Chapter 6

If a \(60-\mathrm{kg}\) woman is riding in a car traveling at \(90 \mathrm{~km} / \mathrm{h}\), what is her linear momentum relative to (a) the ground and (b) the car?

The linear momentum of a runner in a 100 -m dash is \(7.5 \times 10^{2} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). If the runner's speed is \(10 \mathrm{~m} / \mathrm{s}\), what is his mass?

Find the magnitude of the linear momentum of (a) a \(7.1-\mathrm{kg}\) bowling ball traveling at \(12 \mathrm{~m} / \mathrm{s}\) and \((\mathrm{b})\) a \(1200-\mathrm{kg}\) automobile traveling at \(90 \mathrm{~km} / \mathrm{h}\).

In a football game, a lineman usually has more mass than a running back. (a) Will a lineman always have greater linear momentum than a running back? Why? (b) Who has greater linear momentum, a 75-kg running back running at \(8.5 \mathrm{~m} / \mathrm{s}\) or a 120 -kg lineman moving at \(5.0 \mathrm{~m} / \mathrm{s} ?\)

A 0.150 -kg baseball traveling with a horizontal speed of \(4.50 \mathrm{~m} / \mathrm{s}\) is hit by a bat and then moves with a speed of \(34.7 \mathrm{~m} / \mathrm{s}\) in the opposite direction. What is the change in the ball's momentum?

A 15.0 - \(\mathrm{g}\) rubber bullet hits a wall with a speed of \(150 \mathrm{~m} / \mathrm{s}\). If the bullet bounces straight back with a speed of \(120 \mathrm{~m} / \mathrm{s}\), what is the change in momentum of the bullet?

\(A\) 5.0-g bullet with a speed of \(200 \mathrm{~m} / \mathrm{s}\) is fired horizontally into a \(0.75-\mathrm{kg}\) wooden block at rest on a table. If the block containing the bullet slides a distance of \(0.20 \mathrm{~m}\) before coming to rest, (a) what is the coefficient of kinetic friction between the block and the table? (b) What fraction of the bullet's energy is dissipated in the collision?

Two runners of mass \(70 \mathrm{~kg}\) and \(60 \mathrm{~kg}\), respectively, have a total linear momentum of \(350 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). The heavier runner is running at \(2.0 \mathrm{~m} / \mathrm{s}\). Determine the possible velocities of the lighter runner.

A 0.20 -kg billiard ball traveling at a speed of \(15 \mathrm{~m} / \mathrm{s}\) strikes the side rail of a pool table at an angle of \(60^{\circ}\) (vFig. 6.31). If the ball rebounds at the same speed and angle, what is the change in its momentum?

A 2.0 -kg mud ball drops from rest at a height of \(15 \mathrm{~m}\). If the impact between the ball and the ground lasts \(0.50 \mathrm{~s}\), what is the average net force exerted by the ball on the ground?

In football practice, two wide receivers run different pass receiving patterns. One with a mass of \(80.0 \mathrm{~kg}\) runs at \(45^{\circ}\) northeast at a speed of \(5.00 \mathrm{~m} / \mathrm{s}\). The second receiver (mass of \(90.0 \mathrm{~kg}\) ) runs straight down the field (due east) at \(6.00 \mathrm{~m} / \mathrm{s}\). (a) What is the direction of their total momentum: (1) exactly northeast, (2) to the north of northeast, (3) exactly east, or (4) to the east of northeast? (b) Justify your answer in part (a) by actually computing their total momentum.

A major league catcher catches a fastball moving at \(95.0 \mathrm{mi} / \mathrm{h}\) and his hand and glove recoil \(10.0 \mathrm{~cm}\) in bringing the ball to rest. If it took 0.00470 s to bring the ball (with a mass of \(250 \mathrm{~g}\) ) to rest in the glove, (a) what are the magnitude and direction of the change in momentum of the ball? (b) Find the average force the ball exerts on the hand and glove.

At a basketball game, a 120 -lb cheerleader is tossed vertically upward with a speed of \(4.50 \mathrm{~m} / \mathrm{s}\) by a male cheerleader. (a) What is the cheerleader's change in momentum from the time she is released to just before being caught if she is caught at the height at which she was released? (b) Would there be any difference if she were caught \(0.30 \mathrm{~m}\) below the point of release? If so, what is the change then?

A ball of mass \(200 \mathrm{~g}\) is released from rest at a height of \(2.00 \mathrm{~m}\) above the floor and it rebounds straight up to a height of \(0.900 \mathrm{~m}\). (a) Determine the ball's change in momentum due to its contact with the floor. (b) If the contact time with the floor was \(0.0950 \mathrm{~s}\), what was the average force the floor exerted on the ball, and in what direction?

When tossed upward and hit horizontally by a batter, a \(0.20-\mathrm{kg}\) softball receives an impulse of \(3.0 \mathrm{~N} \cdot \mathrm{s}\). With what horizontal speed does the ball move away from the bat?

An automobile with a linear momentum of \(3.0 \times 10^{4} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) is brought to a stop in \(5.0 \mathrm{~s}\). What is the magnitude of the average braking force?

A pool player imparts an impulse of \(3.2 \mathrm{~N} \cdot \mathrm{s}\) to a stationary \(0.25-\mathrm{kg}\) cue ball with a cue stick. What is the speed of the ball just after impact?

When bunting, a baseball player uses the bat to change both the speed and direction of the baseball. (a) Will the magnitude of the change in momentum of the baseball before and after the bunt be (1) greater than the magnitude of the momentum of the baseball either before or after the bunt, (2) equal to the difference between the magnitudes of momenta of the baseball before and after the bunt, or (3) equal to the sum of the magnitudes of momenta of the baseball before and after the bunt? Why? (b) The baseball has a mass of \(0.16 \mathrm{~kg}\); its speeds before and after the bunt are \(15 \mathrm{~m} / \mathrm{s}\) and \(10 \mathrm{~m} / \mathrm{s}\) respectively; the bunt lasts \(0.025 \mathrm{~s}\). What is the change in momentum of the baseball? (c) What is the average force on the ball by the bat?

A car with a mass of \(1500 \mathrm{~kg}\) is rolling on a level road at \(30.0 \mathrm{~m} / \mathrm{s}\). It receives an impulse with a magnitude of \(2000 \mathrm{~N} \cdot \mathrm{s}\) and its speed is reduced as much as possible by an impulse of this size. (a) Was this impulse caused by (1) the driver hitting the accelerator, (2) the driver putting on the brakes, or (3) the driver turning the steering wheel? (b) What was the car's speed after the impulse was applied?

An astronaut (mass of \(100 \mathrm{~kg},\) with equipment) is headed back to her space station at a speed of \(0.750 \mathrm{~m} / \mathrm{s}\) but at the wrong angle. To correct her direction, she fires rockets from her backpack at right angles to her motion for a brief time. These directional rockets exert a constant force of \(100.0 \mathrm{~N}\) for only \(0.200 \mathrm{~s}\). [Neglect the small loss of mass due to burning fuel and assume the impulse is at right angles to her initial momentum.] (a) What is the magnitude of the impulse delivered to the astronaut? (b) What is her new direction (relative to the initial direction)? (c) What is her new speed?

A volleyball is traveling toward you. (a) Which action will require a greater force on the volleyball, your catching the ball or your hitting the ball back? Why? (b) A 0.45 -kg volleyball travels with a horizontal velocity of \(4.0 \mathrm{~m} / \mathrm{s}\) over the net. You jump up and hit the ball back with a horizontal velocity of \(7.0 \mathrm{~m} / \mathrm{s}\). If the contact time is \(0.040 \mathrm{~s}\), what was the average force on the ball?

A \(0.45-\mathrm{kg}\) piece of putty is dropped from a height of \(2.5 \mathrm{~m}\) above a flat surface. When it hits the surface, the putty comes to rest in \(0.30 \mathrm{~s}\). What is the average force exerted on the putty by the surface?

A \(50-\mathrm{kg}\) driver sits in her car waiting for the traffic light to change. Another car hits her from behind in a head-on, rear-end collision and her car suddenly receives an acceleration of \(16 \mathrm{~m} / \mathrm{s}^{2}\). If all of this takes place in \(0.25 \mathrm{~s},\) (a) what is the impulse on the driver? (b) What is the average force exerted on the driver, and what exerts this force?

An incoming 0.14 -kg baseball has a speed of \(45 \mathrm{~m} / \mathrm{s}\). The batter hits the ball, giving it a speed of \(60 \mathrm{~m} / \mathrm{s}\). If the contact time is \(0.040 \mathrm{~s},\) what is the average force of the bat on the ball?

At a shooting competition, a contestant fires and a 12.0-g bullet leaves the rifle with a muzzle speed of \(130 \mathrm{~m} / \mathrm{s}\). The bullet hits the thick target backing and stops after traveling \(4.00 \mathrm{~cm}\). Assuming a uniform acceleration, (a) what is the impulse on the target? (b) What is the average force on the target?

A 15000 -N automobile travels at a speed of \(45 \mathrm{~km} / \mathrm{h}\) northward along a street, and a 7500 -N sports car travels at a speed of \(60 \mathrm{~km} / \mathrm{h}\) eastward along an intersecting street. (a) If neither driver brakes and the cars collide at the intersection and lock bumpers, what will the velocity of the cars be immediately after the collision? (b) What percentage of the initial kinetic energy will be lost in the collision?

In a simulated head-on crash test, a car impacts a wall at \(25 \mathrm{mi} / \mathrm{h}(40 \mathrm{~km} / \mathrm{h})\) and comes abruptly to rest. A 120-lb passenger dummy (with a mass of \(55 \mathrm{~kg}\) ), without a seatbelt, is stopped by an air bag, which exerts a force on the dummy of 2400 lb. How long was the dummy in contact with the air bag while coming to a stop?

A \(60-\mathrm{kg}\) astronaut floating at rest in space outside a space capsule throws his \(0.50-\mathrm{kg}\) hammer such that it moves with a speed of \(10 \mathrm{~m} / \mathrm{s}\) relative to the capsule. What happens to the astronaut?

In a pairs figure-skating competition, a \(65-\mathrm{kg}\) man and his \(45-\mathrm{kg}\) female partner stand facing each other on skates on the ice. If they push apart and the woman has a velocity of \(1.5 \mathrm{~m} / \mathrm{s}\) eastward, what is the velocity of her partner? (Neglect friction.)

To get off a frozen, frictionless lake, a 65.0-kg person takes off a 0.150 -kg shoe and throws it horizontally, directly away from the shore with a speed of \(2.00 \mathrm{~m} / \mathrm{s}\). If the person is \(5.00 \mathrm{~m}\) from the shore, how long does he take to reach it?

An object initially at rest explodes and splits into three fragments. The first fragment flies off to the west, and the second fragment flies off to the south. The third fragment will fly off toward a general direction of (1) southwest, (2) north of east, (3) either due north or due east. Why? (b) If the object has a mass of \(3.0 \mathrm{~kg},\) the first fragment has a mass of \(0.50 \mathrm{~kg}\) and a speed of \(2.8 \mathrm{~m} / \mathrm{s},\) and the second fragment has a mass of \(1.3 \mathrm{~kg}\) and a speed of \(1.5 \mathrm{~m} / \mathrm{s},\) what are the speed and direction of the third fragment?

Consider two string-suspended balls, both with a mass of \(0.15 \mathrm{~kg}\). (Similar to the arrangement in Fig. \(6.15,\) but with only two balls.) One ball is pulled back in line with the other so it has a vertical height of 10 \(\mathrm{cm},\) and is then released. (a) What is the speed of the ball just before hitting the stationary one? (b) If the collision is completely inelastic, to what height do the balls swing?

A cherry bomb explodes into three pieces of equal mass. One piece has an initial velocity of \(10 \mathrm{~m} / \mathrm{s} \hat{\mathrm{x}}\). Another piece has an initial velocity of \(6.0 \mathrm{~m} / \mathrm{s} \hat{\mathrm{x}}-3.0 \mathrm{~m} / \mathrm{s} \hat{\mathrm{y}} .\) What is the velocity of the third piece?

Two ice skaters not paying attention collide in a completely inelastic collision. Prior to the collision, skater 1 , with a mass of \(60 \mathrm{~kg},\) has a velocity of \(5.0 \mathrm{~km} / \mathrm{h}\) eastward, and moves at a right angle to skater \(2,\) who has a mass of \(75 \mathrm{~kg}\) and a velocity of \(7.5 \mathrm{~km} / \mathrm{h}\) southward. What is the velocity of the skaters after collision?

Two balls of equal mass \((0.50 \mathrm{~kg})\) approach the origin along the positive \(x\) - and \(y\) -axes at the same speed \((3.3 \mathrm{~m} / \mathrm{s})\). (a) What is the total momentum of the system? (b) Will the balls necessarily collide at the origin? What is the total momentum of the system after both balls have passed through the origin?

A \(1200-\mathrm{kg}\) car moving to the right with a speed of \(25 \mathrm{~m} / \mathrm{s}\) collides with a \(1500-\mathrm{kg}\) truck and locks bumpers with the truck. Calculate the velocity of the combination after the collision if the truck is initially (a) at rest, (b) moving to the right with a speed of \(20 \mathrm{~m} / \mathrm{s},\) and (c) moving to the left with a speed of \(20 \mathrm{~m} / \mathrm{s}\).

A \(10-\mathrm{g}\) bullet moving horizontally at \(400 \mathrm{~m} / \mathrm{s}\) penetrates a \(3.0-\mathrm{kg}\) wood block resting on a horizontal surface. If the bullet slows down to \(300 \mathrm{~m} / \mathrm{s}\) after emerging from the block, what is the speed of the block immediately after the bullet emerges ( \(\mathbf{v}\) Fig. 6.33 )?

An explosion of a \(10.0-\mathrm{kg}\) bomb releases only two separate pieces. The bomb was initially at rest and a 4.00-kg piece travels westward at \(100 \mathrm{~m} / \mathrm{s}\) immediately after the explosion. (a) What are the speed and direction of the other piece immediately after the explosion? (b) How much kinetic energy was released in this explosion?

A 1600 -kg (empty) truck rolls with a speed of \(2.5 \mathrm{~m} / \mathrm{s}\) under a loading bin, and a mass of \(3500 \mathrm{~kg}\) is deposited into the truck. What is the truck's speed immediately after loading?

A new crowd control method utilizes "rubber" bullets instead of real ones. Suppose that, in a test, one of these "bullets" with a mass of \(500 \mathrm{~g}\) is traveling at \(250 \mathrm{~m} / \mathrm{s}\) to the right. It hits a stationary target head-on. The target's mass is \(25.0 \mathrm{~kg}\) and it rests on a smooth surface. The bullet bounces backward (to the left) off the target at \(100 \mathrm{~m} / \mathrm{s}\). (a) Which way must the target move after the collision: (1) right, (2) left, (3) it could be stationary, or (4) you can't tell from the data given? (b) Determine the recoil speed of the target after the collision.

For a movie scene, a 75-kg stuntman drops from a tree onto a \(50-\mathrm{kg}\) sled that is moving on a frozen lake with a velocity of \(10 \mathrm{~m} / \mathrm{s}\) toward the shore. (a) What is the speed of the sled after the stuntman is on board? (b) If the sled hits the bank and stops, but the stuntman keeps on going, with what speed does he leave the sled? (Neglect friction.)

A 90-\mathrm{kg}\( astronaut is stranded in space at a point \)6.0 \mathrm{~m}\( from his spaceship, and he needs to get back in 4.0 min to control the spaceship. To get back, he throws a \)0.50-\mathrm{kg}\( piece of equipment so that it moves at a speed of \)4.0 \mathrm{~m} / \mathrm{s}$ directly away from the spaceship. (a) Does he get back in time? (b) How fast must he throw the piece of equipment so he gets back in time?

A small asteroid (mass of \(10 \mathrm{~g}\) ) strikes a glancing blow at a satellite in empty space. The satellite was initially at rest and the asteroid was traveling at \(2000 \mathrm{~m} / \mathrm{s}\). The satellite's mass is \(100 \mathrm{~kg} .\) The asteroid is deflected \(10^{\circ}\) from its original direction and its speed decreases to \(1000 \mathrm{~m} / \mathrm{s},\) but neither object loses mass. Determine the (a) direction and (b) speed of the satellite after the collision.

A ballistic pendulum is a device used to measure the velocity of a projectile- for example, the muzzle velocity of a rifle bullet. The projectile is shot horizontally into, and becomes embedded in, the bob of a pendulum, as illustrated in \(>\) Fig. \(6.35 .\) The pendulum swings upward to some height \(h,\) which is measured. The masses of the block and the bullet are known. Using the laws of momentum and energy, show that the initial velocity of the projectile is given by \(v_{\mathrm{o}}=[(m+M) / m] 22 g h\).

A proton of mass \(m\) moving with a speed of \(3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\) undergoes a head-on elastic collision with an alpha particle of mass \(4 m\), which is initially at rest. What are the velocities of the two particles after the collision?

A \(4.0-\mathrm{kg}\) ball with a velocity of \(4.0 \mathrm{~m} / \mathrm{s}\) in the \(+x\) -direction collides head-on elastically with a stationary \(2.0-\mathrm{kg}\) ball. What are the velocities of the balls after the collision?

A dropped rubber ball hits the floor with a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) and rebounds to a height of \(0.25 \mathrm{~m} .\) What fraction of the initial kinetic energy was lost in the collision?

At a county fair, two children ram each other headon while riding on the bumper cars. Jill and her car, traveling left at \(3.50 \mathrm{~m} / \mathrm{s}\), have a total mass of \(325 \mathrm{~kg}\). Jack and his car, traveling to the right at \(2.00 \mathrm{~m} / \mathrm{s}\), have a total mass of \(290 \mathrm{~kg}\). Assuming the collision to be elastic, determine their velocities after the collision.

In a high-speed chase, a policeman's car bumps a criminal's car directly from behind to get his attention. The policeman's car is moving at \(40.0 \mathrm{~m} / \mathrm{s}\) to the right and has a total mass of \(1800 \mathrm{~kg}\). The criminal's car is initially moving in the same direction at \(38.0 \mathrm{~m} / \mathrm{s}\). His car has a total mass of \(1500 \mathrm{~kg}\). Assuming an elastic collision, determine their two velocities immediately after the bump.

Shows a bird catching a fish. Assume that initially the fish jumps up and that the bird coasts horizontally and does not touch the water with its feet or flap its wings. (a) Is this kind of collision (1) elastic, (2) inelastic, or (3) completely inelastic? Why? (b) If the mass of the bird is \(5.0 \mathrm{~kg}\), the mass of the fish is \(0.80 \mathrm{~kg}\), and the bird coasts with a speed of \(6.5 \mathrm{~m} / \mathrm{s}\) before grabbing, what is the speed of the bird after grabbing the fish?