(a) What is the magnitude of the momentum of a $10,000-k g$ truck whose speed is 12.0 $\mathrm{m} / \mathrm{s} ?$ (b) What speed would a $2,000-\mathrm{kg}$ SUV have to attain in order to have (i) the same momentum? (ii) the same kinetic energy?

João Gabriel A.

Numerade Educator

In Conceptual Example 8.1 (Section 8.1 ), show that the iceboat with mass $2 \mathrm{~m}$ has $\sqrt{2}$ times as much momentum at the finish line as does the iceboat with mass $m$.

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(a) Show that the kinetic energy $K$ and the momentum magnitude $p$ of a particle with mass $m$ are related by $K=p^{2} / 2 m .$ (b) A $0.040-\mathrm{kg}$ cardinal (Richmondena cardinalis) and a $0.145-\mathrm{kg}$ baseball have the same kinetic energy. Which has the greater magnitude of momentum? What is the ratio of the cardinal's magnitude of momentum to the bascball's? A $700-N$ man and a $450-N$ woman have the same momentum. Who has the greater kinetic energy? What is the ratio of the man's kinetic energy to that of the woman?

João Gabriel A.

Numerade Educator

In a certain men's track and field event, the shotput has a mass of 7.30 $\mathrm{kg}$ and is released with a speed of 15.0 $\mathrm{m} / \mathrm{s}$ at $40.0^{\circ}$ above the horizontal over a man's straight left leg. What are the initial horizontal and vertical components of the momentum of this shotput?

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One $110-\mathrm{kg}$ football lineman is running to the right at 2.75 $\mathrm{m} / \mathrm{s}$

while another $125-\mathrm{kg}$ lineman is running directly toward him at 2.60 $\mathrm{m} / \mathrm{s}$ . What are (a) the magnitude and direction of the net momentinm of these two athletes, and (b) their total kinetic energy?

João Gabriel A.

Numerade Educator

Two vehicles are approaching an intersection. One is a 2500 -kg pickup traveling at 14.0 $\mathrm{m} / \mathrm{s}$ from east to west (the $-x$ -direction), and the other is a $1500-\mathrm{kg}$ sedan going from south to north (the 1 y-direction at 23.0 $\mathrm{m} / \mathrm{s} )$ (a) Find the $x$ - and $y-$ components of the net momentum of this system. (b) What are the magnitude and direction of the net momentum?

João Gabriel A.

Numerade Educator

Force of a Golf Swing. A $0.0450-\mathrm{kg}$ golf ball initially at rest is given a speed of 25.0 $\mathrm{m} / \mathrm{s}$ when a club strikes. If the club and ball are in contact for 2.00 $\mathrm{ms}$ , what average force acts on the ball? Is the effect of the ball's weight during the time of contact significant? Why or why not?

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Force of a Baseball Swing. A baseball has mass 0.145 $\mathrm{kg}$ , (a) If the velocity of a pitched ball has a magnitude of 45.0 $\mathrm{m} / \mathrm{s}$ and the batted ball's velocity is 55.0 $\mathrm{m} / \mathrm{s}$ in the opposite direction, find the magnitude of the change in momentum of the ball and of the impulse applied to it by the bat. (b) If the ball remains in contact with the bat for 2.00 $\mathrm{ms}$ , find the magnitude of the average force applied by the bat.

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A0.160-kg hockey puck is moving on an icy, frictionless, horizontal sufface. At $t=0$ , the puck is moving to the right at 3.00 $\mathrm{m} / \mathrm{s}$ . (a) Calculate the velocity of the puck (magnitude and direction) after a force of 25.0 $\mathrm{N}$ directed to the right has been applied for 0.050 $\mathrm{s} .(\mathrm{b}) \mathrm{If}$ , instead, a force of 12.0 $\mathrm{N}$ directed to the left is applied from $t=0$ to $t=0.050 \mathrm{s}$ , what is the final velocity of the puck?

João Gabriel A.

Numerade Educator

An engine of the orbital maneuvering system (OMS) on a space shuttle exerts a force of $(26,700 \mathrm{N}) \}$ for 3.90 $\mathrm{s}$ , exhausting a negligible mass of fuel relative to the $95,000-\mathrm{kg}$ mass of the shuttle. (a) What is the impulse of the force for this 3.90 s? (b) What is

the shuttle's change in momentum from this impulse? (c) What is the shutle's change in velocity from this impulse? (d) Why ean't we find the resulting change in the kinetic energy of the shuttle?

João Gabriel A.

Numerade Educator

At time $t=0,$ a $2150-\mathrm{kg}$ rocket in outer space fires an engine that exerts an increasing force on it in the $+x$ -direction. This force obeys the equation $F_{x}=A t^{2},$ where $t$ is time, and has a magnitude of 781.25 $\mathrm{N}$ when $t=1.25 \mathrm{s}$ . (a) Find the SI value of the constant $A,$ including its units. $(\mathrm{b})$ What impulse does the engine exert on the rocket during the 1.50 -s interval starting 2.00 s after the engine is fired? (c) By how much does the rocket's velocity change during this interval?

João Gabriel A.

Numerade Educator

A bat strikes a $0.145-\mathrm{kg}$ 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 $\mathrm{m} / \mathrm{s}$ . 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.

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A $2.00-\mathrm{kg}$ stone is sliding to the right on a frictionless horizontal surface at 5.00 $\mathrm{m} / \mathrm{s}$ when it is suddenly struck by an object that exerts a large horizontal force on it for a short period of time. The graph in Fig. 8.34 shows the magnitude of this force as a function of time. (a) What impulse does this force exert on the stone? (b) Just after the force stops acting, find the magnitude and direction of the stone's velocity if the force acts (i) to the right or (ii) to the left.

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A 68.5 -kg astronaut is doing a repair in space on the orbiting space station. She throws a $2.25-\mathrm{kg}$ tool away from her at 3.20 $\mathrm{m} / \mathrm{s}$ relative to the space station. With what speed and in what direction will she begin to move?

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Squids and octopuses propel themselves by expelling water. They do this by keeping water in a cavity and then suddenly contracting the cavity to force out the water through an opening. $A 6.50-k g$ squid (including the water in the cavity) at rest suddenly sees a dangerous predator. (a) If the squid has 1.75 $\mathrm{kg}$ of water in its cavity, at what speed must it expel this water to suddenly achieve a speed of 2.50 $\mathrm{m} / \mathrm{s}$ to escape the predator? Neglect any drag effects of the surrounding water. (b) How much kinetic energy does the squid create by this maneuver?

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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 after-ward? (b) If the ball hits you and bounces off your chest, so after-ward it is moving horizontally at 8.0 $\mathrm{m} / \mathrm{s}$ in the opposite direction, what is your speed after the collision?

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On a frictionless, horizontal air table, puck $A$ (with mass 0.250 $\mathrm{kg}$ is moving 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.

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When cars are equipped with flexible bumpers, they will bounce off each other during low-speed collisions, thus causing less damage. In one such accident, a $1750-\mathrm{kg}$ car traveling to the right at 1.50 $\mathrm{m} / \mathrm{s}$ collides with a $1450-\mathrm{kg}$ car going to the left at 1.10 $\mathrm{m} / \mathrm{s}$ . Measurements show that the heavier car's speed just after the collision was 0.250 $\mathrm{m} / \mathrm{s}$ in its original direction. You can ignore any road friction during the collision. (a) What was the speed of the lighter car just after the collision? (b) Calculate the

change in the combined kinetic energy of the two-car system during this collision.

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The expanding gases that leave the muzzle of a rifle also contribute to the recoil. A 30 -caliber bullet has mass 0.00720 $\mathrm{kg}$ and a speed of 601 $\mathrm{m} / \mathrm{s}$ relative to the muzzle when fired from a rifle that has mass 2.80 kg. The loosely held rifle recoils at a speed of 1.85 $\mathrm{m} / \mathrm{s}$ relative to the earth. Find the momentum of the propellant gases in a coordinate system attached to the earth as they leave the muzzle of the rifle.

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Block A in Fig. 8.35 has mass 1.00 $\mathrm{kg}$ , and block $B$ has mass 3.00 kg. The blocks are forced together, compressing a spring $S$ between them; then the system is released from rest on a level, frictionless surface. The spring, which has negligible mass, is not fastened to either block and drops to the surface after it has expanded. Block $B$ acquires a speed of 1.20 $\mathrm{m} / \mathrm{s}$ , (a) What is the final speed of block $A ?$ How much potential energy was stored in the compressed spring?

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A hunter on a frozen, essentially frictionless pond uses a rifle that shoots $4.20-\mathrm{g}$ bullets at 965 $\mathrm{m} / \mathrm{s}$ . The mass of the hunter (including his gun) is 72.5 $\mathrm{kg}$ , and the hunter holds tight to the gun after firing it. Find the recoil velocity of the hunter if he fires the

rifle (a) horizontally and $(b)$ at $56.0^{\circ}$ above the horizontal.

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An atomic nucleus suddenly bursts apart (fissions) into two pieces. Piece $A,$ of mass $m_{A},$ travels off to the left with speed $v_{A}$ . Piece $B$ , of mass $m_{B}$ , travels off to the right with speed $v_{B}$ . (a) Use conservation of momenturn to solve for $v_{B}$ in terms of $m_{A}, m_{B},$ and

$v_{A}-(b)$ Use the results of part (a) to show that $K_{A} / K_{B}=m_{B} / m_{A}$ where $K_{\mathrm{A}}$ and $K_{B}$ are the kinetic energies of the two pieces.

João Gabriel A.

Numerade Educator

The nucleus of $^{214} \mathrm{Po}$ decays radioactively by emitting an alpha particle (mass $6.65 \times 10^{-27} \mathrm{kg}$ ) with kinetic energy $1.23 \times 10^{-12} \mathrm{J},$ as measured in the laboratory reference frame Assuming that the Po was initially at rest in this frame, find the recoil velocity of the nuclens that remains after the decay.

João Gabriel A.

Numerade Educator

You are standing on a large sheet of frictionless ice and holding a large rock. In order to get off the ice, you throw the rock so it has velocity 12.0 $\mathrm{m} / \mathrm{s}$ relative to the earth at an angle of $35.0^{\circ}$ above the borizontal. If your mass is 70.0 $\mathrm{kg}$ and the rock's mass is 15.0 kg, what is your speed after you throw the rock (see Discussion Question $Q 8.7$ ?

João Gabriel A.

Numerade Educator

Two ice skaters, Daniel (mass 65.0 $\mathrm{kg}$ ) and Rebeca (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 $53.1^{\circ} \mathrm{from}$ her initial direction. Both skaters move on the frictionless, horizontal 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?

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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 $\mathrm{m} / \mathrm{s} .$ What is the mass of this canister?

João Gabriel A.

Numerade Educator

Changing Mass. An open-topped freight car with mass $24,000 \mathrm{kg}$ is coasting without friction along a level track. It is raining very hard, and the rain is falling vertically downward. Originally, the car is empty and moving with a speed of 4.00 $\mathrm{m} / \mathrm{s}$ . What is the speed of the car after it has collected 3000 $\mathrm{kg}$ of rainwater?

João Gabriel A.

Numerade Educator

Two asteroids of equal mass in the asteroid belt between Mars and Jupiter collide with a glancing

blow. Asteroid $A$ , which was initially traveling at $40.0 \mathrm{m} / \mathrm{s},$ is deflected $30.0^{\circ}$ from its original direction, while asteroid $B$ travels at $45.0^{\circ}$ to the original direction of $A(\text { Fig. } .8 .36) .$ (a) Find the speed of each asteroid after the collision. (b) What fraction of the original kinetic energy of asteroid A dissipates during this collision?

João Gabriel A.

Numerade Educator

A $15.0-\mathrm{kg}$ fish swimming at 1.10 $\mathrm{m} / \mathrm{s}$ suddenly gobbles up a $4.50-\mathrm{kg}$ fish that is initially stationary. Neglect any drag effects of the water. (a) Find the speed of the large fish just after it eats the small one. (b) How much mechanical energy was dissipated during

this meal?

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Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 $\mathrm{kg}$ , is shiding to the left at 5.00 $\mathrm{m} / \mathrm{s}$ , while the other, of mass 5.75 $\mathrm{kg}$ , is slipping to the right at 6.00 $\mathrm{m} / \mathrm{s}$ . They hold fast to each other after they collide. (a) Find the magnitude and direction

of the velocity of these free-spirited otters right after they collide. (b) How much mechanical energy dissipates during this play?

João Gabriel A.

Numerade Educator

In July $2005,$ NASA's "Deep Impact" mission crashed a $372-\mathrm{kg}$ probe directly onto the surface of the comet Tempel 1 , hitting the surface at $37,000 \mathrm{km} / \mathrm{h}$ . The original speed of the comet at that time was about $40,000 \mathrm{km} / \mathrm{h}$ , and its mass was estimated to be in the range $(0.10-2.5) \times 10^{14} \mathrm{kg} .$ Use the smallest value of the estimated mass. (a) What change in the comet's velocity did this collision produce? Would this change be notiveable? (b) Suppose this comet were to hit the earth and fuse with it. By how much would it change our planet's velocity? Would this change be noticeable? (The mass of the earth is $5.97 \times 10^{24} \mathrm{kg}$ )

João Gabriel A.

Numerade Educator

A $1050-\mathrm{kg}$ sports car is moving westbound at 15.0 $\mathrm{m} / \mathrm{s}$ on a level road when it collides with a 6320 $\mathrm{kg}$ truck driving east on the same road at 10.0 $\mathrm{m} / \mathrm{s}$ . The two vehicles remain locked together after the collision. (a) What is the velocity (magnitude and direction) of the two vehicles just after the collision? (b) At what speed should the truck have been moving so that it and car are both stopped in the collision? (c) Find the change in kinetic energy of the system of two vehicles for the situations of part (a) and part (b). For which situation is the change in kinetic energy greater in magnitude?

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On a very muddy football field, a $110-\mathrm{kg}$ linebacker tackles an $85-\mathrm{kg}$ halfback. Immediately before the collision, the line-backer 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?

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Two skaters collide and grab on to each other on frictionless ice. One of them, of mass $70.0 \mathrm{kg},$ is moving to the right at 2.00 $\mathrm{m} / \mathrm{s}$ , while the other. of mass 65.0 $\mathrm{kg}$ , is moving to the left at 2.50 $\mathrm{m} / \mathrm{s}$ . What are the magnitude and direction of the velocity of these skaters just after they collide?

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Two cars, one a compact with mass 1200 $\mathrm{kg}$ and the other a large gas guzler with mass 3000 $\mathrm{kg}$ , collide head on at typical freeway speeds. (a) Which car has a greater magnitude of momentum change? Which car has a greater velocity change? (b) If the larger car changes its velocity by $\Delta v,$ calculate the change in the velocity of the small car in terms of $\Delta v .(\text { c) which car's occupants }$ would you expect to sustain greater injuries? Explain.

João Gabriel A.

Numerade Educator

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 $\mathrm{m} / \mathrm{s}$ hit a $1.50-\mathrm{kg}$ raven flying at 9.0 $\mathrm{m} / \mathrm{s}$ . The falcon hit the raven at right angles to its original path and bounced back at 5.0 $\mathrm{m} / \mathrm{s}$ . (These figures were

estimated by the author as he watched this attack occur in northern New Mexico.) By what angle did the falcon change the raven's direction of motion? (b) What was the raven's speed right after the collision?

Meghan M.

Numerade Educator

At the interscetion of Texas Avenue and University Drive, a yellow subcompact car with mass 950 $\mathrm{kg}$ traveling east on University collides with a red pickup truck with mass 1900 $\mathrm{kg}$ that is traveling north on Texas and ran a red light (Fig. 8.37$) .$ The two vehicles stick

together as a result of the collsion, and the wreckage slides at 16.0 $\mathrm{m} / \mathrm{s}$ in the direction $24,0^{\circ}$ east of north. Calculate the speed of each vehicle before the collision. The collision occurs during a heavy rainstorm; you can ignore friction forces between the vehicles and the wet road.

João Gabriel A.

Numerade Educator

A $5.00-8$ bullet is fired borizontally 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 $\mathrm{m}$ m along the surface before stopping. What was the initial speed of the bullet?

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A 12.0 -g rifle bullet is fired with a speed of 380 $\mathrm{m} / \mathrm{s}$ into a ballistic pendulum with maxs 6.00 $\mathrm{kg}$ , suspended from a cord 70.0 $\mathrm{cm}$ long (see Example 8.8 in Section 8.3 . Compute (a) the vertical height through which the pendulum rises, (b) the initial kinetic energy of the bullet, and (c) the kinetic energy of the bullet and pendulum immediately after the bullet becomes embedded in the pendalum.

João Gabriel A.

Numerade Educator

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 ar $23.0^{\circ}$ 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 energon

of the two people decrease during the collision?

Rashmi S.

Numerade Educator

Blocks $A$ (mass 2.00 $\mathrm{kg} )$ and $B(\mathrm{mass} 10.00 \mathrm{kg})$ move on a frictionless, horizontal surface. Initially, block $B$ is at rest and block $A$ is moving toward it at 2.00 $\mathrm{m} / \mathrm{s}$ . The blocks are equipped with ideal spring bumpers, as in Example 8.10 . The collision is head-on, so all motion before and after the collision is along a straight line. (a) Find the maximum energy stored in the spring bumpers and the velocity of each block at that time. (b) Find the

velocity of each block after they have moved apart.

João Gabriel A.

Numerade Educator

A 0.150 -kg glider is moving to the right on a frictionless. horizontal air track with a speed of 0.80 $\mathrm{m} / \mathrm{s}$ . It has a head-on collision with a $0.300-\mathrm{kg}$ glider that is moving to the left with a speed of 2.20 $\mathrm{m} / \mathrm{s}$ . Find the final velocity (magnitude and direction) of each glider if the collision is elastic.

Ajay S.

Numerade Educator

A 10.0 -g marble slides to the left with a velocity of magnitude 0.400 $\mathrm{m} / \mathrm{s}$ on the

frictionless, horizontal surface of an icy New York sidewalk. and has a head-on, elastic collision with a larger $30.0-\mathrm{g}$ marblesliding to the right with a velocity of magnitude 0.200 $\mathrm{m} / \mathrm{s}$ (Fig. 8.38$) .$ (a) Find the velocity of each marble (magnitude and direction) after the collision. (Since the collision is head-on, all the motion is along a line. (b) Calculate the change in momentum (that is, the momentum after the collision minus the momentum before the collision) for each marble. Compare the values you get for each marble. (c) Calculate the change in kinetic energy (that is, the kinetic energy after the collision minus the kinetic energy before the collision) for each marble. Compare the values you get for each marble.

João Gabriel A.

Numerade Educator

Supply the details of the calculation of $\alpha$ and $\beta$ in Example 8.12$(\text { Section } 8.4) .$

João Gabriel A.

Numerade Educator

Moderators. Canadian nuclear reactors use heavy water moderators in which elastic collisions occur between the neutrons and deuterons of mass 2.0 u (see Example 8.11 in Section 8.4 ) (a) What is the speed of a neutron, expressed as a fraction of its original speed, after a head-on, elastic collision with a deuteron that is initially at rest? (b) What is its kinetic energy, expressed as a fraction of its original kinetic energy? (c) How many such successive collisions will reduce the speed of a neutron to $1 / 59,000$ of its original value?

João Gabriel A.

Numerade Educator

You are at the controls of a particle accelerator, sending a beam of $1.50 \times 10^{7} \mathrm{m} / \mathrm{s}$ protons (mass $m )$ at a gas target of an unknown element. Your detector tells you that some protons bounce straight back after a collision with one of the nuclei of the unknown element. All such protons rebound with a speed of $1.20 \times 10^{7} \mathrm{m} / \mathrm{s}$ . Assume that the initial speed of the target nucleus is negligible and the collision is elastic. (a) Find the mass of one nucleus of the unknown element. Express your answer in terms of the proton mass $m .(b)$ What is the speed of the unknown nucleus immediately after such a collision?

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

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Find the position of the center of mass of the system of the sun and Jupiter. (Since Jupiter is more massive than the rest of the planets combined, this is essentially the position of the center of mass of the solar system.) Does the center of mass lie inside or out-side the sun? Use the data in Appendix F.

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Pluto's diameter is approximately $2370 \mathrm{km},$ and the diameter of its satellite Charon is 1250 $\mathrm{km}$ . Although the distance varies, they are often about $19,700 \mathrm{km}$ apart, center-to-center. Assuming that both Pluto and Charon have the same composition and hence the same average density, find the location of the center of mass of this system relative to the center of Pluto.

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A 1200 -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 (Fig. 8.39$) .$ (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. (o) 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).

Vidhi B.

Numerade Educator

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 $2.00-\mathrm{kg}$ ball at one end (Fig. 8.40$) .$ By what distance will the center of mass of this part move horizontally and vertically if the vertical bar is

pivoted through $90^{\circ}$ to make the entire part horizontal?

João Gabriel A.

Numerade Educator

At one instant, the center of mass of a system of two particles is located on the $x$ -axis at $x=2.0 \mathrm{m}$ and has a velocity of $(5.0 \mathrm{m} / \mathrm{s})$ ) One of the particles is at the origin. The other particle has a mass of 0.10 $\mathrm{kg}$ and is at rest on the $x$ -axis at $x=8.0 \mathrm{m}$ . (a) What is the mass of the particle at the origin? (b) Calculate the total momentum of this system. (c) What is the velocity of the particle at the origin?

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In Example 8.14 (Section $8.5 ),$ Ramon pulls on the rope to give himself a speed of 0.70 $\mathrm{m} / \mathrm{s}$ . What is James's speed?

João Gabriel A.

Numerade Educator

A system consists of two particles. At $t=0$ one particle is at the origin; the other, which has a mass of 0.50 $\mathrm{kg}$ , is on the $y$ -axis at $y=6.0 \mathrm{m}$ . At $t=0$ the center of mass of the system is on the $y$ -axis at $y=2.4 \mathrm{m} .$ The velocity of the center of mass is given by $\left(0.75 \mathrm{m} / \mathrm{s}^{3}\right) t^{2} \hat{\mathrm{i}}$ , (a) Find the total mass of the system. (b) Find the acceleration of the center of mass at any time $t$ . (c) Find the net external force acting on the system at $t=3.0 \mathrm{s} .$

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A radio-controlled model airplane has a momentum given by $\left[\left(-0.75 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{3}\right) t^{2}+\right.$ $(3.0 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}) ] \hat{\mathrm{i}}+\left(0.25 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2}\right) \hat{t}$ What are the $x-y$ , and z-components of the net force on the airplane?

João Gabriel A.

Numerade Educator

A small rocket burns 0.0500 $\mathrm{kg}$ of fuel per second, ejecting it as a gas with a velocity relative to the rocket of magnitude 1600 $\mathrm{m} / \mathrm{s}$ . (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?

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A $70-\mathrm{kg}$ astronaut floating in space in a 110$\cdot \mathrm{kg}$ 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?

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A rocket is fired in deep space, where gravity is segligible. If the rocket has an initial mass of 6000 $\mathrm{kg}$ and ejccts 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 $\mathrm{m} / \mathrm{s}^{2} ?$

João Gabriel A.

Numerade Educator

A rocket is fired in deep space, where gravity is negligible. In the first second it ejects $\frac{1}{160}$ of its mass as exhaust gas and has an accolcration of 15.0 $\mathrm{m} / \mathrm{s}^{2} .$ What is the spoced of tho cxhaust gas relative to the rocket?

João Gabriel A.

Numerade Educator

A $\mathrm{C} 6-5$ model rocket engine has an impolse of 10.0 $\mathrm{N}$ . for 1.70 $\mathrm{s}$ , while buming 0.0125 $\mathrm{kg}$ of propellant. It has a maximum thrust of 13.3 $\mathrm{N}$ . The initial mass of the engine plus propellant is 0.0258 $\mathrm{kg}$ , (a) What fraction of the maximum thrust is the average thrust? (b) Calculate the relative speed of the exhaust gases, assuming it is constant. (c) Assuming that the relative speed of the exhaust gases is constant, find the final speed of the engine if it was attached to a very light frame and fired from rest in gravity-free outer space.

João Gabriel A.

Numerade Educator

A single-stage rocket is fired from rest from a deep-space platform, where gravity is negligible. If the rocket burns its fuel in 50.0 $\mathrm{s}$ and the relative speed of the exhaust gas is $v_{\mathrm{ex}}=2100 \mathrm{m} / \mathrm{s}$ , what must the mass ratio $m_{0} / m$ be for a foral speed $v$ of 8.00 $\mathrm{km} / \mathrm{s}$ (about equal to the orbital speed of an earth satellite)?

João Gabriel A.

Numerade Educator

Obviously, we can make rockets to go very fast, but what is a reasonable top speed? Assume that a rocket is fired from rest at a space station in deep space, where gravity is negligible. (a) If the rocket ejects gas at a relative speed of 2000 $\mathrm{m} / \mathrm{s}$ and you want the rocket's speed eventually to be $1.00 \times 10^{-3} c$ , where $c$ is the speed of light, what fraction of the initial mass of the rocket and fuel is not fuel? (b) What is this fraction if the final speed is to be 3000 $\mathrm{m} / \mathrm{s} ?$

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A steel ball with mass 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 impact. (b) If the ball is in contact with the slab for 2.00 $\mathrm{ms}$ , find

the average force on the ball during impact.

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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 $\mathrm{m}$ directly north of the point of the explosion. Where will the other fragment land? Neglect any air resistance.

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Just before it is struck by a racket, a tennis ball weighing 0.560 $\mathrm{N}$ has a velocity of $(20.0 \mathrm{m} / \mathrm{s}) \hat{\imath}-(4.0 \mathrm{m} / \mathrm{s}) \hat{\mathrm{J}}$ . During the

3.00 $\mathrm{ms}$ that the racket and ball are in contact, the net force on the ball is constant and equal to $-(380 \mathrm{N}) \hat{\imath}+(110 \mathrm{N}) \mathrm{J}$ . (a) What are the $x$ -and $y$ components of the impulse of the net force applied to the ball? $(b)$ What are the $x-$ and $y$ -components of the final velocity of the ball?

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Three coupled railroad cars roll along and couple with a fourth car, which is initially at rest. These four cars roll along and couple with a fifth car initially at rest. This process continues until the speed of the final collection of railroad cars is one-fifth the speed of the initial three railroad cars. All the cars are identical. Ignoring friction, how many cars are in the final collection?

Ajay S.

Numerade Educator

A $1500-\mathrm{kg}$ blue convertible is traveling south, and a $2000-\mathrm{kg}$ red SUV is traveling west. If the total momentum of the system consisting of the two cars is 8000 $\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ directed at $60.0^{\circ}$ west of south, what is the speed of each vehicle?

Sachin R.

Numerade Educator

Three identical pucks on a horizontal air table have repelling magnets. They are held together and then released simultaneously. Each has the same speed at any instant. One puck moves due west. What is the direction of the velocity of each of the other two pucks?

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Spheres $A(\operatorname{mas} 0.020 \mathrm{kg}), B(\mathrm{mass} 0.030 \mathrm{kg}),$ and $C$ $(\text { mass } 0.050 \mathrm{kg})$ are approaching the origin as they slide on a frictionless air table (Fig. 8.41$)$ . The initial velocities of $A$ and $B$ are given in the figure. All three spheres arrive at the origin at the same time and stick together. (a) What must the $x$ - and $y$ -components of the initial velocity of $C$ be if all three objects are to end up moving at 0.50 $\mathrm{m} / \mathrm{s}$ in the $+x$ -direction after the collision? $(b)$ If $C$ has the velocity found in part (a), what is the change in the kinetic energy of the system of three spheres as a result of the collision?

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A railroad handcar is moving along straight, frictionless tracks with negligible air resistance. In the following cases, the car initially has a total mass (car and contents) of 200 $\mathrm{kg}$ and is traveling east with a velocity of magnitude 5.00 $\mathrm{m} / \mathrm{s} .$ Find the final velocity of the car in each case, assuming that the handcar does not leave the tracks. (a) A $25.0-\mathrm{kg}$ mass is thrown sideways out of the car with a velocity of magnitude 2.00 $\mathrm{m} / \mathrm{s}$ relative to the car's initial velocity. $(b) A 25.0-k g$ mass is thrown backward out of the car with a velocity of 5.00 $\mathrm{m} / \mathrm{s}$ relative to the initial motion of the car. (c) A 25.0 $\mathrm{kg}$ mass is thrown into the car with a velocity of 6.00 $\mathrm{m} / \mathrm{s}$ relative to the ground and opposite in direction to the initial velocity of the car.

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A railroad hopper car filled with sand is rolling with un initial speed of 15.0 $\mathrm{m} / \mathrm{s}$ on straight, horizontal tracks. You can ignore frictional forces on the railroad car. The total mass of the car plus sand is $85,000 \mathrm{kg}$ . The hopper door is not fully closed so sand leaks out the bottom. After 20 $\mathrm{min}$ , $13,000 \mathrm{kg}$ of sand has leaked out. Then what is the speed of the railroad car? (Compare your analysis with that used to solve Exercise $8.27 . )$

Paul A.

California State Polytechnic University, Pomona

At a classic auto show, a $840-\mathrm{kg}$ 1955 Nash Metropolitan motors by at 9.0 $\mathrm{m} / \mathrm{s}$ , followed by a $1620-\mathrm{kg} 1957$ Packard Clipper purring past at 5.0 $\mathrm{m} / \mathrm{s}$ , (a) Which car has the greater kinetic energy? What is the ratio of the kinetic energy of the Nash to that of the Packard? (b) Which car has the greater magnitude of momentum? What is the ratio of the magnitude of momentum of the Nash to that of the Packard? (c) Let $F_{N}$ be the net force required to stop the Nash in time $t$ , and let $F_{\mathrm{p}}$ be the net force required to stop the Packard in the same time. Which is larger: $F_{\mathrm{N}}$ or $F_{\mathrm{P}}$ ? What is the ratio $F_{\mathrm{N}} / F_{\mathrm{p}}$ of these two forces? (d) Now let $F_{\mathrm{N}}$ be the net force required to stop the Nash in a distance $d$ , and let $F_{\mathrm{p}}$ be the net force required to stop the Packard in the same distance. Which is

larger. $F_{\mathrm{N}}$ or $F_{\mathrm{P}}$ ? What is the ratio $F_{\mathrm{N}} / F_{\mathrm{P}} ?$

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A soldier on a firing range fires an eight-shot burst from an assault weapon at a full automatic rate of 1000 rounds per minute. Each bullet has a mass of 7.45 $\mathrm{g}$ and a speed of 293 $\mathrm{m} / \mathrm{s}$ relative to the ground as it leaves the barrel of the weapon. Calculate the average recoil force exerted on the weapon during that burst.

Artemisa M.

Numerade Educator

A $0.150-\mathrm{kg}$ frame, when suspended from a coil spring, stretches the spring 0.050 $\mathrm{m}$ . A $0.200-\mathrm{kg}$ lump of putty is dropped from rest onto the frame from a height of 30.0 $\mathrm{cm}$ (Fig. 8.42$)$ . Find the maximum distance the frame moves downward from its initial position.

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A rife bullet with mass 8.00 $\mathrm{g}$ strikes and embeds itself in a block with mass 0.992 kg that rests on a frictionless, horizontal surface and is attached to a coil spring (Fig. 8.43$)$ . 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 $\mathrm{cm} .(\mathrm{a})$ Find the magnitude of the block's velocity just after impact. (b) What was the initial speed of the bullet?

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$0.100-\mathrm{kg}$ stone rests on a friction-less, horizontal surface. A bullet of mass 6.00 $\mathrm{g}$ , traveling horizontally at 350 $\mathrm{m} / \mathrm{s}$ , strikes the stone and rebounds horizontally at right angles to its original direction with a speed of 250 $\mathrm{m} / \mathrm{s}$ . (a) Compute the magnitude and direction of the velocity of the stone after it is struck. (b) Is the collision perfectly elastic?

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A movie stuntman (mass 80.0 $\mathrm{kg}$ ) stands on a window ledge 5.0 $\mathrm{m}$ above the floor (Fig. 8.44 ). 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 5.0 $\mathrm{m} .$ He releases the rope just as be 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 $\mu_{k}=0.250$ , how far do they slide?

Meghan M.

Numerade Educator

Two identical masses are released from rest in a smooth hemispherical bowl of radius $R$ , from the positions shown in Fig. 8.45 . You can ignore friction between the masses and the surface of the bowl. If they stick together when they collide, how high above the bottom of the bowl will the masses go after colliding?

Guilherme B.

Numerade Educator

A ball with mass $M,$ moving horizontally at 5.00 $\mathrm{m} / \mathrm{s}$ , collides elastically with a block with mass 3 $\mathrm{M}$ that is initially hanging at rest from the ceiling on the end of a 50.0 -m wire. Find the maximum angle through which the block swings after it is hit.

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A $20.00-\mathrm{kg}$ lead sphere is hanging from a hook by a thin wire 3.50 $\mathrm{m}$ long, and is free to swing in a complete circle. Suddenly it is struck horizontally by a $5.00-\mathrm{kg}$ steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision?

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An $8.00-\mathrm{kg}$ ball, hanging from the ceiling by a light wire 135 $\mathrm{cm}$ long, is struck in an elastic collision by a $2.00-\mathrm{kg}$ ball moving horizontally at 5.00 $\mathrm{m} / \mathrm{s}$ just before the collision. Find the tension in the wire just after the collision.

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A rubber ball of mass $m$ is released from rest at height $h$ above the floor. After its first bounce, it rises to 90$\%$ of its original height. What impulse (magnitude and direction) does the floor exert on this ball during its first bounce? Express your answer in terms of the variables $m$ and $h$ .

Prashant B.

Numerade Educator

A $4.00-\mathrm{g}$ bullet, traveling horizontally with a velocity of magnitude 400 $\mathrm{m} / \mathrm{s}$ , is fired into a wooden block with mass 0.800 $\mathrm{kg}$ , initially at rest on a level surface. The bullet passes through the block and emerges with its speed reduced to 120 $\mathrm{m} / \mathrm{s} .$ The block shides a distance of 45.0 $\mathrm{cm}$ along the surface from its initial position. (a) What is the coefficient of kinetic friction between block and surface? (b) What is the decrease in kinetic energy of the bullet? (c) What is the kinetic energy of the block at the instant after

the bullet passes through it?

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A $5.00-\mathrm{g}$ bullet is shot through a $1.00-\mathrm{kg}$ wood block suspended on a string 2.00 $\mathrm{m}$ long. The center of mass of the block rises a distance of 0.45 $\mathrm{cm} .$ Find the speed of the bullet as it emerges from the block if its initial speed is 450 $\mathrm{m} / \mathrm{s}$ .

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A neutron with mass $m$ makes a head-on, elastic collision with a nucleus of mass $M$ . which is initially at rest. (a) Show that if the neutron's initial kinetic energy is $K_{\infty}$ the kinetic energy that it loses during the collision is 4$m M K_{0} /(M+m)^{2} .$ (b) For what value of $M$ does the incident neutron lose the most energy? (c) When $M$ has the value calculated in part (b), what is the speed of

the neutron after the collision?

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A stationary object with mass $m_{B}$ is struck head-on by an object with mass $m_{A}$ that is moving initially at speed $v_{0}$ , (a) If the collision is elastic, what percentage of the original energy does each object have after the collision? (b) What does your answer in part (a) give for the special cases $(1) m_{A}=m_{B}$ and $(i i) m_{A}=5 m_{B} ?(c)$ For what values, if any, of the mass ratio $m_{A} / m_{B}$ is the original kinetic energy shared equally by the two objects after the collision?

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In a shipping company distribution center, an open cart of mass 50.0 $\mathrm{kg}$ is rolling to the left

at a speed of 5.00 $\mathrm{m} / \mathrm{s}(\text { Fig } .8 .46)$ . You can ignore friction between the cart and the floor. A 15.0 $\mathrm{kg}$ package slides down a chute that is inclined at $37^{\circ}$ from the horizontal and leaves the end of the chute with a speed of 3.00 $\mathrm{m} / \mathrm{s}$ .

The package lands in the cart and they roll off together. If the lower end of the chute is a vertical distance of 4.00 $\mathrm{m}$ above the bottom of the cart, what are (a) the speed of the package just before it lands in the cart and (b) the final speed of the cart?

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A blue puck with mass 0.0400 $\mathrm{kg}$ , sliding with a velocity of magnitude 0.200 $\mathrm{m} / \mathrm{s}$ on a frictionless, horizontal air table, makes a perfectly elastic, head-on collision with a red puck with mass $m$ , initially at rest. After the collision, the velocity of the blue puck is 0.050 $\mathrm{m} / \mathrm{s}$ in the same direction as its initial velocity. Find (a) the velocity (magnitude and dircction) of the red puck after the collision; and (b) the mass $m$ of the red puck.

Prashant B.

Numerade Educator

Two asteroids with masses $m_{A}$ and $m_{B}$ are moving with velocities $\vec{v}_{A}$ and $\vec{v}_{B}$ with respect to an astronomer in a space vehicle. (a) Show that the total kinetic energy as measured by the astronomer is

$$K=\frac{1}{2} M v_{\mathrm{cm}}^{2}+\frac{1}{2}\left(m_{A} v_{A}^{\prime 2}+m_{B} v_{B}^{\prime 2}\right)$$

with $\vec{v}_{\mathrm{cm}}$ and $M$ defined as in Section $8.5, \vec{v}_{\mathrm{A}}^{\prime}=\vec{v}_{\mathrm{A}}-\overrightarrow{\boldsymbol{v}}_{\mathrm{cm}},$ and $\vec{v}_{B}^{\prime}=\vec{b}_{B}-\vec{b}_{\mathrm{cm}},$ In this expression the total kinetic energy of the two asteroids is the energy associated with their center of mass plus the energy associated with the internal motion relative to the center of mass. (b) If the asteroids collide, what is the minimum possible kinetic energy they can have after the collision, as measured by the astronomer? Explain.

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Suppose you hold a small ball in contact with, and directly over, the center of a large ball. If you then drop the small ball a short time after dropping the large ball, the small ball rebounds with surprising speed. To show the extreme case, ignore air resistance and suppose the large ball makes an clastic collision

with the floor and then rebounds to make an elastic collision with the still-descending small ball. Just before the collision between the two balls, the large ball is moving upward with velocity $\overrightarrow{\boldsymbol{v}}$ and the small ball has velocity $-\overrightarrow{\boldsymbol{v}}$ . (Do you see why? Assume the large ball has a much greater mass than the small ball. (a) What is the velocity of the small ball immediately after its collision with the large hall? (b) From the answer to part (a), what is the ratio of the small ball's rebound distance to the distance it fell before the collision?

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Jack and Sill are standing on a crate at rest on the frictionless, horizontal surface of a frozen pond. Jack has mass 75.0 $\mathrm{kg}$ , Jill has mass 45.0 $\mathrm{kg}$ and the crate has mass 15.0 $\mathrm{kg}$ . They remember that they must fetch a pail of water, so each jumps horizontally from

the top of the crate. Just after each jumps, that person is moving away from the crate with a speed of 4.00 $\mathrm{m} / \mathrm{s}$ relative to the crate. (a) What is the final speed of the crate if both Jack and sill jump simultancously und in the same direction? (Hint: Use an inertial coordinate system attached to the ground, (b) What is the final speed of the crate if Jack jumps first and then a few seconds later Jill jumps in the same direction? (c) What is the final speed of the crate if fill jumps first and then Jack, again in the same direction?

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An object with mass $m,$ initially at rest, explodes into two fragments, one with mass $m_{A}$ and the other with mass $m_{B},$ where $m_{A}+m_{B}=m .$ (a) If energy $Q$ is released in the explosion, how much kinetic energy does each fragment have immediately after the collision? (b) What percentage of the total energy released does each fragment get when one fragment has four times the mass of the other?

Ajay S.

Numerade Educator

A neutron at rest decays (breaks up) to a proton and an electron. Energy is released in the decay and appears as kinetic energy of the proton and electron. The mass of a proton is 1836 times the mass of an electron. What fraction of the total energy released goes into the kinetic energy of the proton?

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$\mathrm{A}^{232} \mathrm{Th}$ (thorium) nucleus at rest decays to a $^{228} \mathrm{Ra}$ (radium) nucleus with the emission of an alpha particle. The total kinetic energy of the decay fragments is $6.54 \times 10^{-13} \mathrm{J}$ . An alpha particle has 1.76$\%$ of the mass of a 28 Ra nucleus. Calculate the kinetic energy of (a) the recoiling 28 Ra nucleus and $(b)$ the alpha particle.

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In beta decay, a nucleus emits an electron. A 210 Bi (bismuth) nucleus at rest undergoes beta decay to $^{210} \mathrm{Po}$ (polonium). Suppose the emitted electron moves to the right with a

momentum of $5.60 \times 10^{-22} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ . The 210 $\mathrm{Po}$ nucleus, with mass $3.50 \times 10^{-25} \mathrm{kg},$ recoils to the left at a speed of $1.14 \times 10^{-3} \mathrm{m} / \mathrm{s}$ Momentum conservation requires that a second particle, called an antineutrino, must also be emitted. Calculate the magnitude and direction of the momentum of the antineutrino that is emitted in this decay.

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A proton moving with speed $v_{A 1}$ in the $+x$ -direction makes an elastic, off-center collision with an identical proton originally at rest. After impact, the first proton moves with speed $v_{A 2}$ in the first

quadrant at an angle $\alpha$ with the $x$ -axis, and the second moves with speed $v_{B 2}$ in the fourth quadrant at an angle $\beta$ with the $x$ -axis $(\text { Fig. } 8.13)$ . (a) Write the equations expressing conservation of linear momentum in the $x$ - and $y$ -directions. (b) Square the equations

from part $(a)$ and add them. (c) Now introduce the fact that the collision is elastic. (d) Prove that $\alpha+\beta=\pi / 2 .$ (You have shown that this equation is obeyed in any elastic, off-center collision between objects of equal mass when one object is initially at rest.)

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Hockey puck $B$ rests on a smooth ice surface and is struck by a second puck $A$ , which has the same mass. Puck $A$ is initially traveling at 15.0 $\mathrm{m} / \mathrm{s}$ and is defiected $25.0^{\circ}$ from its initial direction. Assume that the collision is perfectly elastic. Find the final speed

of each puck and the direction of $B^{\prime}$ s velocity after the collision. [Hint: Use the relationship derived in part (d) of Problem 8.96 .

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Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathan's weight is 800 $\mathrm{N}$ , Jane's weight is 600 $\mathrm{N}$ , and that of the sleigh and immediately see a poisonous spider on the floor of the sleigh and immediately jump off. Jonathan jumps to the left with a velocity of 5.00 $\mathrm{m} / \mathrm{s}$ at $30.0^{\circ}$ above the horizontal (relative to the ice), and Jane jumps to the right at 7.00 $\mathrm{m} / \mathrm{s}$ at $36.9^{\circ}$ above the horizontal (relative to the ice). Calculate the sleigh's horizontal velocity (magnitude and direction) after they jump out.

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The objects in Fig. 8.47 are constructed of uniform wire bent into the shapes shown. Find the position of the center of mass of each.

Khoobchandra A.

Numerade Educator

A 45.0 -kg woman stands up in a 60.0 $\mathrm{kg}$ canoe 5.00 $\mathrm{m}$ long. She walks from a point 1.00 $\mathrm{m}$ from one end to a point 1.00 $\mathrm{m}$ from the other end (Fig. 8.48$) .$ If you ignore resistance to motion of the canoe in the water, how far does the canoe move during this

process?

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You are standing on a concrete slab that in turn is resting on a frozen lake. Assume there is no friction between the slab and the ice. The slab has a weight five times your weight. If you begin walking forward at 2.00 $\mathrm{m} / \mathrm{s}$ relative to the ice, with what speed, relative to the ice, does the slab move?

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A 20.0 -kg projectile is fired at an angle of $60.0^{\circ}$ above the horizontal with a speed of 80.0 $\mathrm{m} / \mathrm{s}$ . At the highest point of its trajectory, the projectile explodes into two fragments with equal mass, one of which falls vertically with zero initial speed. You can ignore air resistance. (a) How far from the point of firing does the other fragment strike if the terrain is level? (b) How much energy is released during the explosion?

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A fireworks rocket is fired vertically upward. At its maximum height of 80.0 $\mathrm{m}$ , it cxplodes and breaks into two picces, one with mass 1.40 $\mathrm{kg}$ and the other with mass 0.28 $\mathrm{kg}$ . In the explosion, 860 $\mathrm{J}$ of chemical energy is converted to kinetic energy of the two fragments. (a) What is the speed of each fragment just after the explosion? $(b)$ It is observed that the two fragments hit the ground at the same time. What is the distance between the points on the ground where they land? Assume that the ground is level and air resistance can be ignored.

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A 12.0 kg shell is launched at an angle of $55.0^{\circ}$ above the horizontal with an initial speed of 150 $\mathrm{m} / \mathrm{s}$ . When it is at its highest point, the shell exploded into two fragments, one three times heavier than the other. The two fragments reach the ground at the same time. Assume that air resistance can be ignored. If the heavier fragment lands back at the same point from which the shell was launched, where will the lighter fragment land how much energy was released in the explosion?

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Fission, the process that supplies energy in nuclear power plants, occurs when a heavy nucleus is

split into two medium-sized nuclei. One such reaction occurs when a neutron colliding witha $^{235} \mathrm{U}$ (uranium) nucleus splits that nucleus into a $^{141} \mathrm{Ba}$ (barium) nucleus and a $^{92} \mathrm{Kr}$ (krypton) nucleus. In this reaction, two neutrons also are split off from the original $^{235} \mathrm{U}$ Before the collision, the arrangement is as shown in Fig. 8.49 $\mathrm{a}$ . After the collision, the $^{141} \mathrm{Ba}$ nucleus is moving in the $+z$ -direction and the $^{92} \mathrm{Kr}$ nucleus in the $-z$ -direction. The three neutrons are moving in the $x y$ -plane, as shown in Fig. 8.49 $\mathrm{b}$ . If the incoming neutron has an initial velocity of magnitude $3.0 \times 10^{3} \mathrm{m} / \mathrm{s}$ and a final velocity of magnitude $20 \times 10^{3} \mathrm{m} / \mathrm{s}$ in the directions shown, what are the speeds of the other two neutrons, and what can you say about the speeds of the 141 $\mathrm{Ba}$ and $^{92} \mathrm{Kr}$ nuclei? (The mass of the 14 $\mathrm{Ba}$ nucleus is approximately $23 \times 10^{-25} \mathrm{kg},$ and the mass of $^{92} \mathrm{Kr}$ is about $1.5 \times 10^{-25} \mathrm{kg} . )$

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Puck $A$ (mass $m_{A} )$ is moving on a frictionless, horizontal air table in the $+x$ -direction

with velocity $\vec{v}_{A 1}$ and makes an elastic, headon collision with puck $B\left(\text { mass } m_{B}\right),$ which is initially at rest. After the collision, both pucks are moving along the $x$ -axis. (a) Calculate the velocity of the center of mass of the two-puck system before the collision. (b) Consider a coordinate system whose origin is at the center of mass and moves with it. Is this an inertial reference frame? (c) What are the initial velocities $\vec{u}_{\Delta 1}$ and $\vec{u}_{B 1}$ of the two pucks in this center-of-mass reference frame? What is the total momentum in this frame? (d) Use conservation of momentum and energy, applied in the center-of-mass reference frame, to relate the final momentum of each puck to its initial momentum and thus the final velocity of each puck to its initial velocity. Your results should show that a one-dimensional, elastic collision has a very simple description in center-of-mass coordinates. (e) Let $m_{A}=0.400 \mathrm{kg}$ ,

$m_{B}=0.200 \mathrm{kg},$ and $v_{A 1}=6.00 \mathrm{m} / \mathrm{s}$ . Find the center-of-mass velocities $\vec{u}_{A 1}$ and $\vec{u}_{B 1},$ apply the simple result found in part (d),

and transform back to velocities in a stationary frame to find the final velocities of the pucks. Does your result agree with Eqs. $(8.24)$ and $(8.25) ?$

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The coefficient of restitution $\epsilon$ for a head-on collision is defined as the ratio of the relative speed after the collision to the relative speed before. (a) What is e for a completely inelastic collision? (b) What is $\epsilon$ for an elastic collision? (c) A ball is dropped from a height $h$ onto a stationary surface and rebounds back to a height $H_{1}$ . Show that $e=\sqrt{H_{1}} / h .$ (d) A properly inflated basket-ball should have a coefficient of restitution of $0.85 .$ When dropped from a height of 1.2 $\mathrm{m}$ above a solid wood floor, to what height should a properly inflated basketball bounce? (e) The height of the first bounce is $H_{1}$ . If $\epsilon$ is constant, show that the height of the $n$ th bounce is $H_{n}=\epsilon^{2 n} h .(f)$ If $\epsilon$ is constant, what is the height of the eighth bounce of a properly inflated basketball dropped from 1.2 $\mathrm{m} ?$

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When two hydrogen atoms of mass $m$ combine to form a diatomic hydrogen molecule $\left(H_{2}\right),$ the potential energy of the system after they combine is $-\Delta,$ where $\Delta$ is a positive quantity called the binding energy of the molecule. (a) Show that in a collision that involves only two hydrogen atoms, it is impossible to form an $\mathrm{H}_{2}$ molecule because momentum and energy cannot simultaneously be conserved. (Hint: If you can show this to be true in one frame of reference, then it is true in all frames of reference. Can you see why?) (b) An $\mathrm{H}_{2}$ molecule can be formed in a collision that involves three hydrogen atoms. Suppose that before such a collision, each of the three atoms has speed $1.00 \times 10^{3} \mathrm{m} / \mathrm{s}$ , and they are approaching at $120^{\circ}$ angles so that at any instant, the atoms lie at the comers of an equilateral triangle. Find the speeds of the $\mathrm{H}_{2}$ molecule and of the single hydrogen atom that remains after the collision. The binding energy of $\mathrm{H}_{2}$ is $\Delta=7.23 \times 10^{-19} \mathrm{J},$ and the mass of the bindrogen

atom is $1.67 \times 10^{-27} \mathrm{kg}$ .

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A wagon with two boxes of gold, having total mass 300 $\mathrm{kg}$ . is cut loose from the horses by an outlaw when the wagon is at rest 50 $\mathrm{m}$ up a $6.0^{\circ}$ slope (Fig. 8.50$) .$ The outlaw plans to have the wagon roll down the slope and across the level ground. and then fall into a canyon where his confederates wait. But in a tree 40 $\mathrm{m}$ from the canyon edge wait the Lone Ranger (mass 75.0 $\mathrm{kg}$ ) and Tonto (mass 60.0 $\mathrm{kg}$ ). They drop vertically into the wagon as it passes beneath them. (a) If they require 5.0 $\mathrm{s}$ to grab the gold and jump out, will they make it before the wagon goes over the edge? The wagon rolls with negligible friction. (b) When the two heroes drop into the wagon, is the kinetic energy of the system of the heroes plus the wagon conserved? If not, does it increase or decrease, and by how much?

Vidhi B.

Numerade Educator

In Section $8.6,$ we considered a rocket fired in outer space where there is no air resistance and where gravity is negligible. Suppose instead that the rocket is accelerating vertically upward from rest on the earth's surface. Continue to ignore air resistance and consider only that part of the motion where the altitude of the rocket is small so that $g$ may be assumed to be constant. (a) How is Eq. $(8.37)$ modified by the presence of the gravity force? (b) Derive an expression for the acceleration $a$ of the rocket, analogous to Eq. $(8.39)$ . (c) What is the acceleration of the rocket in Example $\& 15$ (Section 8.6 if it is near the earth's surface rather than in outer space? You can ignore air resistance. (d) Find the speed of the rocket in Example 8.16 (Section 8.6$)$ after 90 s if the rocket is fired from the earth's surface rather than in outer space. You can ignore air resistance. How does your answer compare with the rocket speed calculated in Example 8.16$?$

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Suppose the first stage of a two stage rocket has total mass $12,000 \mathrm{kg}$ , of which 9000 $\mathrm{kg}$ is fuel. The total mass of the second stage is $1000 \mathrm{kg},$ of which 700 $\mathrm{kg}$ is fuel. Assume that the relative speed $\boldsymbol{U}_{\mathbf{e x}}$ of ejected material is constant, and ignore any effect of gravity. (The effect of gravity is small during the firing period if the rate of fuel consumption is large) (a) Suppose the entire fuel supply carried by the two-stage rocket is utilized in a single-stage rocket with the same total mass of $13,000 \mathrm{kg}$ . In terms of $v_{e x}$ what is the speed of the rocket, starting from rest, when its fuel is exhausted? (b) For the two-stage rocket, what is the speed when the fuel of the first stage is exhausted if the first stage carries the second stage with it to this point? This speed then becomes the initial speed of the second stage. At this point, the second stage separates from the first stage.(c) What is the final speed of the second stage? (d) What value of $v_{\mathrm{ex}}$ is required to give the second stage of the rocket a speed of 7.00 $\mathrm{km} / \mathrm{s}$ ?

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For the rocket described in Examples 8.15 and 8.16 (Section 8.6$)$ , the mass of the rocket as a function of time is

$$m(t)=\left\{\begin{array}{ll}{m_{0}} & {\text { for } t<0} \\ {m_{0}\left(1-\frac{t}{120 \mathrm{s}}\right)} & {\text { for } 0 \leq t \leq 90 \mathrm{s}} \\ {m_{\mathrm{d}} / 4} & {\text { for } t \geq 90 \mathrm{s}}\end{array}\right.$$

(a) Calculate and graph the velocity of the rocket as a function of time from $t=0$ to $t=100 \mathrm{s}$ . (b) Calculate and graph the acceleration of the rocket as a function of time from $t=0$ to $t=100 \mathrm{s}$ (c) A $75-k g$ astronaut lies on a reclined chair during the firing of the rocket. What is the maximum net force exerted by the chair on the astronaut during the firing? How does your answer compare with her weight on earth?

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In Section 8.5 we calculated the center of mass by considering objects composed of a finite number of point masses or objects that, by symmetry, could be represented by a finite number of point masses. For a solid object whose mass distribution does not allow for a simple determination of the center of mass by symmetry, the sums of Eqs. $(8.28)$ must be generalized to integrals

$$x_{\mathrm{em}}=\frac{1}{M} \int x d m \quad y_{\mathrm{em}}=\frac{1}{M} \int y d m$$

where $x$ and $y$ are the coordinates of the small piece of the object that has mass $d m$ . The integration is over the whole of the object. Consider a thin rod of length $L,$ mass $M,$ and cross-sectional area A. Let the origin of the coordinates be at the left end of the rod and the positive $x$ -axis lie along the rod. (a) If the density $\rho=M / V$ of show that the $x$ -coordinate of the center of mass of the rod is at its geometrical center. (b) If the density of the object varies linearly with $x-$ that is, $\rho=\alpha x$ , where $\alpha$ is a positive constant - calculate the $x$ -coordinate of the rod's center of mass.

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Use the methods of Challenge Problem 8.113 to calculate the $x-$ and $y$ -coordinates of the center of mass of a semicircular metal plate with uniform density $\rho$ and thickness $t$ . Let the radius of the plate be $a$ . The mass of the plate is thus $M=\frac{1}{2} \rho \pi a^{2} t$ . Use the coordinate system indicated in Fig. $8.51 .$

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One-fourth of a rope of length $l$ is hanging down over the edge of a frictionless table. The rope has a uniform, linear density (mass per unit length) $\lambda$ (Greek lambda), and the end already on the table is held by a person. How much work does the person do when she pulls on the rope to raise he rest of the rope slowly onto the table? Do the problem in two ways as follows. (a) Find the force that the person must exert to raise the rope and from this the work done. Note that this force is variable because at different times, different amounts of rope are hanging over the edge. (b) Suppose the segment of the rope initially hanging over the edge of the table has all of its mass concentrated at its center of mass. Find the work necessary to raise this to table height. You will probably find this approach simpler than that of part (a). How do the answers compare, and why is this so?

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In a rocket-propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is

$$F_{\mathrm{ext}}=\frac{d p}{d t}=m \frac{d v}{d t}+v \frac{d m}{d t}$$

Suppose the mass of the raindrop depends on the distance $x$ that it has fallen. Then $m=k x,$ where $k$ is a constant, and $d m / d t=k v$ . This gives, since $F_{\text { ext }}=m g .$

$$m g=m \frac{d v}{d t}+v(k v)$$

Or, dividing by $k$

$$x g=x \frac{d v}{d t}+v^{2}$$

This is a differential equation that has a solution of the form $v=a t,$ where $a$ is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for $v,$ find the acceleration $a$ . (b) Find the distance the raindrop has fallen in $t=3.00 \mathrm{s}$ . (c) Given that $k=2.00 \mathrm{g} / \mathrm{m}$ , find the mass of the raindrop at $t=3.00 \mathrm{s}$ . For many more intriguing aspects of this problem, see $\mathbf{K} .$ S. Krane, Amer Jour. Phys, Vol. 49$(1981)$ pp. $113-117$

Khoobchandra A.

Numerade Educator