Chapter 8

A 2646 lb car is moving on the freeway at 68 mph. (a) Find the magnitude of its momentum and its kinetic energy in SI units. (b) At what speed, in \(\mathrm{m} / \mathrm{s}\), will the car's momentum be half of what it is in part (a)? (c) At what speed, in \(\mathrm{m} / \mathrm{s}\), will the car's kinetic energy be half of what it is in part (a)?

The speed of the fastest-pitched baseball was \(45 \mathrm{~m} / \mathrm{s},\) and the ball's mass was \(145 \mathrm{~g}\). (a) What was the magnitude of the momentum of this ball, and how many joules of kinetic energy did it have? (b) How fast would a 57 gram ball have to travel to have the same amount of (i) kinetic energy, and (ii) momentum?

Cart \(A\) has a mass of \(5 \mathrm{~kg}\) and is moving in the \(+x\) direction at \(2 \mathrm{~m} / \mathrm{s}\). Cart \(B\) has a mass of \(2 \mathrm{~kg}\) and is moving in the \(+y\) direction at \(5 \mathrm{~m} / \mathrm{s}\). (a) Do the two carts have the same momentum? Explain. (b) Is the magnitude of the momentum of each cart the same? Explain. (c) Is the kinetic energy of each cart the same? Explain.

The magnitude of the momentum of a cat is \(p\). What would be the magnitude of the momentum (in terms of \(p\) ) of a dog having three times the mass of the cat if it had (a) the same speed as the cat, and (b) the same kinetic energy as the cat?

Two figure skaters, one weighing \(625 \mathrm{~N}\) and the other \(725 \mathrm{~N}\), push off against each other on frictionless ice. (a) If the heavier skater travels at \(1.50 \mathrm{~m} / \mathrm{s},\) how fast will the lighter one travel? (b) How much kinetic energy is "created" during the skaters' maneuver, and where does this energy come from?

On a frictionless air track, a \(0.150 \mathrm{~kg}\) glider moving at \(1.20 \mathrm{~m} / \mathrm{s}\) to the right collides with and sticks to a stationary \(0.250 \mathrm{~kg}\) glider. (a) What is the net momentum of this two-glider system before the collision? (b) What must be the net momentum of this system after the collision? Why? (c) Use your answers in parts (a) and (b) to find the speed of the gliders after the collision. (d) Is kinetic energy conserved during the collision?

You are standing on a sheet of ice that covers the football stadium parking lot in Buffalo; there is negligible friction between your feet and the ice. A friend throws you a \(0.400 \mathrm{~kg}\) ball that is traveling horizontally at \(10.0 \mathrm{~m} / \mathrm{s}\). Your mass is \(70.0 \mathrm{~kg}\). (a) If you catch the ball, with what speed do you and the ball move afterward? (b) If the ball hits you and bounces off your chest, so that afterward it is moving horizontally at \(8.00 \mathrm{~m} / \mathrm{s}\) in the opposite direction, what is your speed after the collision?

On a frictionless, horizontal air table, puck \(A\) (with mass \(0.250 \mathrm{~kg}\) ) is moving to the right toward puck \(B\) (with mass \(0.350 \mathrm{~kg}\) ), which is initially at rest. After the collision, puck \(A\) has a velocity of \(0.120 \mathrm{~m} / \mathrm{s}\) to the left, and puck \(B\) has a velocity of \(0.650 \mathrm{~m} / \mathrm{s}\) to the right. (a) What was the speed of puck \(A\) before the collision? (b) Calculate the change in the total kinetic energy of the system that occurs during the collision.

A \(750 \mathrm{~kg}\) car is stalled on an icy road during a snowstorm. A \(1000 \mathrm{~kg}\) car traveling eastbound at \(10 \mathrm{~m} / \mathrm{s}\) collides with the rear of the stalled car. After being hit, the \(750 \mathrm{~kg}\) car slides on the ice at \(4 \mathrm{~m} / \mathrm{s}\) in a direction \(30^{\circ}\) north of east. (a) What are the magnitude and direction of the velocity of the \(1000 \mathrm{~kg}\) car after the collision? (b) Calculate the ratio of the kinetic energy of the two cars just after the collision to that just before the collision. (You may ignore the effects of friction during the collision.)

You (mass \(55 \mathrm{~kg}\) ) are riding your frictionless skateboard (mass \(5.0 \mathrm{~kg}\) ) in a straight line at a speed of \(4.5 \mathrm{~m} / \mathrm{s}\) when a friend standing on a balcony above you drops a \(2.5 \mathrm{~kg}\) sack of flour straight down into your arms. (a) What is your new speed, while holding the flour sack? (b) since the sack was dropped vertically, how can it affect your horizontal motion? Explain. (c) Suppose you now try to rid yourself of the extra weight by throwing the flour sack straight up. What will be your speed while the sack is in the air? Explain.

A 4.25 g bullet traveling horizontally with a velocity of magnitude \(375 \mathrm{~m} / \mathrm{s}\) is fired into a wooden block with mass \(1.12 \mathrm{~kg},\) initially at rest on a level frictionless surface. The bullet passes through the block and emerges with its speed reduced to \(122 \mathrm{~m} / \mathrm{s} .\) How fast is the block moving just after the bullet emerges from it?

A ball with a mass of \(0.600 \mathrm{~kg}\) is initially at rest. It is struck by a second ball having a mass of \(0.400 \mathrm{~kg}\), initially moving with a velocity of \(0.250 \mathrm{~m} / \mathrm{s}\) toward the right along the \(x\) axis. After the collision, the \(0.400 \mathrm{~kg}\) ball has a velocity of \(0.200 \mathrm{~m} / \mathrm{s}\) at an angle of \(36.9^{\circ}\) above the \(x\) axis in the first quadrant. Both balls move on a frictionless, horizontal surface. (a) What are the magnitude and direction of the velocity of the \(0.600 \mathrm{~kg}\) ball after the collision? (b) What is the change in the total kinetic energy of the two balls as a result of the collision?

Combining conservation laws. A \(5.00 \mathrm{~kg}\) chunk of ice is sliding at \(12.0 \mathrm{~m} / \mathrm{s}\) on the floor of an ice-covered valley when it collides with and sticks to another \(5.00 \mathrm{~kg}\) chunk of ice that is initially at rest. (See Figure \(8.39 .\) ) Since the valley is icy, there is no friction. After the collision, the blocks slide partially up a hillside and then slide back down. How fast are they moving when they reach the valley floor again? (Hint: Break this problem into two parts - the collision and the behavior after the collision - and apply the appropriate conservation law to each part.)

A \(15.0 \mathrm{~kg}\) block is attached to a very light horizontal spring of force constant \(500.0 \mathrm{~N} / \mathrm{m}\) and is resting on a frictionless horizontal table. (See Figure \(8.40 .\) ) Suddenly it is struck by a \(3.00 \mathrm{~kg}\) stone traveling horizontally at \(8.00 \mathrm{~m} / \mathrm{s}\) to the right, whereupon the stone rebounds at \(2.00 \mathrm{~m} / \mathrm{s}\) horizontally to the left. Find the maximum distance that the block will compress the spring after the collision. (Hint: Break this problem into two parts the collision and the behavior after the collision -and apply the appropriate conservation law to each part.)

Three identical boxcars are coupled together and are moving at a constant speed of \(20.0 \mathrm{~m} / \mathrm{s}\) on a level track. They collide with another identical boxcar that is initially at rest and couple to it, so that the four cars roll on as a unit. Friction is small enough to be ignored. (a) What is the speed of the four cars? (b) What percentage of the kinetic energy of the boxcars is dissipated in the collision? What happened to this energy?

2On a highly polished, essentially frictionless lunch counter, a \(0.500 \mathrm{~kg}\) submarine sandwich moving \(3.00 \mathrm{~m} / \mathrm{s}\) to the left collides with a \(0.250 \mathrm{~kg}\) grilled cheese sandwich moving \(1.20 \mathrm{~m} / \mathrm{s}\) to the right. (a) If the two sandwiches stick together, what is their final velocity? (b) How much mechanical energy dissipates in the collision? Where did this energy go?

A \(2 \mathrm{~kg}\) block is moving at \(5 \mathrm{~m} / \mathrm{s}\) along a frictionless table and collides with a second \(2 \mathrm{~kg}\) block that is initially at rest. After the collision, the two blocks stick together and then slide up a \(45^{\circ}\) frictionless inclined plane, as shown in Figure \(8.41 .\) Calculate the maximum distance \(L\) that the two blocks travel up the incline.

On a very muddy football field, a \(110 \mathrm{~kg}\) linebacker tackles an \(85 \mathrm{~kg}\) halfback. Immediately before the collision, the linebacker is slipping with a velocity of \(8.8 \mathrm{~m} / \mathrm{s}\) north and the halfback is sliding with a velocity of \(7.2 \mathrm{~m} / \mathrm{s}\) east. What is the velocity (magnitude and direction) at which the two players move together immediately after the collision?

A \(5.00 \mathrm{~g}\) bullet is fired horizontally into a \(1.20 \mathrm{~kg}\) wooden block resting on a horizontal surface. The coefficient of kinetic friction between block and surface is \(0.20 .\) The bullet remains embedded in the block, which is observed to slide \(0.230 \mathrm{~m}\) along the surface before stopping. What was the initial speed of the bullet?

A hungry \(11.5 \mathrm{~kg}\) predator fish is coasting from west to east at \(75.0 \mathrm{~cm} / \mathrm{s}\) when it suddenly swallows a \(1.25 \mathrm{~kg}\) fish swimming from north to south at \(3.60 \mathrm{~m} / \mathrm{s}\). Find the magnitude and direction of the velocity of the large fish just after it snapped up this meal. Ignore any effects due to the drag of the water.

To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a \(600 \mathrm{~g}\) falcon flying at \(20.0 \mathrm{~m} / \mathrm{s}\) flew into a \(1.5 \mathrm{~kg}\) raven flying at \(9.0 \mathrm{~m} / \mathrm{s}\). The falcon hit the raven at right angles to its original path and bounced back with a speed of \(5.0 \mathrm{~m} / \mathrm{s} .\) By what angle did the falcon change the raven's direction of motion?

A \(0.300 \mathrm{~kg}\) glider is moving to the right on a frictionless, horizontal air track with a speed of \(0.80 \mathrm{~m} / \mathrm{s}\) when it makes a head-on collision with a stationary \(0.150 \mathrm{~kg}\) glider. (a) Find the magnitude and direction of the final velocity of each glider if the collision is elastic. (b) Find the final kinetic energy of each glider.

On a cold winter day, a penny (mass \(2.50 \mathrm{~g}\) ) and a nickel (mass \(5.00 \mathrm{~g}\) ) are lying on the smooth (frictionless) surface of a frozen lake. With your finger, you flick the penny toward the nickel with a speed of \(2.20 \mathrm{~m} / \mathrm{s}\). The coins collide head-on elastically; calculate the final velocities (speed and direction) of both.

On an air track, a 400.0 g glider moving to the right at \(2.00 \mathrm{~m} / \mathrm{s}\) collides elastically with a 500.0 g glider moving in the opposite direction at \(3.00 \mathrm{~m} / \mathrm{s}\). Find the velocity of each glider after the collision.

A \(2 \mathrm{~kg}\) block is moving at a speed of \(10 \mathrm{~m} / \mathrm{s}\) and makes a perfectly elastic collision with a second block of mass \(M\), which is initially at rest. After the collision, the \(2 \mathrm{~kg}\) block bounces straight back at \(3 \mathrm{~m} / \mathrm{s}\). (a) Determine the mass \(M\) of the second block. (b) Determine the speed of the second block after the collision.

A catcher catches a 145 g baseball traveling horizontally at \(36.0 \mathrm{~m} / \mathrm{s}\). (a) How large an impulse does the ball give to the catcher? (b) If the ball takes \(20 \mathrm{~ms}\) to stop once it is in contact with the catcher's glove, what average force did the ball exert on the catcher?

A block of ice with a mass of \(2.50 \mathrm{~kg}\) is moving on a frictionless, horizontal surface. At \(t=0,\) the block is moving to the right with a velocity of magnitude \(8.00 \mathrm{~m} / \mathrm{s}\). Calculate the magnitude and direction of the velocity of the block after each of the following forces has been applied for \(5.00 \mathrm{~s}\) : (a) a force of \(5.00 \mathrm{~N}\) directed to the right; (b) a force of \(7.00 \mathrm{~N}\) directed to the left.

Biomechanics. The mass of a regulation tennis ball is \(57.0 \mathrm{~g}\) (although it can vary slightly), and tests have shown that the ball is in contact with the tennis racket for \(30 \mathrm{~ms}\). (This number can also vary, depending on the racket and swing.) We assume a \(30.0 \mathrm{~ms}\) contact time in this problem. In the 2011 Davis Cup competition, Ivo Karlovic made one of the fastest recorded serves in history, which was clocked at \(156 \mathrm{mph}(70 \mathrm{~m} / \mathrm{s}) .\) (a) What impulse and what average force did Karlovic exert on the tennis ball in his record serve? (b) If his opponent returned this serve with a speed of \(55.0 \mathrm{~m} / \mathrm{s},\) what impulse and what average force did his opponent exert on the ball, assuming purely horizontal motion?

To warm up for a match, a tennis player hits the \(57.0 \mathrm{~g}\) ball vertically with her racket. If the ball is stationary just before it is hit and goes \(5.50 \mathrm{~m}\) high, what impulse did she impart to it?

Your little sister (mass \(25.0 \mathrm{~kg}\) ) is sitting in her little red wagon (mass \(8.50 \mathrm{~kg}\) ) at rest. You begin pulling her forward and continue accelerating her with a constant force for \(2.35 \mathrm{~s}\), at the end of which time she's moving at a speed of \(1.80 \mathrm{~m} / \mathrm{s}\). (a) Calculate the impulse you imparted to the wagon and its passenger. (b) With what force did you pull on the wagon?

A 270 caliber hunting rifle fires an 8.5 g bullet, which exits the gun barrel at a speed of \(900 \mathrm{~m} / \mathrm{s}\). (a) What impulse does the burning gunpowder impart to the bullet? (b) If it takes \(2 \mathrm{~ms}\) for the bullet to travel the length of the barrel, what is the average force on the bullet? Express your answer in pounds.

Three odd-shaped blocks of chocolate have the following masses and center-of- mass coordinates: \(\begin{array}{lll}\text { (1) } 0.300 \text { kg. }\end{array}\) \((0.200 \mathrm{~m}, 0.300 \mathrm{~m});$$\begin{array}{lll}\text { (2) } 0.400 \text { kg. }\end{array}$$(0.100 \mathrm{~m},-0.400 \mathrm{~m});\)(3) \(0.200 \mathrm{~kg},(-0.300 \mathrm{~m}, 0.600 \mathrm{~m})\). Find the coordinates of the center of mass of the system of three chocolate blocks.

To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is \(92.0 \mathrm{~cm}\) long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a \(70.0 \mathrm{~kg}\) person, the mass of the upper leg is \(8.60 \mathrm{~kg},\) while that of the lower leg (including the foot) is \(5.25 \mathrm{~kg}\). Find the location of the center of mass of this leg, relative to the hip joint, if it is (a) fully extended and (b) bent at the knee to form a right angle with the upper leg.

A \(1200 \mathrm{~kg}\) station wagon is moving along a straight highway at \(12.0 \mathrm{~m} / \mathrm{s}\). Another car, with mass \(1800 \mathrm{~kg}\) and speed \(20.0 \mathrm{~m} / \mathrm{s}\), has its center of mass \(40.0 \mathrm{~m}\) ahead of the center of mass of the station wagon. (See Figure \(8.46 .\) ) (a) Find the position of the center of mass of the system consisting of the two automobiles. (b) Find the magnitude of the total momentum of the system from the given data. (c) Find the speed of the center of mass of the system. (d) Find the total momentum of the system, using the speed of the center of mass. Compare your result with that of part (b).

A small rocket burns \(0.0500 \mathrm{~kg}\) of fuel per second, ejecting it as a gas with a velocity of magnitude \(1600 \mathrm{~m} / \mathrm{s}\) relative to the rocket. (a) What is the thrust of the rocket? (b) Would the rocket operate in outer space, where there is no atmosphere? If so, how would you steer it? Could you brake it?

A rocket is fired in deep space, where gravity is negligible. If the rocket has an initial mass of \(6000 \mathrm{~kg}\) and ejects gas at a relative velocity of magnitude \(2000 \mathrm{~m} / \mathrm{s}\), how much gas must it eject in the first second to have an initial acceleration of \(25.0 \mathrm{~m} / \mathrm{s}^{2} ?\)

A rocket is fired in deep space, where gravity is negligible. In the first second, it ejects \(1 / 160\) of its mass as exhaust gas and has an acceleration of \(15.0 \mathrm{~m} / \mathrm{s}^{2}\). What is the speed of the exhaust gas relative to the rocket?

A \(70 \mathrm{~kg}\) astronaut floating in space in a \(110 \mathrm{~kg}\) MMU (manned maneuvering unit) experiences an acceleration of \(0.029 \mathrm{~m} / \mathrm{s}^{2}\) when he fires one of the MMU's thrusters. (a) If the speed of the escaping \(\mathrm{N}_{2}\) gas relative to the astronaut is \(490 \mathrm{~m} / \mathrm{s},\) how much gas is used by the thruster in \(5.0 \mathrm{~s} ?\) (b) What is the thrust of the thruster?

In \(1.00 \mathrm{~s}\), an automatic paint ball gun can fire 15 balls, each with a mass of \(0.113 \mathrm{~g}\), at a muzzle velocity of \(88.5 \mathrm{~m} / \mathrm{s}\). Calculate the average recoil force experienced by the player who's holding the gun.

In a volcanic eruption, a \(2400-\mathrm{kg}\) boulder is thrown vertically upward into the air. At its highest point, it suddenly explodes (due to trapped gases) into two fragments, one being three times the mass of the other. The lighter fragment starts out with only horizontal velocity and lands \(274 \mathrm{~m}\) directly north of the point of the explosion. Where will the other fragment land? Ignore any air resistance.

A \(0.4 \mathrm{~kg}\) stone is thrown horizontally at a speed of \(20 \mathrm{~m} / \mathrm{s}\) from a \(40-\mathrm{m}\) -tall building. (a) Determine the \(x\) and \(y\) components of the stone's momentum the moment after it is thrown. (b) What are the components of its momentum just before it hits the ground? What impulse did gravity impart to the stone?

A stone with a mass of \(0.100 \mathrm{~kg}\) rests on a frictionless, horizontal surface. A bullet of mass \(2.50 \mathrm{~g}\) traveling horizontally at \(500 \mathrm{~m} / \mathrm{s}\) strikes the stone and rebounds horizontally at right angles to its original direction with a speed of \(300 \mathrm{~m} / \mathrm{s}\). (a) Compute the magnitude and direction of the velocity of the stone after it is struck. (b) Is the collision perfectly elastic?

A steel ball with a mass of \(40.0 \mathrm{~g}\) is dropped from a height of \(2.00 \mathrm{~m}\) onto a horizontal steel slab. The ball rebounds to a height of \(1.60 \mathrm{~m}\). (a) Calculate the impulse delivered to the ball during the impact. (b) If the ball is in contact with the slab for \(2.00 \mathrm{~ms},\) find the magnitude and direction of the average force on the ball during the impact. (c) How much mechanical energy was lost in the impact with the steel slab? (d) How high would the ball have rebounded if the impact had been perfectly elastic?

A movie stuntman (mass \(80.0 \mathrm{~kg}\) ) stands on a window ledge \(5.0 \mathrm{~m}\) above the floor (Figure 8.48 ). Grabbing a rope attached to a chandelier, he swings down to grapple with the movie's villain (mass \(70.0 \mathrm{~kg}\) ), who is standing directly under the chandelier. (Assume that the stuntman's center of mass moves downward \(4.0 \mathrm{~m}\). He releases the rope just as he reaches the villain.) (a) With what speed do the entwined foes start to slide across the floor? (b) If the coefficient of kinetic friction of their bodies with the floor is \(\mu_{\mathrm{k}}=0.250\), how far do they slide?

Tennis players sometimes leap into the air to return a volley. (a) If a \(57 \mathrm{~g}\) tennis ball is traveling horizontally at \(72 \mathrm{~m} / \mathrm{s}\) (which does occur), and a \(61 \mathrm{~kg}\) tennis player leaps vertically upward and hits the ball, causing it to travel at \(45 \mathrm{~m} / \mathrm{s}\) in the reverse direction, how fast will her center of mass be moving horizontally just after hitting the ball? (b) If, as is reasonable, her racket is in contact with the ball for \(30.0 \mathrm{~ms}\), what force does her racket exert on the ball? What force does the ball exert on the racket?

A mass \(m\) is placed at the rim of a frictionless hemispherical bowl with a radius \(R\) and released from rest (Figure 8.49 ). It then slides down and undergoes a perfectly elastic collision with a second mass \(3 m\) sitting at rest at the bottom of the bowl. (a) What are the direction and speed of each mass just after the collision? (b) In terms of \(R\), to what maximum height will each mass travel after the collision?

Two identical \(1.50 \mathrm{~kg}\) masses are pressed against opposite ends of a light spring of force constant \(1.75 \mathrm{~N} / \mathrm{cm}\), compressing the spring by \(20.0 \mathrm{~cm}\) from its normal length. Find the speed of each mass when it has moved free of the spring on a frictionless, horizontal lab table.

A rifle bullet with mass \(8.00 \mathrm{~g}\) strikes and embeds itself in a block with a mass of \(0.992 \mathrm{~kg}\) that rests on a frictionless, horizontal surface and is attached to a coil spring. (See Figure \(8.50 .\) ) The impact compresses the spring \(15.0 \mathrm{~cm}\). Calibration of the spring shows that a force of \(0.750 \mathrm{~N}\) is required to compress the spring \(0.250 \mathrm{~cm} .\) (a) Find the magnitude of the block's velocity just after impact. (b) What was the initial speed of the bullet?

A \(5.00 \mathrm{~g}\) bullet traveling horizontally at \(450 \mathrm{~m} / \mathrm{s}\) is shot through a \(1.00 \mathrm{~kg}\) wood block suspended on a string \(2.00 \mathrm{~m}\) long. If the center of mass of the block rises a distance of \(0.450 \mathrm{~cm},\) find the speed of the bullet as it emerges from the block.

Forensic scientists can measure the muzzle velocity of a gun by firing a bullet horizontally into a large hanging block that absorbs the bullet and swings upward. (See Figure \(8.52 .\) ) The measured maximum angle of swing can be used to calculate the speed of the bullet. In one such test, a rifle fired a \(4.20 \mathrm{~g}\) bullet into a \(2.50 \mathrm{~kg}\) block hanging by a thin wire \(75.0 \mathrm{~cm}\) long, causing the block to swing upward to a maximum angle of \(34.7^{\circ}\) from the vertical. What was the original speed of this bullet?