blue bead slides on a frictionless, curved wire, starting from rest at point A in Figure P9.79, where h 5 1.50 m. At point B, the blue bead collides elastically with a 0.600-kg green bead at rest. Find the maximum height the green bead rises as it moves up the wire.

A small block of mass m1 5 0.500 kg is released from rest at the top of a frictionless, curve-shaped wedge of mass m2 5 3.00 kg, which sits on a frictionless, horizontal surface as shown in Figure P9.80a. When the block leaves the wedge, its velocity is measured to be 4.00 m/s to the right as shown in Figure P9.80b. (a) What is the velocity of the wedge after the block reaches the horizontal surface? (b) What is the height h of the wedge?

Review. A bullet of mass m 5 8.00 g is fired into a block of mass M 5 250 g that is initially at rest at the edge of a table of height h 5 1.00 m (Fig. P9.81). The bullet remains in the block, and after the impact the block lands d 5 2.00 m from the bottom of the table. Determine the initial speed of the bullet.

Review. A bullet of mass m is fired into a block of mass M initially at rest at the edge of a frictionless table of height h (Fig. P9.81). The bullet remains in the block, and after impact the block lands a distance d from the bottom of the table. Determine the initial speed of the bullet.

A 0.500-kg sphere moving with a velocity given by 12.00 i ^ 2 3.00 j ^ 1 1.00k^ 2 m/s strikes another sphere of mass 1.50 kg moving with an initial velocity of 121.00 i ^ 1 2.00 j ^ 2 3.00k^ 2 m/s. (a) The velocity of the 0.500-kg sphere after the collision is 121.00 i ^ 1 3.00 j ^ 2 8.00k^ 2 m/s. Find the final velocity of the 1.50-kg sphere and identify the kind of collision (elastic, inelastic, or perfectly inelastic). (b) Now assume the velocity of the 0.500-kg sphere after the collision is (20.250 i ^ 1 0.750 j ^ 2 2.00k^ ) m/s. Find the final velocity of the 1.50-kg sphere and identify the kind of collision. (c) What If? Take the velocity of the 0.500-kg sphere after the collision as 121.00 i ^ 1 3.00 j ^ 1 a k^ 2 m/s. Find the value of a and the velocity of the 1.50-kg sphere after an elastic collision.

A 75.0-kg firefighter slides down a pole while a constant friction force of 300 N retards her motion. A horizontal 20.0-kg platform is supported by a spring at the bottom of the pole to cushion the fall. The firefighter starts from rest 4.00 m above the platform, and the spring constant is 4 000 N/m. Find (a) the firefighters speed just before she collides with the platform and (b) the maximum distance the spring is compressed. Assume the friction force acts during the entire motion.

George of the Jungle, with mass m, swings on a light vine hanging from a stationary tree branch. A second vine of equal length hangs from the same point, and a gorilla of larger mass M swings in the opposite direction on it. Both vines are horizontal when the primates start from rest at the same moment. George and the gorilla meet at the lowest point of their swings. Each is afraid that one vine will break, so they grab each other and hang on. They swing upward together, reaching a point where the vines make an angle of 35.08 with the vertical. Find the value of the ratio m/M.

Review. A student performs a ballistic pendulum experiment using an apparatus similar to that discussed in Example 9.6 and shown in Figure P9.68. She obtains the following average data: h 5 8.68 cm, projectile mass m1 5 68.8 g, and pendulum mass m2 5 263 g. (a) Determine the initial speed v1A of the projectile. (b) The second part of her experiment is to obtain v1A by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its final horizontal position x and distance of fall y (Fig. P9.86). What numerical value does she obtain for v1A based on her measured values of x 5 257cm and y 5 85.3 cm? (c) What factors might account for the difference in this value compared with that obtained in part(a)?

Review. A light spring of force constant 3.85 N/m is compressed by 8.00 cm and held between a 0.250-kg block on the left and a 0.500-kg block on the right. Both blocks are at rest on a horizontal surface. The blocks are released simultaneously so that the spring tends to push them apart. Find the maximum velocity each block attains if the coefficient of kinetic friction between each block and the surface is (a) 0, (b) 0.100, and (c) 0.462. Assume the coefficient of static friction is greater than the coefficient of kinetic friction in every case

Consider as a system the Sun with the Earth in a circular orbit around it. Find the magnitude of the change in the velocity of the Sun relative to the center of mass of the system over a six-month period. Ignore the influence of other celestial objects. You may obtain the necessary astronomical data from the endpapers of the book.

A 5.00-g bullet moving with an initial speed of vi 5 400 m/s is fired into and passes through a 1.00-kg block as shown in Figure P9.89. The block, initially at rest on a frictionless, horizontal surface, is connected to a spring with force constant 900 N/m. The block moves d 5 5.00 cm to the right after impact before being brought to rest by the spring. Find (a) the speed at which the bullet emerges from the block and (b) the amount of initial kinetic energy of the bullet that is converted into internal energy in the bullet block system during the collision

Review. There are (one can say) three coequal theories of motion for a single particle: Newtons second law, stating that the total force on the particle causes its acceleration; the workkinetic energy theorem, stating that the total work on the particle causes its change in kinetic energy; and the impulsemomentum theorem, stating that the total impulse on the particle causes its change in momentum. In this problem, you compare predictions of the three theories in one particular case. A 3.00-kg object has velocity 7.00 j ^ m/s. Then, a constant net force 12.0 i ^ N acts on the object for 5.00 s. (a) Calculate the objects final velocity, using the impulsemomentum theorem. (b) Calculate its acceleration from a S 5 1 v Sf 2 v Si 2/Dt. (c) Calculate its acceleration from a S 5 g F S /m. (d) Find the objects vector displacement from Dr S 5 v S it 1 1 2 a St 2 . (e) Find the work done on the object from W 5 F S ?Dr S. (f) Find the final kinetic energy from 1 2mvf 2 5 1 2mv S f ? v S f . (g) Find the final kinetic energy from 1 2mvi 2 1 W. (h) State the result of comparing the answers to parts (b) and (c), and the answers to parts (f) and (g).

A 2.00-g particle moving at 8.00 m/s makes a perfectly elastic head-on collision with a resting 1.00-g object. (a) Find the speed of each particle after the collision. (b) Find the speed of each particle after the collision if the stationary particle has a mass of 10.0 g. (c) Find the final kinetic energy of the incident 2.00-g particle in the situations described in parts (a) and (b). In which case does the incident particle lose more kinetic energy?

In the 1968 Olympic games, University of Oregon jumper Dick Fosbury introduced a new technique of high jumping called the Fosbury flop. It contributed to raising the world record by about 30 cm and is currently used by nearly every world-class jumper. In this technique, the jumper goes over the bar face-up while arching her back as much as possible as shown in Figure P9.92a. This action places her center of mass outside her body, below her back. As her body goes over the bar, her center of mass passes below the bar. Because a given energy input implies a certain elevation for her center of mass, the action of arching her back means that her body is higher than if her back were straight. As a model, consider the jumper as a thin uniform rod of length L. When the rod is straight, its center of mass is at its center. Now bend the rod in a circular arc so that it subtends an angle of 90.08 at the center of the arc as shown in Figure P9.92b. In this configuration, how far outside the rod is the center of mass?

Two particles with masses m and 3m are moving toward each other along the x axis with the same initial speeds vi . Particle m is traveling to the left, and particle 3m is traveling to the right. They undergo an elastic glancing collision such that particle m is moving in the negative y direction after the collision at a right angle from its initial direction. (a) Find the final speeds of the two particles in terms of vi . (b) What is the angle u at which the particle 3m is scattered?

Sand from a stationary hopper falls onto a moving conveyor belt at the rate of 5.00 kg/s as shown in Figure P9.94. The conveyor belt is supported by frictionless rollers and moves at a constant speed of v 5 0.750 m/s under the action of a constant horizontal external force F S ext supplied by the motor that drives the belt. Find (a) the sands rate of change of momentum in the horizontal direction, (b) the force of friction exerted by the belt on the sand, (c) the external force F S ext , (d) the work done by F S ext in 1 s, and (e) the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion. (f) Why are the answers to parts (d) and (e) different?

On a horizontal air track, a glider of mass m carries a G-shaped post. The post supports a small dense sphere, also of mass m, hanging just above the top of the glider on a cord of length L. The glider and sphere are initially at rest with the cord vertical. (Figure P9.57 shows a cart and a sphere similarly connected.) A constant horizontal force of magnitude F is applied to the glider, moving it through displacement x1; then the force is removed. During the time interval when the force is applied, the sphere moves through a displacement with horizontal component x2. (a) Find the horizontal component of the velocity of the center of mass of the glidersphere system when the force is removed. (b) After the force is removed, the glider continues to move on the track and the sphere swings back and forth, both without friction. Find an expression for the largest angle the cord makes with the vertical.

Review. A chain of length L and total mass M is released from rest with its lower end just touching the top of a table as shown in Figure P9.96a. Find the force exerted by the table on the chain after the chain has fallen through a distance x as shown in Figure P9.96b. (Assume each link comes to rest the instant it reaches the table.)