Chapter 8

\(\cdot \mathrm{A} 1200 \mathrm{kg}\) cur is moving on the freeway at 65 \(\mathrm{mph}\) (a) Find the magnitude of its momentum and its kinetic energy in SI units. (b) If a 2400 \(\mathrm{kg}\) SUV has the same speed as the 1200 \(\mathrm{kg}\) car, how much momentum and kinetic energy does it have?

\(\cdot\) 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 mo- mentum 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?

\(\bullet\) Some useful relationships. The following relationships between the momentum and kinetic energy of an object can be very useful for calculations: If an object of mass \(m\) has momentum of magnitude \(p\) and kinetic energy \(K,\) show that (a) \(K=\left(p^{2} / 2 m\right),\) and \((b) p=\sqrt{2 m K}\) . (c) Find the momen- tum of a 1.15 kg ball that has 15.0 J of kinetic energy. (d) Find the kinetic energy of a 3.50 kg cat that has 0.220 \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) of momentum.

\(\bullet\) 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?

\(\cdot\) 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?

Recoil speed of the earth. In principle, any time someone jumps up, the earth moves in the opposite direction. To see why we are unaware of this motion, calculate the recoil speed of the earth when a 75 kg person jumps upward at a speed of 2.0 \(\mathrm{m} / \mathrm{s} .\) Consult Appendix E as needed.

\(\bullet\) 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 glid- ers after the collision. (d) Is kinetic energy conserved during the collision?

Baseball. A regulation 145 g baseball can be hit at speeds of 100 mph. If a line drive is hit essentially horizontally at this speed and is caught by a 65 \(\mathrm{kg}\) player who has leapt directly upward into the air, what horizontal speed (in \(\mathrm{cm} / \mathrm{s} )\) does he acquire by catching the ball?

\(\cdot\) 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 afterwards? (b) If the ball hits you and bounces off your chest, so that afterwards it is moving horizontally at 8.00 \(\mathrm{m} / \mathrm{s}\) in the opposite direction, what is your speed after the collision?

\(\bullet\) 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 veloc- ity 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.

\(\bullet\) Two ice skaters, Daniel (mass 65.0 \(\mathrm{kg}\) ) and Rebecca (mass \(45.0 \mathrm{kg} ),\) are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 \(\mathrm{m} / \mathrm{s}\) before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 \(\mathrm{m} / \mathrm{s}\) at an angle of \(53.1^{\circ}\) from her initial direction. Both skaters move on the frictionless, hori- zontal surface of the rink. (a) What are the magnitude and direction of Daniel's velocity after the collision? (b) What is the change in total kinetic energy of the two skaters as a result of 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.

\(\bullet\) 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 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?

\(\bullet\) 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 \(8.37 . )\) since the valley is icy, there is no friction. After the collision, how high above the valley floor will the combined chunks go? (Hint: Break this problem into two parts - the collision and the behavior after the collision and apply the appropriate conservation law to each part.)

Combining conservation laws. A 15.0 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 \(8.38 .\) ) 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.)

\(\cdot\) 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 neglected. (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?

\(\cdot\) On a highly polished, essentially frictionless lunch counter, a 0.500 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?

\(\bullet\) An astronaut in space cannot use a scale or balance to weigh objects because there is no gravity. But she does have devices to measure distance and time accurately. She knows her own mass is \(78.4 \mathrm{kg},\) but she is unsure of the mass of a large gas canister in the airless rocket. When this canister is approaching her at \(3.50 \mathrm{m} / \mathrm{s},\) she pushes against it, which slows it down to 1.20 \(\mathrm{m} / \mathrm{s}\) (but does not reverse it) and gives her a speed of 2.40 \(\mathrm{m} / \mathrm{s}\) . (a) What is the mass of this canister? (b) How much kinetic energy is "lost" in this collision, and what happens to that energy?

\(\bullet\) On a very muddy football field, a \(110-\) kg linebacker tack- les an 85 -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 \(\mathrm{A} 5.00\) 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?

. You and your friends are doing physics experiments on a frozen pond that serves as a frictionless, horizontal surface. Sam, with mass 80.0 \(\mathrm{kg}\) , is given a push and slides eastward. Abigail, with mass \(50.0 \mathrm{kg},\) is sent sliding northward. They collide, and after the collision Sam is moving at \(37.0^{\circ}\) north of east with a speed of 6.00 \(\mathrm{m} / \mathrm{s}\) and Abigail is moving at \(23.0^{\circ}\) south of east with a speed of 9.00 \(\mathrm{m} / \mathrm{s}\) . (a) What was the speed of each person before the collision? (b) By how much did the total kinetic energy of the two people decrease during the collision?

\(\bullet\) A hungry 11.5 \(\mathrm{kg}\) predator fish is coasting from west to east to east at 75.0 \(\mathrm{cm} / \mathrm{s}\) when it suddenly swallows a 1.25 \(\mathrm{kg}\) fish swim- ming 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. Neglect any effects due to the drag of the water.

\(\bullet\) Bird defense. 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 gram falcon flying at 20.0 \(\mathrm{m} / \mathrm{s}\) ran 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 \(\mathrm{m} / \mathrm{s}\) . (These figures were estimated by one of the authors \((\mathrm{WRA})\) as he watched this attack occur in northern New Mexico.) By what angle did the falcon change the raven's direction of motion?

Accident analysis. Two cars collide at an intersection. Car \(A\), with a mass of \(2000 \mathrm{~kg}\), is going from west to east, while car \(B\), of mass \(1500 \mathrm{~kg}\), is going from north to south at \(15 \mathrm{~m} / \mathrm{s}\). As a result of this collision, the two cars become enmeshed and move as one afterward. In your role as an expert witness, you inspect the scene and determine that, after the collision, the enmeshed cars moved at an angle of \(65^{\circ}\) south of east from the point of impact. (a) How fast were the enmeshed cars moving just after the collision? (b) How fast was car \(A\) going just before the collision?

A 0.300 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.

\(\bullet\) On a cold winter day, a penny (mass 2.50 g) and a nickel (mass 5.00 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 elastically; calculate both their final velocities (speed and direction).

\(\bullet\) 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.

. Two identical objects traveling in opposite directions with the same speed \(V\) make a head-on collision. Find the speed of each object after the collision if (a) they stick together and (b) if the collision is perfectly elastic.

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 fric- tionless, 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 s: (a) a force of 5.00 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 g (although it can vary slightly), and tests have shown that the ball is in contact with the tennis racket for 30 ms. (This number can also vary, depending on the racket and swing.) We shall assume a 30.0 \(\mathrm{ms}\) contact time throughout this problem. The fastest-known served tennis ball was served by "Big Bill" Tilden in \(1931,\) and its speed was measured to be 73.14 \(\mathrm{m} / \mathrm{s}\) .(a) What impulse and what force did Big Bill exert on the ten- nis ball in his record serve? (b) If Big Bill's opponent returned his serve with a speed of \(55 \mathrm{m} / \mathrm{s},\) what force and what impulse did he exert on the ball, assuming only horizontal motion?

\(\bullet\) To warm up for a match, a tennis player hits the 57.0 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?

. Bone fracture. Experimental tests have shown that bone will rupture if it is subjected to a force density of \(1.0 \times\) \(10^{8} \mathrm{N} / \mathrm{m}^{2} .\) Suppose a 70.0 \(\mathrm{kg}\) person carelessly roller-skates into an overhead metal beam that hits his forehead and completely stops his forward motion. If the area of contact with the person's forehead is \(1.5 \mathrm{cm}^{2},\) what is the greatest speed with which he can hit the wall without breaking any bone if hiss head is in contact with the beam for 10.0 \(\mathrm{ms}\) ?

A bat strikes a 0.145 kg baseball. Just before impact, the ball is traveling horizontally to the right at \(50.0 \mathrm{m} / \mathrm{s},\) and it leaves the bat traveling to the left at an angle of \(30^{\circ}\) above horizontal with a speed of 65.0 \(\mathrm{m} / \mathrm{s}\) . (a) What are the horizontal and vertical components of the impulse the bat imparts to the ball? (b) If the ball and bat are in contact for 1.75 \(\mathrm{ms}\) , find the horizontal and vertical components of the average force on the ball.

\(\bullet\) Detecting planets around other stars. Roughly 500 planets have so far been detected beyond our solar system. This is accomplished by looking for the effect the planet has on the star. The star is not truly stationary; instead, it and its planets orbit around the center of mass of the system. Astronomers can measure this wobble in the position of a star.(a) For a star with the mass and size of our sun and having a planet with five times the mass of Jupiter, where would the center of mass of this system be located, relative to the center of the star, if the distance from the star to the planet was the same as the distance from Jupiter to our sun? (Consult Appendix E.) (b) If the planet had earth's mass, where would the center of mass of the system be located if the planet was just as far from the star as the earth is from the sun? (c) In view of your results in parts (a) and (b), why is it much easier to detect stars having large planets rather than small ones?

\(\cdot\) Three odd-shaped blocks of chocolate have the following masses and center-of-mass coordinates: \((1) \quad 0.300\) kg, \((0.200 \mathrm{m}, 0.300 \mathrm{m}) ;\) (2) \(0.400 \mathrm{kg}, \quad(0.100 \mathrm{m},-0.400 \mathrm{m})\) (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.

A machine part consists of a thin, uniform \(4.00-\mathrm{kg}\) bar that is 1.50 \(\mathrm{m}\) long, hinged perpendicular to a similar vertical bar of mass 3.00 \(\mathrm{kg}\) and length 1.80 \(\mathrm{m} .\) The longer bar has a small but dense 2.00 -kg ball at one end (Fig. 8.42\()\) By what distance will the center of mass of this part move horizontally and vertically if the vertical bar is pivoted counterclockwise through \(90^{\circ}\) to make the entire part horizontal?

. Changing your center of mass. 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 he upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a 70.0 kg per- son, the mass of the upper leg would be 8.60 \(\mathrm{kg}\) , while that of the lower leg (including the foot) would be 5.25 \(\mathrm{kg}\) . Find the location of the center of mass of this leg, relative to the hip joint, if it is (a) stretched out horizontally and (b) bent at the knee to form a right angle with the upper leg remaining horizontal.

\(\cdot\) A 1200 kg station wagon is moving along a straight high-way 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 8.43.) (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. 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 \(\mathrm{m} / \mathrm{s}^{2}\) .

\(\bullet\) In outer space, where gravity is negligible, a \(75,000 \mathrm{kg}\) rocket (including \(50,000 \mathrm{kg}\) of fuel) expels this fuel at a steady rate of 135 \(\mathrm{kg} / \mathrm{s}\) with a speed of 1200 \(\mathrm{m} / \mathrm{s}\) relative to the rocket. (a) Find the thrust of the rocket. (b) What are the initial acceleration and the maximum acceleration of the rocket? (c) After the fuel runs out, what happens to this rocket's acceleration? Does it (i) remain the same as it was just as the fuel ran out, (ii) suddenly become zero, or (iii) gradually drop to zero? Explain your reasoning. (d) After the fuel runs out, what happens to the rocket's speed? Does it (i) remain the same as it was just as the fuel ran out, (ii) suddenly become zero, or (iii) gradually drop to zero? Explain your reasoning.

\(\bullet\) 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?

\(\bullet\) In 1.00 second an automatic paintball 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.

\(\bullet\) In a volcanic eruption, a 2400 -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 m directly north of the point of the explosion. Where will the other fragment land? Neglect any air resistance.

\(\bullet\) A 5.00 kg ornament is hanging by a 1.50 \(\mathrm{m}\) wire when it is suddenly hit by a 3.00 \(\mathrm{kg}\) missile traveling horizontally at 12.0 \(\mathrm{m} / \mathrm{s} .\) The missile embeds itself in the ornament during the collision. What is the tension in the wire immediately after the collision?

\(\bullet\) 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?

\(\bullet\) 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 average force on the ball during the impact.