Chapter 7

In many old Western movies, a bandit is knocked back \(3 \mathrm{~m}\) after being shot by a sheriff. Which statement best describes what happened to the sheriff after he fired his gun? a) He remained in the same position. b) He was knocked back a step or two. c) He was knocked back approximately \(3 \mathrm{~m}\). d) He was knocked forward slightly. e) He was pushed upward.

The value of the momentum for a system is the same at a later time as at an earlier time if there are no a) collisions between particles within the system. b) inelastic collisions between particles within the system. c) changes of momentum of individual particles within the system. d) internal forces acting between particles within the system. e) external forces acting on particles of the system.

Consider these three situations: (i) A ball moving to the right at speed \(v\) is brought to rest. (ii) The same ball at rest is projected at speed \(v\) toward the left. (iii) The same ball moving to the left at speed \(v\) speeds up to \(2 v\). In which situation(s) does the ball undergo the largest change in momentum? a) situation (i) d) situations (i) and (ii) b) situation (ii) e) all three situations c) situation (iii)

Consider two carts, of masses \(m\) and \(2 m,\) at rest on a frictionless air track. If you push the lower-mass cart for \(35 \mathrm{~cm}\) and then the other cart for the same distance and with the same force, which cart undergoes the larger change in momentum? a) The cart with mass \(m\) has the larger change. b) The cart with mass \(2 m\) has the larger change. c) The change in momentum is the same for both carts. d) It is impossible to tell from the information given.

Consider two carts, of masses \(m\) and \(2 m\), at rest on a frictionless air track. If you push the lower-mass cart for \(3 \mathrm{~s}\) and then the other cart for the same length of time and with the same force, which cart undergoes the larger change in momentum? a) The cart with mass \(m\) has the larger change b) The cart with mass \(2 m\) has the larger change. c) The change in momentum is the same for both carts. d) It is impossible to tell from the information given.

Which of the following statements about car collisions are true and which are false? a) The essential safety benefit of crumple zones (parts of the front of a car designed to receive maximum deformation during a head-on collision) results from absorbing kinetic energy, converting it into deformation, and lengthening the effective collision time, thus reducing the average force experienced by the driver. b) If car 1 has mass \(m\) and speed \(v\), and car 2 has mass \(0.5 m\) and speed \(1.5 v\), then both cars have the same momentum. c) If two identical cars with identical speeds collide head on, the magnitude of the impulse received by each car and each driver is the same as if one car at the same speed had collided head on with a concrete wall. d) Car 1 has mass \(m,\) and car 2 has mass \(2 m .\) In a head-on collision of these cars while moving at identical speeds in opposite directions, car 1 experiences a bigger acceleration than car 2 . e) Car 1 has mass \(m\), and car 2 has mass \(2 m\). In a headon collision of these cars while moving at identical speeds in opposite directions, car 1 receives an impulse of bigger magnitude than that received by car 2.

A fireworks projectile is launched upward at an angle above a large flat plane. When the projectile reaches the top of its flight, at a height of \(h\) above a point that is a horizontal distance \(D\) from where it was launched, the projectile explodes into two equal pieces. One piece reverses its velocity and travels directly back to the launch point. How far from the launch point does the other piece land? a) \(D\) b) \(2 D\) c) \(3 D\) d) \(4 D\)

An astronaut becomes stranded during a space walk after her jet pack malfunctions. Fortunately, there are two objects close to her that she can push to propel herself back to the International Space Station (ISS). Object A has the same mass as the astronaut, and Object \(\mathrm{B}\) is 10 times more massive. To achieve a given momentum toward the ISS by pushing one of the objects away from the ISS, which object should she push? That is, which one requires less work to produce the same impulse? Initially, the astronaut and the two objects are at rest with respect to the ISS.

A bungee jumper is concerned that his elastic cord might break if it is overstretched and is considering replacing the cord with a high-tensile- strength steel cable. Is this a good idea?

To solve problems involving projectiles traveling through the air by applying the law of conservation of momentum requires evaluating the momentum of the system immediately before and immediately after the collision or explosion. Why?

Using momentum and force principles, explain why an air bag reduces injury in an automobile collision.

A rocket works by expelling gas (fuel) from its nozzles at a high velocity. However, if we take the system to be the rocket and fuel, explain qualitatively why a stationary rocket is able to move.

When hit in the face, a boxer will "ride the punch"; that is, if he anticipates the punch, he will allow his neck muscles to go slack. His head then moves back easily from the blow. From a momentum-impulse standpoint, explain why this is much better than stiffening his neck muscles and bracing himself against the punch.

An open train car moves with speed \(v_{0}\) on a flat frictionless railroad track, with no engine pulling it. It begins to rain. The rain falls straight down and begins to fill the train car. Does the speed of the car decrease, increase, or stay the same? Explain.

Rank the following objects from highest to lowest in terms of momentum and from highest to lowest in terms of energy. a) an asteroid with mass \(10^{6} \mathrm{~kg}\) and speed \(500 \mathrm{~m} / \mathrm{s}\) b) a high-speed train with a mass of \(180,000 \mathrm{~kg}\) and a speed of \(300 \mathrm{~km} / \mathrm{h}\) c) a 120 -kg linebacker with a speed of \(10 \mathrm{~m} / \mathrm{s}\) d) a 10 -kg cannonball with a speed of \(120 \mathrm{~m} / \mathrm{s}\) e) a proton with a mass of \(6 \cdot 10^{-27} \mathrm{~kg}\) and a speed of \(2 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\).

A car of mass \(1200 \mathrm{~kg}\), moving with a speed of \(72 \mathrm{mph}\) on a highway, passes a small SUV with a mass \(1 \frac{1}{2}\) times bigger, moving at \(2 / 3\) of the speed of the car. a) What is the ratio of the momentum of the SUV to that of the car? b) What is the ratio of the kinetic energy of the SUV to that of the car?

The electron-volt, \(\mathrm{eV},\) is a unit of energy \((1 \mathrm{eV}=\) \(\left.1.602 \cdot 10^{-19} \mathrm{~J}, 1 \mathrm{MeV}=1.602 \cdot 10^{-13} \mathrm{~J}\right) .\) Since the unit of \(\mathrm{mo}\) mentum is an energy unit divided by a velocity unit, nuclear physicists usually specify momenta of nuclei in units of \(\mathrm{MeV} / c,\) where \(c\) is the speed of light \(\left(c=2.998 \cdot 10^{9} \mathrm{~m} / \mathrm{s}\right) .\) In the same units, the mass of a proton \(\left(1.673 \cdot 10^{-27} \mathrm{~kg}\right)\) is given as \(938.3 \mathrm{MeV} / \mathrm{c}^{2} .\) If a proton moves with a speed of \(17,400 \mathrm{~km} / \mathrm{s}\) what is its momentum in units of \(\mathrm{MeV} / \mathrm{c}\) ?

A soccer ball with a mass of \(442 \mathrm{~g}\) bounces off the crossbar of a goal and is deflected upward at an angle of \(58.0^{\circ}\) with respect to horizontal. Immediately after the deflection, the kinetic energy of the ball is \(49.5 \mathrm{~J} .\) What are the vertical and horizontal components of the ball's momentum immediately after striking the crossbar?

A billiard ball of mass \(m=0.250 \mathrm{~kg}\) hits the cushion of a billiard table at an angle of \(\theta_{1}=60.0^{\circ}\) at a speed of \(v_{1}=27.0 \mathrm{~m} / \mathrm{s}\) It bounces off at an angle of \(\theta_{2}=71.0^{\circ}\) and a speed of \(v_{2}=10.0 \mathrm{~m} / \mathrm{s}\). a) What is the magnitude of the change in momentum of the billiard ball? b) In which direction does the change of momentum vector point?

In the movie Superman, Lois Lane falls from a building and is caught by the diving superhero. Assuming that Lois, with a mass of \(50.0 \mathrm{~kg}\), is falling at a terminal velocity of \(60.0 \mathrm{~m} / \mathrm{s}\), how much force does Superman exert on her if it takes \(0.100 \mathrm{~s}\) to slow her to a stop? If Lois can withstand a maximum acceleration of \(7 g^{\prime}\) s, what minimum time should it take Superman to stop her after he begins to slow her down?

One of the events in the Scottish Highland Games is the sheaf toss, in which a \(9.09-\mathrm{kg}\) bag of hay is tossed straight up into the air using a pitchfork. During one throw, the sheaf is launched straight up with an initial speed of \(2.7 \mathrm{~m} / \mathrm{s}\). a) What is the impulse exerted on the sheaf by gravity during the upward motion of the sheaf (from launch to maximum height)? b) Neglecting air resistance, what is the impulse exerted by gravity on the sheaf during its downward motion (from maximum height until it hits the ground)? c) Using the total impulse produced by gravity, determine how long the sheaf is airborne.

An 83.0 -kg running back leaps straight ahead toward the end zone with a speed of \(6.50 \mathrm{~m} / \mathrm{s}\). A 115 -kg linebacker, keeping his feet on the ground, catches the running back and applies a force of \(900 . \mathrm{N}\) in the opposite direction for 0.750 s before the running back's feet touch the ground. a) What is the impulse that the linebacker imparts to the running back? b) What change in the running back's momentum does the impulse produce? c) What is the running back's momentum when his feet touch the ground? d) If the linebacker keeps applying the same force after the running back's feet have touched the ground, is this still the only force acting to change the running back's momentum?

A baseball pitcher delivers a fastball that crosses the plate at an angle of \(7.25^{\circ}\) relative to the horizontal and a speed of \(88.5 \mathrm{mph}\). The ball (of mass \(0.149 \mathrm{~kg}\) ) is hit back over the head of the pitcher at an angle of \(35.53^{\circ}\) with respect to the horizontal and a speed of \(102.7 \mathrm{mph}\). What is the magnitude of the impulse received by the ball?

Although they don't have mass, photons-traveling at the speed of light-have momentum. Space travel experts have thought of capitalizing on this fact by constructing solar sails-large sheets of material that would work by reflecting photons. Since the momentum of the photon would be reversed, an impulse would be exerted on it by the solar sail, and-by Newton's Third Law-an impulse would also be exerted on the sail, providing a force. In space near the Earth, about \(3.84 \cdot 10^{21}\) photons are incident per square meter per second. On average, the momentum of each photon is \(1.30 \cdot 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\). For a \(1000 .-\mathrm{kg}\) spaceship starting from rest and attached to a square sail \(20.0 \mathrm{~m}\) wide, how fast could the ship be moving after 1 hour? One week? One month? How long would it take the ship to attain a speed of \(8000 . \mathrm{m} / \mathrm{s}\), roughly the speed of the space shuttle in orbit?

In a severe storm, \(1.00 \mathrm{~cm}\) of rain falls on a flat horizontal roof in \(30.0 \mathrm{~min}\). If the area of the roof is \(100 \mathrm{~m}^{2}\) and the terminal velocity of the rain is \(5.00 \mathrm{~m} / \mathrm{s}\), what is the average force exerted on the roof by the rain during the storm?

NASA has taken an increased interest in near Earth asteroids. These objects, popularized in recent blockbuster movies, can pass very close to Earth on a cosmic scale, sometimes as close as 1 million miles. Most are small-less than \(500 \mathrm{~m}\) across \(-\) and while an impact with one of the smaller ones could be dangerous, experts believe that it may not be catastrophic to the human race. One possible defense system against near Earth asteroids involves hitting an incoming asteroid with a rocket to divert its course. Assume a relatively small asteroid with a mass of \(2.10 \cdot 10^{10} \mathrm{~kg}\) is traveling toward the Earth at a modest speed of \(12.0 \mathrm{~km} / \mathrm{s}\). a) How fast would a large rocket with a mass of \(8.00 \cdot 10^{4} \mathrm{~kg}\) have to be moving when it hit the asteroid head on in order to stop the asteroid? b) An alternative approach would be to divert the asteroid from its path by a small amount to cause it to miss Earth. How fast would the rocket of part (a) have to be traveling at impact to divert the asteroid's path by \(1.00^{\circ}\) ? In this case, assume that the rocket hits the asteroid while traveling along a line perpendicular to the asteroid's path.

A sled initially at rest has a mass of \(52.0 \mathrm{~kg}\), including all of its contents. A block with a mass of \(13.5 \mathrm{~kg}\) is ejected to the left at a speed of \(13.6 \mathrm{~m} / \mathrm{s} .\) What is the speed of the sled and the remaining contents?

Stuck in the middle of a frozen pond with only your physics book, you decide to put physics in action and throw the 5.00 -kg book. If your mass is \(62.0 \mathrm{~kg}\) and you throw the book at \(13.0 \mathrm{~m} / \mathrm{s}\), how fast do you then slide across the ice? (Assume the absence of friction.)

Astronauts are playing baseball on the International Space Station. One astronaut with a mass of \(50.0 \mathrm{~kg}\), initially at rest, hits a baseball with a bat. The baseball was initially moving toward the astronaut at \(35.0 \mathrm{~m} / \mathrm{s},\) and after being hit, travels back in the same direction with a speed of \(45.0 \mathrm{~m} / \mathrm{s}\). The mass of a baseball is \(0.14 \mathrm{~kg}\). What is the recoil velocity of the astronaut?

An automobile with a mass of \(1450 \mathrm{~kg}\) is parked on a moving flatbed railcar; the flatbed is \(1.5 \mathrm{~m}\) above the ground. The railcar has a mass of \(38,500 \mathrm{~kg}\) and is moving to the right at a constant speed of \(8.7 \mathrm{~m} / \mathrm{s}\) on a frictionless rail. The automobile then accelerates to the left, leaving the railcar at a speed of \(22 \mathrm{~m} / \mathrm{s}\) with respect to the ground. When the automobile lands, what is the distance \(D\) between it and the left end of the railcar? See the figure.

Two bumper cars moving on a frictionless surface collide elastically. The first bumper car is moving to the right with a speed of \(20.4 \mathrm{~m} / \mathrm{s}\) and rear-ends the second bumper car, which is also moving to the right but with a speed of \(9.00 \mathrm{~m} / \mathrm{s} .\) What is the speed of the first bumper car after the collision? The mass of the first bumper car is \(188 \mathrm{~kg}\), and the mass of the second bumper car is \(143 \mathrm{~kg}\). Assume that the collision takes place in one dimension.

A satellite with a mass of \(274 \mathrm{~kg}\) approaches a large planet at a speed \(v_{i, 1}=13.5 \mathrm{~km} / \mathrm{s}\). The planet is moving at a speed \(v_{i, 2}=10.5 \mathrm{~km} / \mathrm{s}\) in the opposite direction. The satellite partially orbits the planet and then moves away from the planet in a direction opposite to its original direction (see the figure). If this interaction is assumed to approximate an elastic collision in one dimension, what is the speed of the satellite after the collision? This so-called slingshot effect is often used to accelerate space probes for journeys to distance parts of the solar system (see Chapter 12).

An alpha particle (mass \(=4.00 \mathrm{u}\) ) has a head-on, elastic collision with a nucleus (mass \(=166 \mathrm{u}\) ) that is initially at rest. What percentage of the kinetic energy of the alpha particle is transferred to the nucleus in the collision?

Cosmic rays from space that strike Earth contain some charged particles with energies billions of times higher than any that can be produced in the biggest accelerator. One model that was proposed to account for these particles is shown schematically in the figure. Two very strong sources of magnetic fields move toward each other and repeatedly reflect the charged particles trapped between them. (These magnetic field sources can be approximated as infinitely heavy walls from which charged particles get reflected elastically.) The high- energy particles that strike the Earth would have been reflected a large number of times to attain the observed energies. An analogous case with only a few reflections demonstrates this effect. Suppose a particle has an initial velocity of \(-2.21 \mathrm{~km} / \mathrm{s}\) (moving in the negative \(x\) -direction, to the left), the left wall moves with a velocity of \(1.01 \mathrm{~km} / \mathrm{s}\) to the right, and the right wall moves with a velocity of \(2.51 \mathrm{~km} / \mathrm{s}\) to the left. What is the velocity of the particle after six collisions with the left wall and five collisions with the right wall?

Here is a popular lecture demonstration that you can perform at home. Place a golf ball on top of a basketball, and drop the pair from rest so they fall to the ground. (For reasons that should become clear once you solve this problem, do not attempt to do this experiment inside, but outdoors instead!) With a little practice, you can achieve the situation pictured here: The golf ball stays on top of the basketball until the basketball hits the floor. The mass of the golf ball is \(0.0459 \mathrm{~kg}\), and the mass of the basketball is \(0.619 \mathrm{~kg}\). you can achieve the situation pictured here: The golf ball stays on top of the basketball until the basketball hits the floor. The mass of the golf ball is \(0.0459 \mathrm{~kg}\), and the mass of the basketball is \(0.619 \mathrm{~kg}\). a) If the balls are released from a height where the bottom of the basketball is at \(0.701 \mathrm{~m}\) above the ground, what is the absolute value of the basketball's momentum just before it hits the ground? b) What is the absolute value of the momentum of the golf ball at this instant? c) Treat the collision of the basketball with the floor and the collision of the golf ball with the basketball as totally elastic collisions in one dimension. What is the absolute magnitude of the momentum of the golf ball after these collisions? d) Now comes the interesting question: How high, measured from the ground, will the golf ball bounce up after its collision with the basketball?

A hockey puck with mass \(0.250 \mathrm{~kg}\) traveling along the blue line (a blue-colored straight line on the ice in a hockey rink) at \(1.50 \mathrm{~m} / \mathrm{s}\) strikes a stationary puck with the same mass. The first puck exits the collision in a direction that is \(30.0^{\circ}\) away from the blue line at a speed of \(0.750 \mathrm{~m} / \mathrm{s}\) (see the figure). What is the direction and magnitude of the velocity of the second puck after the collision? Is this an elastic collision?

A ball with mass \(m=0.210 \mathrm{~kg}\) and kinetic energy \(K_{1}=2.97 \mathrm{~J}\) collides elastically with a second ball of the same mass that is initially at rest. \(m_{1}\) After the collision, the first ball moves away at an angle of \(\theta_{1}=\) \(30.6^{\circ}\) with respect to the horizontal, as shown in the figure. What is the kinetic energy of the first ball after the collision?

When you open the door to an air-conditioned room, you mix hot gas with cool gas. Saying that a gas is hot or cold actually refers to its average energy; that is, the hot gas molecules have a higher kinetic energy than the cold gas molecules. The difference in kinetic energy in the mixed gases decreases over time as a result of elastic collisions between the gas molecules, which redistribute the energy. Consider a two-dimensional collision between two nitrogen molecules \(\left(\mathrm{N}_{2},\right.\) molecular weight \(=28.0 \mathrm{~g} / \mathrm{mol}\) ). One molecule moves at \(30.0^{\circ}\) with respect to the horizontal with a velocity of \(672 \mathrm{~m} / \mathrm{s} .\) This molecule collides with a second molecule moving in the negative horizontal direction at \(246 \mathrm{~m} / \mathrm{s}\). What are the molecules' final velocities if the one that is initially more energetic moves in the vertical direction after the collision?

Current measurements and cosmological theories suggest that only about \(4 \%\) of the total mass of the universe is composed of ordinary matter. About \(22 \%\) of the mass is composed of dark matter, which does not emit or reflect light and can only be observed through its gravitational interaction with its surroundings (see Chapter 12). Suppose a galaxy with mass \(M_{\mathrm{G}}\) is moving in a straight line in the \(x\) -direction. After it interacts with an invisible clump of dark matter with mass \(M_{\mathrm{DM}}\), the galaxy moves with \(50 \%\) of its initial speed in a straight line in a direction that is rotated by an angle \(\theta\) from its initial velocity. Assume that initial and final velocities are given for positions where the galaxy is very far from the clump of dark matter, that the gravitational attraction can be neglected at those positions, and that the dark matter is initially at rest. Determine \(M_{\mathrm{DM}}\) in terms of \(M_{\mathrm{G}}, v_{0},\) and \(\theta\).

7.57 A 1439 -kg railroad car traveling at a speed of \(12 \mathrm{~m} / \mathrm{s}\) strikes an identical car at rest. If the cars lock together as a result of the collision, what is their common speed (in \(\mathrm{m} / \mathrm{s})\) afterward?

Bats are extremely adept at catching insects in midair. If a 50.0-g bat flying in one direction at \(8.00 \mathrm{~m} / \mathrm{s}\) catches a \(5.00-\mathrm{g}\) insect flying in the opposite direction at \(6.00 \mathrm{~m} / \mathrm{s}\), what is the speed of the bat immediately after catching the insect?

A small car of mass 1000 . kg traveling at a speed of \(33.0 \mathrm{~m} / \mathrm{s}\) collides head on with a large car of mass \(3000 \mathrm{~kg}\) traveling in the opposite direction at a speed of \(30.0 \mathrm{~m} / \mathrm{s}\). The two cars stick together. The duration of the collision is \(100 . \mathrm{ms}\). What acceleration (in \(g\) ) do the occupants of the small car experience? What acceleration (in \(g\) ) do the occupants of the large car experience?

To determine the muzzle velocity of a bullet fired from a rifle, you shoot the \(2.00-\mathrm{g}\) bullet into a \(2.00-\mathrm{kg}\) wooden block. The block is suspended by wires from the ceiling and is initially at rest. After the bullet is embedded in the block, the block swings up to a maximum height of \(0.500 \mathrm{~cm}\) above its initial position. What is the velocity of the bullet on leaving the gun's barrel?

Two balls of equal mass collide and stick together as shown in the figure. The initial velocity of ball \(\mathrm{B}\) is twice that of ball A. a) Calculate the angle above the horizontal of the motion of mass \(\mathrm{A}+\mathrm{B}\) after the collision. b) What is the ratio of the final velocity of the mass \(A+B\) to the initial velocity of ball \(A, v_{f} / v_{A} ?\) c) What is the ratio of the final energy of the system to the initial energy of the system, \(E_{\mathrm{f}} / E_{\mathrm{i}}\) ? Is the collision elastic or inelastic?

A golf ball is released from rest from a height of \(0.811 \mathrm{~m}\) above the ground and has a collision with the ground, for which the coefficient of restitution is \(0.601 .\) What is the maximum height reached by this ball as it bounces back up after this collision?

A soccer ball rolls out of a gym through the center of a doorway into the next room. The adjacent room is \(6.00 \mathrm{~m}\) by \(6.00 \mathrm{~m}\) with the \(2.00-\mathrm{m}\) wide doorway located at the center of the wall. The ball hits the center of a side wall at \(45.0^{\circ} .\) If the coefficient of restitution for the soccer ball is \(0.700,\) does the ball bounce back out of the room? (Note that the ball rolls without slipping, so no energy is lost to the floor.)

Two Sumo wrestlers are involved in an inelastic collision. The first wrestler, Hakurazan, has a mass of \(135 \mathrm{~kg}\) and moves forward along the positive \(x\) -direction at a speed of \(3.5 \mathrm{~m} / \mathrm{s}\). The second wrestler, Toyohibiki, has a mass of \(173 \mathrm{~kg}\) and moves straight toward Hakurazan at a speed of \(3.0 \mathrm{~m} / \mathrm{s} .\) Immediately after the collision, Hakurazan is deflected to his right by \(35^{\circ}\) (see the figure). In the collision, \(10 \%\) of the wrestlers' initial total kinetic energy is lost. What is the angle at which Toyohibiki is moving immediately after the collision?

A hockey puck \(\left(m=170 . \mathrm{g}\right.\) and \(v_{0}=2.00 \mathrm{~m} / \mathrm{s}\) ) slides without friction on the ice and hits the rink board at \(30.0^{\circ}\) with respect to the normal. The puck bounces off the board at a \(40.0^{\circ}\) angle with respect to the normal. What is the coefficient of restitution for the puck? What is the ratio of the puck's final kinetic energy to its initial kinetic energy?

How fast would a \(5.00-\mathrm{g}\) fly have to be traveling to slow a \(1900 .-\mathrm{kg}\) car traveling at \(55.0 \mathrm{mph}\) by \(5.00 \mathrm{mph}\) if the fly hit the car in a totally inelastic head-on collision?

Attempting to score a touchdown, an 85-kg tailback jumps over his blockers, achieving a horizontal speed of \(8.9 \mathrm{~m} / \mathrm{s} .\) He is met in midair just short of the goal line by a 110 -kg linebacker traveling in the opposite direction at a speed of \(8.0 \mathrm{~m} / \mathrm{s}\). The linebacker grabs the tailback. a) What is the speed of the entangled tailback and linebacker just after the collision? b) Will the tailback score a touchdown (provided that no other player has a chance to get involved, of course)?