In the center-of-mass reference frame, a particle with

Show that if two particles have equal kinetic energies, the magnitudes of their momenta are equal only if they have the same mass.

Particle A has twice the momentum and four times the kinetic energy of particle B. What is the ratio of the mass of particle A to that of particle B? Explain your reasoning.

Using SI units, show that the units of momentum squared divided by those of mass is equivalent to the joule.

True or false: (a) The total linear momentum of a system may be conserved even when the mechanical energy of the system is not. (b) For the total linear momentum of a system to be conserved, there must be no external forces acting on the system. (c) The velocity of the center of mass of a system changes only when there is a net external force on the system.

If a bullet is fired due west, explain how conservation of linear momentum enables you to predict that the recoil of the rifle will be exactly due east. Is kinetic energy conserved here?

Achild jumps from a small boat to a dock. Why does she have to jump with more effort than she would need if she were jumping through an identical displacement, but from a boulder to a tree stump?

Much of the early research in rocket motion was done by Robert Goddard, physics professor at Clark College in Worcester, Massachusetts. A quotation from a 1920 editorial in the New York Times illustrates the publics opinion of his work: That Professor Goddard with his chair at Clark College and the countenance of the Smithsonian Institution does not know the relation between action and reaction, and the need to have something better than a vacuum against which to reactto say that would be absurd. Of course, he only seems to lack the knowledge ladled out daily in high schools.* The belief that a rocket needs something to push against was a prevalent misconception before rockets in space were commonplace. Explain why that belief is wrong. SSM * On page 43 of the July 17, 1969, edition of the New York Times A Correction to their editorial of 1920 was printed. This commentary, which was published three days before mans first walk on the moon, stated that it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error.

Two identical bowling balls are moving with the same center-of-mass velocity, but one just slides down the alley without rotating, whereas the other rolls down the alley. Which ball has more kinetic energy?

A philosopher tells you, Changing motion of objects is impossible. Forces always come in equal but opposite pairs. Therefore, all forces cancel out. Because forces cancel, the momenta of objects can never be changed. Answer his argument.

A moving object collides with a stationary object. Is it possible for both objects to be at rest immediately after the collision? (Assume any external forces acting on this two-object system are negligibly small.) Is it possible for one object to be at rest immediately after the collision? Explain.

Several researchers in physics education claim that part of the cause of physical misconceptions among students comes from special effects they observe in cartoons and movies. Using the conservation of linear momentum, how would you explain to a class of high school physics students what is conceptually wrong with a superhero hovering at rest in midair while tossing massive objects such as cars at villains? Does this action violate the conservation of energy as well? Explain.

A struggling physics student asks, If only external forces can cause the center of mass of a system of particles to accelerate, how can a car move? Doesnt the cars engine supply the force needed to accelerate the car? Explain what external agent produces the force that accelerates the car, and explain how the engine makes that agent do so.

When we push on the brake pedal to slow down a car, a brake pad is pressed against the rotor so that the friction of the pad slows the rotation of the rotor and thus the rotation of the wheel. However, the friction of the pad against the rotor cannot be the force that slows the car down because it is an internal forceboth the rotor and the wheel are parts of the car, so any forces between them are internal, not external forces. What external agent exerts the force that slows down the car? Give a detailed explanation of how this force operates.

Explain why a circus performer falling into a safety net can survive unharmed, while a circus performer falling from the same height onto the hard concrete floor suffers serious injury or death. Base your explanation on the impulsemomentum theorem.

In Problem 14, estimate the ratio of the collision time with the safety net to the collision time with the concrete for the performer falling from a height of Hint: Use the procedure outlined in step 4 of the Problem-Solving Strategy located in Section 8-3.

(a) Why does a drinking glass survive a fall onto a carpet but not a fall onto a concrete floor? (b) On many automobile race tracks, dangerous curves are surrounded by massive bails of hay. Explain how this setup reduces the chances of car damage and driver injury.

True or false: (a) Following any perfectly inelastic collision, the kinetic energy of the system is zero in all inertial reference frames. (b) For a head-on elastic collision, the relative speed of recession equals the relative speed of approach. (c) In a perfectly inelastic head-on collision with one object initially at rest, only some of the systems kinetic energy is dissipated. (d) After a perfectly inelastic head-on collision along the eastwest horizontal axis, the two objects are observed to be moving west. The initial total system momentum was therefore to the west. SSM 25 m.

Under what conditions can all the initial kinetic energy of an isolated system consisting of two colliding objects be lost in a collision? Explain how this result can be, and yet the momentum of the system can be conserved.

Consider a perfectly inelastic collision of two objects of equal mass. (a) Is the loss of kinetic energy greater if the two objects are moving in opposite directions, each moving at speed or if one of the two objects is initially at rest and the other has an initial speed of (b) In which of these situations is the percentage loss in kinetic energy the greatest?

A double-barreled pea shooter is shown in Figure 8-41. Air is blown into the left end of the pea shooter, and identical peas A and B are positioned inside each straw as shown. If the pea shooter is held horizontally while the peas are shot off, which pea, A or B, will travel farther after leaving the straw? Explain. (Base your explanation on the impulsemomentum theorem.) v?

A particle of mass traveling with a speed v makes a head-on elastic collision with a stationary particle of mass In which scenario will the largest amount of energy be imparted to the particle of mass (a) (b) (c) (d) None of the above

ENGINEERING APPLICATION, CONTEXT-RICH Suppose you are in charge of an accident reconstruction team that has reconstructed an accident in which a car was rear-ended, causing the two cars to lock bumpers and skid to a halt. During the trial, you are on the stand as an expert witness for the prosecution and the lawyer for the defense claims that you wrongly neglected friction and the force of gravity during the fraction of a second while the cars collided. Defend your report. Why were you correct in ignoring these forces? You did not ignore these two forces in your skid analysis both before and after the collision. Can you explain to the jury why you did not ignore these two forces during the preand postcollision skids?

Nozzles for a garden hose are often made with a rightangle shape, as shown in Figure 8-42. If you open the nozzle to spray water out, you will find that the nozzle presses against your hand with a pretty strong forcemuch stronger than if you used a nozzle not bent into a right angle. Why is this situation true?

Describe a perfectly inelastic head-on collision between two stunt cars as viewed in the center-of-mass reference frame.

One air-hockey puck is initially at rest. An identical airhockey puck collides with it, striking it with a glancing blow. Assume the collision is elastic and neglect any rotational motion of the pucks. Describe the collision in the center-of-mass frame of the pucks.

A baton with one end more massive than the other is tossed at an angle into the air. (a) Describe the trajectory of the center of mass of the baton in the reference frame of the ground. (b) Describe the motion of the two ends of the baton in the centerof- mass frame of the baton.

Describe the forces acting on a descending Lunar lander as it fires its retrorockets to slow down for a safe landing. (Assume the landers mass loss during the rocket firing is not negligible.)

Arailroad car rolling along by itself is passing by a grain elevator, which is dumping grain into it at a constant rate. (a) Does momentum conservation imply that the railroad car should be slowing down as it passes the grain elevator? Assume that the track is frictionless and perfectly level and that the grain is falling vertically. (b) If the car is slowing down, this situation implies that there is some external force acting on the car to slow it down. Where does this force come from? (c) After passing the elevator, the railroad car springs a leak, and grain starts leaking out of a vertical hole in its floor at a constant rate. Should the car speed up as it loses mass?

To show that even really intelligent people can make mistakes, consider the following problem, which was asked of a freshman class at Caltech on an exam (paraphrased): A sailboat is sitting in the water on a windless day. In order to make the boat move, a misguided sailor sets up a fan in the back of the boat to blow into the sails to make the boat move forward. Explain why the boat will not move. The idea was that the net force of the wind pushing the sail forward would be counteracted by the force pushing the fan back (Newtons third law). However, as one of the students pointed out to his professor, the sailboat could in fact move forward. Why is that?

ENGINEERING APPLICATION A 2000-kg car traveling at crashes into an immovable concrete wall. (a) Estimate the time of collision, assuming that the center of the car travels halfway to the wall with constant acceleration. (Use any plausible length for the car.) (b) Estimate the average force exerted by the wall on the car.

In hand-pumped railcar races, a speed of has been achieved by teams of four people. A car that has a mass equal to is moving at that speed toward a river when Carlos, the chief pumper, notices that the bridge ahead is out. All four people (each with a mass of ) simultaneously jump backward off the car with a velocity that has a horizontal component of relative to the car. The car proceeds off the bank and falls into the water a horizontal distance of from the bank. (a) Estimate the time of the fall of the railcar. (b) What is the horizontal component of the velocity of the pumpers when they hit the ground?

A wooden block and a gun are firmly fixed to opposite ends of a long glider mounted on a frictionless air track (Figure 8-43). The block and gun are a distance L apart. The system 25.0m 4.00m>s 75.0 kg 350 kg 32.0 km>h 90 km>h SSM is initially at rest. The gun is fired and the bullet leaves the gun with a velocity and impacts the block, becoming imbedded in it. The mass of the bullet is and the mass of the gungliderblock system is (a) What is the velocity of the glider immediately after the bullet leaves the gun? (b) What is the velocity of the glider immediately after the bullet comes to rest in the block? (c) How far does the glider move while the bullet is in transit between the gun and the block?

Tyrone, a 85-kg teenager, runs off the end of a horizontal pier and lands on a free-floating 150-kg raft that was initially at rest. After he lands on the raft, the raft, with him on it, moves away from the pier at What was Tyrones speed as he ran off the end of the pier?

A 55-kg woman contestant on a reality television show is at rest at the south end of a horizontal 150-kg raft that is floating in crocodile-infested waters. She and the raft are initially at rest. She needs to jump from the raft to a platform that is several meters off the north end of the raft. She takes a running start. When she reaches the north end of the raft she is running at relative to the raft. At that instant, what is her velocity relative to the water?

A 5.0-kg object and a 10-kg object, both resting on a frictionless table, are connected by a massless compressed spring. The spring is released and the objects fly off in opposite directions. The 5.0-kg object has a velocity of to the left. What is the velocity of the 10-kg object?

Figure 8-44 shows the behavior of a projectile just after it has broken up into three pieces. What was the speed of the projectile the instant before it broke up: (a) (b) (c) (d) (e) (v1 _ v2 _ v3)>4?

A shell of mass m and speed v explodes into two identical fragments. If the shell was moving horizontally with respect to Earth, and one of the fragments is subsequently moving vertically with speed v, find the velocity of the other fragment immediately following the explosion.

For this weeks physics lab, the experimental setup consists of two gliders on a horizontal frictionless air track (see Figure 8-45). Each glider supports a strong magnet centered on top of it, and the magnets are oriented so they attract each other. The mass of glider 1 and its magnet is and the mass of glider 2 and its magnet is You and your lab partners are instructed to take the origin to be at the left end of the track and to center glider 1 at and glider 2 at Glider 1 is long, and glider 2 is long, and each glider has its center of mass at its geometric center. When the two gliders are released from rest, they will move toward each other and stick. (a) Predict the position of the center of each glider when they first touch. (b) Predict the velocity that the two gliders will continue to move with after they stick. Explain the reasoning behind this prediction for your lab partners.

Bored, a boy shoots his pellet gun at a piece of cheese that sits on a massive block of ice. On one particular shot, his 1.2-g pellet gets stuck in the cheese, causing it to slide before coming to a stop. If the muzzle velocity of the gun is known to be and the cheese has a mass of what is the coefficient of friction between the cheese and the ice?

MULTISTEP A wedge of mass M is placed on a frictionless, horizontal surface, and a block of mass m is placed on the wedge, which also has a frictiononless surface (Figure 8-46). The blocks center of mass moves downward a distance h as the block slides from its initial position to the horizontal floor. (a) What are the speeds of the block and of the wedge as they separate from each other and go their own ways? (b) Check your calculation plausibility by considering the limiting case when

MULTISTEP A 3.0-kg block is traveling to the right (the direction) at and a second 3.0-kg block is traveling to the left at (a) Find the total kinetic energy of the two blocks. (b) Find the velocity of the center of mass of the twoblock system. (c) Find the velocity of each block relative to the center of mass. (d) Find the kinetic energy of the blocks relative to the center of mass. (e) Show that your answer for Part (a) is greater than your answer for Part (d) by an amount equal to the kinetic energy associated with the motion of the center of mass.

Repeat Problem 41 with the second 3.0-kg block replaced by a 5.0-kg block moving to the right at

You kick a soccer ball whose mass is The ball leaves your foot with an initial speed of (a) What is the magnitude of the impulse associated with the force of your foot on the ball? (b) If your foot is in contact with the ball for what is the magnitude of the average force exerted by your foot on the ball?

A 0.30-kg brick is dropped from a height of It hits the ground and comes to rest. (a) What is the impulse exerted by the ground on the brick during the collision? (b) If it takes from the time the brick first touches the ground until it comes to rest, what is the average force exerted by the ground on the brick at impact?

Ameteorite that has a mass equal to 30.8 tonnes is exhibited in the American Museum of Natural History in New York City. Suppose that the kinetic energy of the meteorite as it hit the ground was Find the magnitude of the impulse experienced by the meteorite up to the time its kinetic energy was halved (which took about ). Also find the average force exerted on the meteorite during this time interval.

A 0.15-kg baseball traveling horizontally is hit by a bat and its direction is exactly reversed. Its velocity changes from to (a) What is the magnitude of the impulse delivered by the bat to the ball? (b) If the baseball is in contact with the bat for what is the average force exerted by the bat on the ball?

A 60-g handball moving with a speed of strikes the wall at a 40 angle with the normal, and then bounces off with the same speed at the same angle with the normal. It is in contact with the wall for What is the average force exerted by the ball on the wall?

ESTIMATION You throw a 150-g ball straight up to a height of (a) Use a reasonable value for the displacement of the ball while it is in your hand to estimate the time the ball is in your hand while you are throwing it. (b) Calculate the average force exerted by your hand while you are throwing it. (Is it okay to neglect the gravitational force on the ball while it is being thrown?)

A 0.060-g handball is thrown straight toward a wall with a speed of It rebounds straight backward at a speed of (a) What impulse is exerted on the wall? (b) If the ball is in contact with the wall for what average force is exerted on the wall by the ball? (c) The rebounding ball is caught by a player who brings it to rest. In the process, her hand moves back What is the impulse received by the player? (d) What average force was exerted on the player by the ball?

A spherical 0.34-kg orange, in radius, is dropped from the top of a 35-m-tall building. After the orange strikes the pavement, its shape is that of a 0.50-cm-thick pancake. Neglect air resistance and assume that the collision is completely inelastic. (a) How much time did the orange take to completely squish to a stop? (b) What average force did the pavement exert on the orange during the collision?

The pole vault landing pad at an Olympic competition contains what is essentially a bag of air that compresses from its resting height of down to as the vaulter is slowed to a stop. (a) What is the time interval during which a vaulter who has just cleared a height of slows to a stop? (b) What is the time interval if instead the vaulter is brought to rest by a 20-cm layer of sawdust that compresses to when he lands? (c) Qualitatively discuss the difference in average force the vaulter experiences from the two different landing pads. That is, which landing pad would exert the least force on the vaulter and why?

Large limestone caverns have been formed by dripping water. (a) If water droplets of fall from a height of at a rate of 10 droplets per minute, what is the average force exerted on the limestone floor by the droplets of water during a 1.0-min period? (Assume the water does not accumulate on the floor.) (b) Compare this force to the weight of one water droplet.

A 2000-kg car traveling to the right at 30 m/s is chasing a second car of the same mass that is traveling in the same direction at 10 m/s. (a) If the two cars collide and stick together, what is their speed just after the collision? (b) What fraction of the initial kinetic energy of the cars is lost during this collision? Where does it go?

An 85-kg running back moving at makes a perfectly inelastic head-on collision with a 105-kg linebacker who is initially at rest. What is the speed of the players just after their collision?

A 5.0-kg object with a speed of collides head-on with a 10-kg object moving toward it with a speed of The 10-kg object stops dead after the collision. (a) What is the postcollision speed of the 5.0-kg object? (b) Is the collision elastic?

A small superball of mass m moves with speed v to the right toward a much more massive bat that is moving to the left with speed v. Find the speed of the ball after it makes an elastic head-on collision with the bat.

A proton that has a mass m and is moving at undergoes a head-on elastic collision with a stationary carbon nucleus of mass 12m. Find the velocities of the proton and the carbon nucleus after the collision.

A 3.0-kg block moving at has a head-on elastic collision with a stationary block of mass Use conservation of momentum and the fact that the relative speed of recession equals the relative speed of approach to find the velocity of each block after the collision. Check your answer by calculating the initial and final kinetic energies of each block.

A block of mass slides along a frictionless table with a speed of Directly in front of it, and moving in the same direction with a speed of is a block of mass A massless spring that has a force constant is attached to the second block, as in Figure 8-47. (a) What is the velocity of the center of mass of the system? (b) During the collision, the spring is compressed by a maximum amount x. What is the value of (c) The blocks will eventually separate again. What are the velocities of the two blocks measured in the reference frame of the table, after they separate?

A bullet of mass m is fired vertically from below into a thin horizontal sheet of plywood of mass M that is initially at rest, supported by a thin sheet of paper (Figure 8-48). The bullet punches through the plywood, which rises to a height, H, above the paper before falling back down. The bullet continues rising to a height, h, above the paper. (a) Express the upward velocity of the bullet and the plywood immediately after the bullet exits the plywood in terms of h and H. (b) What is the speed of the bullet? (c) What is the mechanical energy of the system before and after the inelastic collision? (d) How much mechanical energy is dissipated during the collision?

A proton of mass m is moving with initial speed directly toward the center of an particle of mass 4m, which is initially at rest. Both particles carry positive charge, so they repel each other. (The repulsive forces are sufficient to prevent the two particles from coming into direct contact.) Find the speed of the particle (a) when the distance between the two particles is a minimum, and (b) later when the two particles are far apart.

An electron collides elastically with a hydrogen atom that is initially at rest. Assume all the motion occurs along a straight line. What fraction of the electrons initial kinetic energy is transferred to the atom? (Take the mass of the hydrogen atom to be 1840 times the mass of an electron.)

A 16-g bullet is fired into the bob of a 1.5-kg ballistic pendulum (Figure 8-18). When the bob is at its maximum height, the strings make an angle of 60 with the vertical. The pendulum strings are long. Find the speed of the bullet prior to impact.

Show that in a one-dimensional elastic collision, if the mass and velocity of object 1 are and and if the mass and velocity of object 2 are and then their final velocities and are given by and

Investigate the plausibility of the results of Problem 64 by calculating the final velocities in the following limits: (a) When the two masses are equal, show that the particles swap velocities: and (b) If and show that and (c) If and show that and v2f _ 2v1i v .

A bullet of mass is fired horizontally with a speed into the bob of a ballistic pendulum of mass The pendulum consists of a bob attached to one end of a very light rod of length L. The rod is free to rotate about a horizontal axis through its other end. The bullet is stopped in the bob. Find the minimum such that the bob will swing through a complete circle.

A bullet of mass is fired horizontally with a speed v into the bob of a ballistic pendulum of mass (Figure 18-19). Find the maximum height h attained by the bob if the bullet passes through the bob and emerges with a speed

A heavy wooden block rests on a flat table and a highspeed bullet is fired horizontally into the block, the bullet stopping in it. How far will the block slide before coming to a stop? The mass of the bullet is the mass of the block is the bullets impact speed is and the coefficient of kinetic friction between the block and the table is 0.220. (Assume that the bullet does not cause the block to spin.)

A 0.425-kg ball with a speed of rolls across a level surface toward an open 0.327-kg box that is resting on its side. The ball enters the box, and the box (with the ball inside it) then slides across the surface a distance of What is the coefficient of kinetic friction between the box and the table?

Tarzan is in the path of a pack of stampeding elephants when Jane swings in to the rescue on a rope vine, hauling him off to safety. The length of the vine is and Jane starts her swing with the rope horizontal. If Janes mass is and Tarzans mass is to what height above the ground will the pair swing after she rescues him? (Assume the rope is vertical when she grabs him.)

Scientists estimate that the meteorite responsible for the creation of Barringer Meteorite Crater in Arizona weighed roughly and was traveling at a speed of Take Earths orbital speed to be about (a) What should the direction of impact be if Earths orbital speed is to be changed by the maximum possible amount? (b) Assuming the condition of collision in Part (a), estimate the maximum percentage change in Earths orbital speed as a result of this collision. (c) What mass of an asteroid, having a speed equal to Earths orbital speed, would be necessary to change Earths orbital speed by 1.00%?

William Tell shoots an apple from his sons head. The speed of the 125-g arrow just before it strikes the apple is and at the time of impact it is traveling horizontally. If the arrow sticks in the apple and the arrow/apple combination strikes the ground behind the sons feet, how massive was the apple? Assume the son is tall.

The beryllium isotope is unstable, decays into two particles and releases of energy. Determine the velocities of the two particles that arise from the decay of a nucleus at rest, assuming that all the energy appears as kinetic energy of the particles.

The light isotope, of lithium is unstable and breaks up spontaneously into a proton and an particle. In this process, of energy are released, appearing as the kinetic energy of the two decay products. Determine the velocities of the proton and the particle that arise from the decay of a nucleus at rest. (Note: The masses of the proton and alpha particle are and m ) a _ 4 mp m _ 6.64 _ 10_27 kg. p _ 1.67 _ 10_27 kg v0 m1 75 A 3.00-kg projectile is fired with an initial speed of at an angle of 30.0 with the horizontal. At the top of its trajectory, the projectile explodes into two fragments of masses and At after the explosion, the 2.00-kg fragment lands on the ground directly below the point of explosion. (a) Determine the velocity of the 1.00-kg fragment immediately after the explosion. (b) Find the distance between the point of firing and the point at which the 1.00-kg fragment strikes the ground. (c) Determine the energy released in the explosion.

A 3.00-kg projectile is fired with an initial speed of at an angle of 30.0 with the horizontal. At the top of its trajectory, the projectile explodes into two fragments of masses and At after the explosion, the 2.00-kg fragment lands on the ground directly below the point of explosion. (a) Determine the velocity of the 1.00-kg fragment immediately after the explosion. (b) Find the distance between the point of firing and the point at which the 1.00-kg fragment strikes the ground. (c) Determine the energy released in the explosion.

The boron isotope is unstable and disintegrates into a proton and two particles. The total energy released as kinetic energy of the decay products is In one such event, with the nucleus at rest prior to decay, the velocity of the proton is measured as If the two particles have equal energies, find the magnitude and the direction of their velocities with respect to the direction of the proton.

ENGINEERING APPLICATION, CONTEXT-RICH You are in charge of measuring the coefficient of restitution for a new alloy of steel. You convince your engineering team to accomplish this task by simply dropping a small ball onto a plate, both ball and plate made from the experimental alloy. If the ball is dropped from a height of and rebounds to a height of what is the coefficient of restitution?

According to official racquetball rules, to be acceptable for tournament play, a ball must bounce to a height of between 173 and when dropped from a height of at room temperature. What is the acceptable range of values for the coefficient of restitution for the racquetballfloor system?

A ball bounces to 80 percent of its original height. (a) What fraction of its mechanical energy is lost each time it bounces? (b) What is the coefficient of restitution of the ballfloor system?

A 2.0-kg object moving to the right at collides head-on with a 4.0-kg object that is initially at rest. After the collision, the 2.0-kg object is moving to the left at (a) Find the velocity of the 4.0-kg object after the collision. (b) Find the energy lost in the collision. (c) What is the coefficient of restitution for these objects?

A 2.0-kg block moving to the right with a speed of collides with a 3.0-kg block that is moving in the same direction at as in Figure 8-49. After the collision, the 3.0-kg block moves to the right at Find (a) the velocity of the 2.0-kg block after the collision, and (b) the coefficient of restitution between the two blocks.

CONTEXT-RICH To keep homerun records and distances consistent from year to year, organized baseball randomly checks the coefficient of restitution between new baseballs and wooden surfaces similar to that of an average bat. Suppose you are in charge of making sure that no juiced baseballs are produced. (a) In a random test, you find one that when dropped from rebounds What is the coefficient of restitution for this ball? (b) What is the maximum distance home run shot you would expect from this ball, neglecting any effects due to air resistance and making reasonable assumptions for bat speeds and incoming pitch speeds? Is this a juiced ball, a normal ball, or a dead ball?

CONCEPTUAL To make puck handling easy, hockey pucks are kept frozen until they are used in the game. (a) Explain why room-temperature pucks would be more difficult to handle on the end of a stick than a frozen puck. (Hint: Hockey pucks are made of rubber.) (b) A room-temperature puck rebounds when dropped onto a wooden surface from If a frozen puck has only half the coefficient of restitution of a room-temperature one, predict how high the frozen puck would rebound under the same conditions.

In Section 8-3, it was proved by using geometry that when a particle elastically collides with another particle of equal mass that is initially at rest, the two postcollision velocities are perpendicular. Here we examine another way of proving this result that illustrates the power of vector notation. (a) Given that square both sides of this equation (obtain the scalar product of each side with itself) to show that (b) Let the momentum of the initially moving particle be and the momenta of the particles after the collision be and Write the vector equation for the conservation of linear momentum and square both sides (obtain the dot product of each side with itself). Compare it to the equation gotten from the elastic collision condition (kinetic energy is conserved) and finally show that these two equations imply that

In a pool game, the cue ball, which has an initial speed of makes an elastic collision with the eight ball, which is initially at rest. After the collision, the eight ball moves at an angle of 30 to the right of the original direction of the cue ball. Assume that the balls have equal mass. (a) Find the direction of motion of the cue ball immediately after the collision. (b) Find the speed of each ball immediately after the collision.

Object A, which has a mass m and a velocity collides with object B, which has a mass 2m and a velocity Following the collision, object B has a velocity of (a) Determine the velocity of object A after the collision. (b) Is the collision elastic? If not, express the change in the kinetic energy in terms of m and

A puck of mass moving at approaches an identical puck that is stationary on frictionless ice. After the collision, the first puck leaves with a speed at 30 to the original line of motion; the second puck leaves with speed at 60, as in Figure 8-50. (a) Calculate the speeds and (b) Was the collision elastic? SSM

Figure 8-51 shows the result of a collision between two objects of unequal mass. (a) Find the speed of the larger mass after the collision; also find the angle (b) Show that the collision is elastic.

A 2.0-kg ball moving at makes an off-center collision with a 3.0-kg ball that is initially at rest. After the collision, the 2.0-kg ball is deflected at an angle of 30 from its original direction of motion and the 3.0-kg ball is moving at Find the speed of the 2.0-kg ball and the direction of the 3.0-kg ball after the collision. Hint:

A particle has initial speed It collides with a second particle with the same mass that is initially at rest. The first particle is deflected through an angle Its speed after the collision is v. The second particle recoils, and its velocity makes an angle with the initial direction of the first particle. (a) Show that (b) Show that if the collision is elastic, then

In the center-of-mass reference frame, a particle with mass and momentum makes an elastic head-on collision with a second particle of mass and momentum After the collision the first particles momentum is Write the total kinetic initial energy in terms of and and the total final energy in terms of and and show that If the particle is merely turned around by the collision and leaves with the speed it had initially. What is the situation for the solution?

MULTISTEP A 3.0-kg block is traveling in the direction at and a 1.0-kg block is traveling in the direction at (a) Find the velocity of the center of mass. (b) Subtract from the velocity of each block to find the velocity of each block in the center-of-mass reference frame. (c) After they make a head-on elastic collision, the velocity of each block is reversed (in the centerof- mass frame). Find the velocity of each block in the center-ofmass frame after the collision. (d) Transform back into the original frame by adding to the velocity of each block. (e) Check your result by finding the initial and final kinetic energies of the blocks in the original frame and comparing them.

Repeat Problem 92 with the second block having a mass of and moving to the right at

ENGINEERING APPLICATION A rocket burns fuel at a rate of and exhausts the gas at a speed of relative to the rocket. Find the magnitude of the thrust of the rocket.

ENGINEERING APPLICATION A rocket has an initial mass of 80 percent of which is the fuel. It burns fuel at a rate of 200 kg/s and exhausts its gas at a relative speed of 1.80 km/s. Find (a) the thrust of the rocket, (b) the time until burnout, and (c) the rocket’s speed at burnout, assuming it moves straight upward near the surface of Earth. Assume g is constant and neglect any effects of air resistence.

ENGINEERING APPLICATION The specific impulse of a rocket propellant is defined as where is the thrust of the propellant, g is the magnitude of free-fall acceleration, and R is the rate at which the propellant is burned. The rate depends predominantly on the type and exact mixture of the propellant. (a) Show that the specific impulse has the dimension of time. (b) Show that where is the relative speed of the exhaust. (c) What is the specific impulse (in seconds) of the propellant used in the Saturn V rocket of Example 8-19.

SPREADSHEET, ENGINEERING APPLICATION The initial thrust-to-weight ratio of a rocket is where is the rockets thrust and the initial mass of the rocket, including the propellant. (a) For a rocket launched straight up from Earths surface, show that where is the initial acceleration of the rocket. For manned rocket flight, cannot be made much larger than 4 for the comfort and safety of the astronauts. (As the rocket lifts off, the astronauts will feel that their weight is equal to times their normal weight.) (b) Show that the final velocity of a rocket launched from Earths surface can, in terms of and (see Problem 96), be written as where is the mass of the rocket (not including the spent propellant). (c) Using a spreadsheet program or graphing calculator, graph as a function of the mass ratio for and for values of the mass ratio from 2 to 10. (Note that the mass ratio cannot be less than 1.) (d) To lift a rocket into orbit, a final velocity after burnout of is needed. Calculate the mass ratio required of a single-stage rocket to do this, using the values of specific impulse and thrust ratio given in Part (b). For engineering reasons, it is difficult to make a rocket with a mass ratio much greater than 10. Can you see why multistage rockets are usually used to put payloads into orbit around Earth?

ENGINEERING APPLICATION The height that a model rocket launched from Earths surface can reach can be estimated by assuming that the burn time is short compared to the total flight time; the rocket is therefore in free-fall for most of the flight. (This estimate neglects the burn time in calculations of both time and displacement.) For a model rocket with specific impulse mass ratio and initial thrustto- weight ratio (these parameters are defined in Problems 96 and 97), estimate (a) the height the rocket can reach, and (b) the total flight time. (c) Justify the assumption used in the estimates by comparing the flight time from Part (b) to the time it takes to consume the fuel.

A 250-g model-train car traveling at links up with a 400-g car that is initially at rest. What is the speed of the cars immediately after they link up? Find the pre- and postcollision kinetic energies of the two-car system.

MULTISTEP A 250-g model-train car traveling at heads toward a 400-g car that is initially at rest. (a) Find the total kinetic energy of the two-car system. (b) Find the velocity of each car in the center-of-mass reference frame, and use these velocities to calculate the kinetic energy of the two-car system in the center-of-mass reference frame. (c) Find the kinetic energy associated with the motion of the center of mass of the system. (d) Compare your answer for Part (a) with the sum of your answers for Parts (b) and (c).

A1500-kg car traveling north at collides at an intersection with a 2000-kg car traveling west at The two cars stick together. (a) What is the total momentum of the system before the collision? (b) What are the magnitude and direction of the velocity of the wreckage just after the collision.

A 60-kg woman stands on the back of a 6.0-m-long, 120-kg raft that is floating at rest in still water. The raft is from a fixed pier, as shown in Figure 8-52. (a) The woman walks to the front of the raft and stops. How far is the raft from the pier now? (b) While the woman walks, she maintains a constant speed of relative to the raft. Find the total kinetic energy of the system (woman plus raft), and compare your answer with the kinetic energy if the woman walked at on a raft tied to the pier. (c) Where do these kinetic energies come from, and where do they go when the woman stops at the front of the raft? (d) On land, the woman puts a lead shot Then, standing at the back of the raft, she aims forward, and puts the shot so that just after it leaves her hand, it has the same velocity relative to her as it did when she threw it from the ground. Approximately, where does her shot land?

A 1.0-kg steel ball and a 2.0-m cord of negligible mass make up a simple pendulum that can pivot without friction about the point O, as in Figure 8-53. This pendulum is released from rest in a horizontal position, and when the ball is at its lowest point it strikes a 1.0-kg block sitting at rest on a shelf. Assume that the collision is perfectly elastic and that the coefficient of kinetic friction between the block and shelf is 0.10. (a) What is the velocity of the block just after impact? (b) How far does the block slide before coming to rest (assuming that the shelf is long enough)?

Figure 8-54 shows a World War I cannon mounted on a railcar and set so that it will project a shell at an angle of 30 above the horizontal. With the car initially at rest on a horizontal frictionless track, the cannon fires a 200-kg projectile at (All values are for the frame of reference of the track.). (a) Will the vector momentum of the carcannonshell system be the same just before and just after the shell is fired? Explain your answer. (b) If the mass of the railcar plus cannon is what will be the recoil velocity of the car along the track after the firing? (c) The shell is observed to rise to a maximum height of as it moves through its trajectory. At this point, its speed is On the basis of this information, calculate the amount of thermal energy produced by air friction on the shell from the cannons mouth to this maximum height.

MULTISTEP One popular, if dangerous, classroom demonstration involves holding a baseball an inch or so directly above a basketball that you are holding a few feet above a hard floor, and dropping the two balls simultaneously. The two balls will collide just after the basketball bounces from the floor; the baseball will then rocket off into the ceiling tiles, while the basketball will stop in midair. (a) Assuming that the collision of the basketball with the floor is elastic, what is the relation between the velocities of the balls just before they collide? (b) Assuming the collision between the two balls is elastic, use the result of Part (a) and the conservation of momentum and the conservation of energy to show that, if the basketball is three times as heavy as the baseball, the final velocity of the basketball will be zero. (This is approximately the true mass ratio, which is why the demonstration is so dramatic.) (c) If the speed of the baseball is v just before the collision, what is its speed just after the collision?

(a) In Problem 105, if we held a third ball above the baseball and basketball and wanted both the baseball and basketball to stop in midair, what should the ratio of the mass of the top ball to the mass of the baseball be? (b) If the speed of the top ball is v just before the collision, what is its speed just after the collision?

In the slingshot effect, the transfer of energy in an elastic collision is used to boost the energy of a space probe so that it can escape from the solar system. All speeds are relative to an inertial frame in which the center of the Sun remains at rest. Figure 8-55 shows a space probe moving at toward Saturn, which is moving at toward the probe. Because of the gravitational attraction between Saturn and the probe, the probe swings around Saturn and heads back in the opposite direction with speed (a) Assuming this collision to be a one-dimensional elastic collision with the mass of Saturn much much greater than that of the probe, find (b) By what factor is the kinetic energy of the probe increased? Where does this energy come from?

A 13-kg block is at rest on a level floor. A 400-g glob of putty is thrown at the block so that the putty travels horizontally, hits the block, and sticks to it. The block and putty slide 15 cm along the floor. If the coefficient of kinetic friction is 0.40, what is the initial speed of the putty?

CONTEXT-RICH Your accident-reconstruction team has been hired by the local police to analyze the following accident. Acareless driver rear-ended a car that was halted at a stop sign. Just before impact, the driver slammed on his brakes, locking the wheels. The driver of the struck car had his foot solidly on the brake pedal, locking his brakes. The mass of the struck car was and that of the initially moving vehicle was On collision, the bumpers of the two cars meshed. Police determine from the skid marks that after the collision the two cars moved together. Tests revealed that the coefficient of kinetic friction between the tires and pavement was 0.92. The driver of the moving car claims that he was traveling at less than as he approached the intersection. Is he telling the truth?

A pendulum consists of a compact 0.40-kg bob attached to a string of length A block of mass m rests on a horizontal frictionless surface. The pendulum is released from rest at an angle of 53 with the vertical. The bob collides elastically with the block at the lowest point in its arc. Following the collision, the maximum angle of the pendulum with the vertical is 5.73. Determine the mass m.

A 1.00-kg block (mass m ) and a second block (mass M) are both initially at rest on a frictionless inclined plane (Figure 8-56). Mass M rests against a spring that has a force constant of \(11.0 \mathrm{kN} / \mathrm{m}\). The distance along the plane between the two blocks is \(4.00 \mathrm{~m}\). The \(1.00-\mathrm{kg}\) block is released, making an elastic collision with the larger block. The \(1.00-\mathrm{kg}\) block then rebounds a distance of \(2.56 \mathrm{~m}\) back up the inclined plane. The block of mass M momentarily comes to rest \(4.00 \mathrm{~cm}\) from its initial position. Find M.

A neutron of mass m makes an elastic head-on collision with a stationary nucleus of mass M. (a) Show that the kinetic energy of the nucleus after the collision is given by where is the initial kinetic energy of the neutron. (b) Show that the fractional change in the kinetic energy of the neutron is given by (c) Show that this expression gives plausible results both if and if What is the best stationary nucleus for the neutron to collide head-on with if the objective is to produce a maximum loss in the kinetic energy of the neutron?

ENGINEERING APPLICATION The mass of a carbon nucleus is approximately 12 times the mass of a neutron. (a) Use the results of Problem 112 to show that after N head-on collisions of a neutron with carbon nuclei at rest, the kinetic energy of the neutron is approximately where is its initial kinetic energy. (b) Neutrons emitted in the fission of a uranium nucleus have kinetic energies of about For such a neutron to cause the fission of another uranium nucleus in a reactor, its kinetic energy must be reduced to about How many head-on collisions are needed to reduce the kinetic energy of a neutron from to assuming elastic head-on collisions with stationary carbon nuclei?

ENGINEERING APPLICATION On average, a neutron actually loses only 63 percent of its energy in an elastic collision with a hydrogen atom (not 100 percent) and 11 percent of its energy in an elastic collision with a carbon atom (not 28 percent). (These numbers are an average over all types of collisions, not just head-on ones. Thus, the results are lower than the ones determined from analyses like that in Problem 112, because most collisions are not head-on.) Calculate the actual number of collisions, on average, needed to reduce the energy of a neutron from to if the neutron collides with (a) stationary hydrogen atoms and (b) stationary carbon atoms.

Two astronauts at rest face each other in space. One, who has mass throws a ball of mass to the other, whose mass is The second astronuat catches the ball and throws it back to the first astronaut. Following each throw, the ball has a speed of v relative to the thrower. After each has made one throw and one catch, (a) how fast are the astronauts moving? (b) How much has the twoastronaut systems kinetic energy changed and where did this energy come from?

A stream of elastic glass beads, each with a mass of 0.50 g, comes out of a horizontal tube at a rate of 100 per second (see Figure 8-57). The beads fall a distance of to a balance pan and bounce back to their original height. How much mass must be placed in the other pan of the balance to keep the pointer at zero?

A dumbbell consisting of two balls of mass m connected by a massless 1.00-m-long rod rests on a frictionless floor against a frictionless wall with one ball directly above the other. The centerto- center distance between the balls is equal to 1.00 m. The dumbbell then begins to slide down the wall, as in Figure 8-58. Find the speed of the bottom ball at the moment when it equals the speed of the top ball.