Chapter 6

Calculate the magnitude of the linear momentum for the following cases: (a) a proton with mass equal to \(1.67 \times 10^{-27} \mathrm{~kg}\), moving with a speed of \(5.00 \times 10^{6}\) \(\mathrm{m} / \mathrm{s} ;\) (b) a \(15.0-\mathrm{g}\) bullet moving with a speed of \(300 \mathrm{~m} / \mathrm{s} ;\) (c) a \(75.0-\mathrm{kg}\) sprinter running with a speed of \(10.0 \mathrm{~m} / \mathrm{s}\); (d) the Earth (mass \(=5.98 \times 10^{24} \mathrm{~kg}\) ) moving with an orbital speed equal to \(2.98 \times 10^{4} \mathrm{~m} / \mathrm{s}\).

A high-speed photograph of a club hitting a golf ball is shown in Figure \(6.3\). The club was in contact with a ball, initially at rest, for about \(0.0020 \mathrm{~s}\). If the ball has a mass of \(55 \mathrm{~g}\) and leaves the head of the club with a speed of \(2.0 \times 10^{2} \mathrm{ft} / \mathrm{s}\), find the average force exerted on the ball by the club.

A pitcher claims he can throw a \(0.145-\mathrm{kg}\) baseball with as much momentum as a \(3.00-g\) bullet moving with a speed of \(1.50 \times 10^{3} \mathrm{~m} / \mathrm{s}\). (a) What must the baseball's speed be if the pitcher's claim is valid? (b) Which has greater kinetic energy, the ball or the bullet?

S A ball of mass \(m\) is thrown straight up into the air with an initial speed \(v_{0}\). (a) Find an expression for the maximum height reached by the ball in terms of \(v_{0}\) and \(g\). (b) Using conservation of energy and the result of part (a), find the magnitude of the momentum of the ball at one-half its maximum height in terms of \(m\) and \(v_{0}\).

Q|C Drops of rain fall perpendicular to the roof of a parked car during a rainstorm. The drops strike the roof with a speed of \(12 \mathrm{~m} / \mathrm{s}\), and the mass of rain per second striking the roof is \(0.035 \mathrm{~kg} / \mathrm{s}\). (a) Assuming the drops come to rest after striking the roof, find the average force exerted by the rain on the roof. (b) If hailstones having the same mass as the raindrops fall on the roof at the same rate and with the same speed, how would the average force on the roof compare to that found in part (a)?

S Show that the kinetic energy of a particle of mass \(m\) is related to the magnitude of the momentum \(p\) of that particle by \(K E=p^{2} / 2 \mathrm{~m}\). (Note: This expression is invalid for particles traveling at speeds near that of light.)

An object has a kinetic energy of \(275 \mathrm{~J}\) and a momentum of magnitude \(25.0 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). Find the (a) speed and (b) mass of the object.

A \(0.280-\mathrm{kg}\) volleyball approaches a player horizontally with a speed of \(15.0 \mathrm{~m} / \mathrm{s}\). The player strikes the ball with her fist and causes the ball to move in the opposite direction with a speed of \(22.0 \mathrm{~m} / \mathrm{s}\). (a) What impulse is delivered to the ball by the player? (b) If the player's fist is in contact with the ball for \(0.0600 \mathrm{~s}\), find the magnitude of the average force exerted on the player's fist.

A man claims he can safely hold on to a \(12.0-\mathrm{kg}\) child in a head-on collision with a relative speed of \(120-\mathrm{mi} / \mathrm{h}\) lasting for \(0.10 \mathrm{~s}\) as long as he has his seat belt on. (a) Find the magnitude of the average force needed to hold onto the child. (b) Based on the result to part (a), is the man's claim valid? (c) What does the answer to this problem say about laws requiring the use of proper safety devices such as seat belts and special toddler seats?

A ball of mass \(0.150 \mathrm{~kg}\) is dropped from rest from a height of \(1.25 \mathrm{~m}\). It rebounds from the floor to reach a height of \(0.960 \mathrm{~m}\). What impulse was given to the ball by the floor?

A car is stopped for a traffic signal. When the light turns green, the car accelerates, increasing its speed from 0 to \(5.20 \mathrm{~m} / \mathrm{s}\) in \(0.832 \mathrm{~s}\). What are (a) the magnitudes of the linear impulse and (b) the average total force experienced by a \(70.0-\mathrm{kg}\) passenger in the car during the time the car accelerates?

A \(65.0-\mathrm{kg}\) basketball player jumps vertically and leases the floor with a velocity of \(1.80 \mathrm{~m} / \mathrm{s}\) upward. (a) What impulse does the player experience? (b) What force does the floor exert on the player before the jump? (c) What is the total average force exerted by the floor on the player if the player is in contact with the floor for \(0.450\) s during the jump?

W A S.00-kg steel ball strikes a massive wall at \(10.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(\theta=60.0^{\circ}\) with the plane of the wall. It bounces off the wall with the same speed and angle (Fig. P6.18). If the ball is in con- tact with the wall for \(0.200 \mathrm{~s}\), what is the average force exerted by the wall on the ball? the ball?

I The front \(1.20 \mathrm{~m}\) of a \(1400-\mathrm{kg}\) car is designed as a "crumple zone" that collapses to absorb the shock of a collision. If a car traveling \(25.0 \mathrm{~m} / \mathrm{s}\) stops uniformly in \(1.20 \mathrm{~m}\), (a) how long does the collision last, (b) what is the magnitude of the average force on the car, and (c) what is the acceleration of the car? Express the acceleration as a multiple of the acceleration of gravity.

\(\mathrm{Q} \mathrm{C}\) A pitcher throws a 0.14-\textrm{kg baseball toward the } batter so that it crosses home plate horizontally and has a speed of \(42 \mathrm{~m} / \mathrm{s}\) just before it makes contact with the bat. The batter then hits the ball straight back at the pitcher with a speed of \(48 \mathrm{~m} / \mathrm{s}\). Assume the ball travels along the same line leaving the bat as it followed before contacting the bat. (a) What is the magnitude of the impulse delivered by the bat to the baseball? (b) If the ball is in contact with the bat for \(0.00508\), what is the magnitude of the average force exerted by the bat on the ball? (c) How does your answer to part (b) compare to the weight of the ball?

Whigh-speed stroboscopic photographs show that the head of a \(200-\mathrm{g}\) golf club is traveling at \(55 \mathrm{~m} / \mathrm{s}\) just before it strikes a \(46-g\) golf ball at rest on a tee. After the collision, the club head travels (in the same direction) at \(40 \mathrm{~m} / \mathrm{s}\). Find the speed of the golf ball just after impact.

A rifle with a weight of \(30 \mathrm{~N}\) fires a \(5.0\)-g bullet with a speed of \(300 \mathrm{~m} / \mathrm{s}\). (a) Find the recoil speed of the rifle. (b) If a \(700-\mathrm{N}\) man holds the rifle firmly against his shoulder, find the recoil speed of the man and rifle.

A \(45.0-\mathrm{kg}\) girl is standing on a \(150-\mathrm{kg}\) plank. The plank, originally at rest, is free to slide on a frozen lake, which is a flat, frictionless surface. The girl begins to walk along the plank at a constant velocity of \(1.50 \mathrm{~m} / \mathrm{s}\) to the right relative to the plank. (a) What is her velocity relative to the surface of the ice? (b) What is the velocity of the plank relative to the surface of the ice?

This is a symbolic version of Problem 23. A girl of mass \(m_{G}\) is standing on a plank of mass \(m_{p}\). Both are originally at rest on a frozen lake that constitutes a frictionless, flat surface. The girl begins to walk along the plank at a constant velocity \(x_{C P}\) to the right relative to the plank. (The subscript \(G P\) denotes the girl relative to plank.) (a) What is the velocity \(v_{P}\) of the plank relative to the surface of the ice? (b) What is the girl's velocity \(v_{C I}\) relative to the ice surface?

A \(6.5 .0-\mathrm{kg}\) person throws a \(0.0450-\mathrm{kg}\) snowball forward with a ground speed of \(30.0 \mathrm{~m} / \mathrm{s} .\) A second person, with a mass of \(60.0 \mathrm{~kg}\), catches the snowball. Both people are on skates. The first person is initially moving forward with a speed of \(2.50 \mathrm{~m} / \mathrm{s}\), and the second person is initially at rest. What are the velocities of the two people after the snowball is exchanged? Disregard friction between the skates and the ice.

amateur skater of mass \(M\) (when fully dressed) is trapped in the middle of an ice rink and is unable to return to the side where there is no ice. Every motion she makes causes her to slip on the ice and remain in the same spot. She decides to try to return to safety by removing her gloves of mass \(m\) and throwing them in the direction opposite the safe side. (a) She throws the gloves as hard as she can, and they leave her hand with a velocity \(\vec{v}_{g b r v a}\). Explain whether or not she moxes. If she does move, calculate her velocity \(\vec{v}_{\text {gill }}\) relative to the Farth after she throws the gloves. (b) Discuss her motion from the point of view of the forces acting on her.

A man of mass \(m_{1}=70.0 \mathrm{~kg}\) is skating at \(v_{1}=\) \(8.00 \mathrm{~m} / \mathrm{s}\) behind his wife of mass \(\mathrm{m}_{2}=50.0 \mathrm{~kg}\), who is skating at \(\tau_{2}=4.00 \mathrm{~m} / \mathrm{s}\). Instead of passing her, he inadvertently collides with her. He grabs her around the waist, and they maintain their balance. (a) Sketch the problem with before-and-after diagrams, representing the skaters as blocks. (b) Is the collision best described as elastic, inelastic, or perfectly inelastic? Why? (c) Write the general equation for conservation of momentum in terms of \(m_{1}, v_{1}, w_{2}, v_{2}\), and final velocity \(v\) - (d) Solve the momentum equation for \(v_{\gamma}\). (e) Substitute values, obtaining the numerical value for \(v_{f}\), their speed after the collision.

An archer shoors an arrow toward a \(300-g\) target that is sliding in her direction at a speed of \(2.50 \mathrm{~m} / \mathrm{s}\) on a smooth, slippery surface. The \(22.5-\mathrm{g}\) arrow is shot with a speed of \(35.0 \mathrm{~m} / \mathrm{s}\) and passes through the target, which is stopped by the impact. What is the speed of the arrow after passing through the target?

Gayle runs at a speed of \(4.00 \mathrm{~m} / \mathrm{s}\) and dives on a sled, initially at rest on the top of a frictionless, snow-covered hill. After she has descended a vertical distance of \(5.00 \mathrm{~m}\), her brother, who is initially at rest, hops on her back, and they continue down the hill together. What is their speed at the bottom of the hill if the total vertical drop is \(15.0 \mathrm{~m}\) ? Gayle's mass is \(50.0 \mathrm{~kg}\), the sled has a mass of \(5.00 \mathrm{~kg}\), and her brother has a mass of \(30.0 \mathrm{~kg}-\)

BIO A \(75.0-\mathrm{kg}\) ice skater mowing at \(10.0 \mathrm{~m} / \mathrm{s}\) crashes into a stationary skater of equal mass. After the collision, the two skaters move as a unit at \(5.00 \mathrm{~m} / \mathrm{s}\). Suppose the average force a skater can experience withour breaking a bone is \(4500 \mathrm{~N}\). If the impact time is \(0.100 \mathrm{~s}\), does a bone break?

A railroad car of mass \(2.00 \times 10^{4} \mathrm{~kg}\) moving at \(3.00 \mathrm{~m} / \mathrm{s}\) collides and couples with two coupled railroad cars, each of the same mass as the single car and moving in the same direction at \(1.20 \mathrm{~m} / \mathrm{s}\). (a) What is the speed of the three coupled cars after the collision? (b) How much kinetic energy is lost in the collision?

S This is a symbolic version of Problem 33. A railroad car of mass \(M\) moving at a speed \(v_{1}\) collides and couples with two coupled railroad cars, each of the same mass \(M\) and moving in the same direction at a speed \(t_{2}\). (a) What is the speed \(v_{y}\) of the three coupled cars after the collision in terms of \(v_{1}\) and \(v_{2}\) ? (b) How much kinetic energy is lost in the collision? Answer in terms of \(M\), \(v_{1}\), and \(v_{2}\).

8 A car of mass \(m\) moving at a speed \(v_{1}\) collides and couples with the back of a truck of mass 2 m moving initially in the same direction as the car at a lower speed \(v_{2}\) - (a) What is the speed \(y_{f}\) of the two vehicles immediately after the collision? (b) What is the change in kinetic energy of the car-truck system in the collision?

A 0.090-kg bullet is Fired vertically at \(200 \mathrm{~m} / \mathrm{s}\) into a \(0.15-\mathrm{kg}\) baseball that is initially at rest. How high does the combined buller and baseball rise after the collision, assuming the bullet embeds itself in the ball?

M An bullet of mass \(w=8.00 \mathrm{~g}\) is fired into a block of mass \(M=250 \mathrm{~g}\) that is initially at rest at the edge of a table of height \(h=1.00 \mathrm{~m}\) (Fig. P6.40). The bullet remains in the block, and after the impact the block lands \(d=2.00 \mathrm{~m}\) from the bottom of the table. Determine the initial speed of the bullet.

A \(12.0-\mathrm{g}\) bullet is fired horizontally into a \(100-\mathrm{g}\) wooden block that is initially at rest on a frictionless horizontal surface and connected to a spring having spring constant \(150 \mathrm{~N} / \mathrm{m}\). The bullet becomes embedded in the block. If the bullet-block system compresses the spring by a maximum of \(80.0 \mathrm{~cm}\), what was the speed of the bullet at impact with the block?

A \(1200-\mathrm{kg}\) car traveling initially with a speed of \(25.0 \mathrm{~m} / \mathrm{s}\) in an easterly direction crashes into the rear end of a \(9000-\mathrm{kg}\) truck mowing in the same direction at \(20.0 \mathrm{~m} / \mathrm{s}\) (Fig. P6.42). The velocity of the car right after the collision is \(18.0 \mathrm{~m} / \mathrm{s}\) to the east. (a) What is the velocity of the truck right after the collision? (b) How much mechanical energy is lost in the collision? Account for this loss in energy-

A boy of mass mo and his girlfriend of mass \(m\), both wearing ice skates, face each other at rest while standing on a frictionless joe rink. The boy pushes the girl, sending her away with velocity \(\tau_{k}\) toward the east. Assume that \(m_{b}>m_{e}\). (a) Describe the subsequent motion of the boy. (b) Find expressions for the final kinetic energy of the girl and the Final kinetic energy of the boy, and show that the girl has greater kinetic energy than the boy. (c) The boy and girl had zero kinetic energy before the boy pushed the girl, but ended up with kinetic energy after the event. How do you account for the appearance of mechanical energy?

A space probe, initially at rest, undergoes an internal mechanical malfunction and breaks into three pieces. One piece of mass \(m_{1}=48.0 \mathrm{~kg}\) travels in the positive \(x\)-direction at \(12.0 \mathrm{~m} / \mathrm{s}\), and a second päece of mass \(m_{2}=62.0 \mathrm{~kg}\) travels in the \(x y\)-plane at an angle of \(105^{\circ}\) at \(15.0 \mathrm{~m} / \mathrm{s}\). The third piece has mass \(\mathrm{m}_{3}=112 \mathrm{~kg}\). (a) Sketch a diagram of the situation, labeling the different masses and their velocities. (b) Write the general expression for conservation of momentum in the \(x\)-and \(y\)-directions in terms of \(m_{1}, m_{2}, m_{3}, v_{1}, \tau_{2}\), and \(\tau_{3}\) and the sines and cosines of the angles, taking \(\theta\) to be the unknown angle. (c) Calculate the final \(x\)-componens of the momenta of \(m_{1}\) and \(m_{2}\). (d) Calculate the final \(y\)-components of the momenta of \(m_{1}\) and \(m_{2}\). (e) Substitute the known momentum components into the general equations of momentum for the \(x\)-and \(y\)-directions, along with the known mass \(m_{3}\). (1) Solve the two momentum equations for \(x_{3} \cos \theta\) and \(v_{3} \sin \theta\), respectively, and use the identity \(\cos ^{2} \theta+\sin ^{2} \theta=1\) to obtain \(x_{3}\) (g) Divide the equation for \(v_{3} \sin \theta\) by that for \(v_{3} \cos \theta\) to obtain tan \(\theta\), then obtain the angle by taking the inverse tangent of both sides. (h) In general, would three such pieces necessarily have to move in the same plane? Why?

A \(25.0-g\) object moving to the right at \(20.0 \mathrm{~cm} / \mathrm{s}\) overtakes and collides elastically with a \(10.0-\mathrm{g}\) object moving in the same direction at \(150 \mathrm{~cm} / \mathrm{s}\). Find the velocity of each object after the collision.

A billiand ball rolling across a table at \(1.50 \mathrm{~m} / \mathrm{s}\) makes a head-on elastic collision with an identical ball. Find the speed of each ball after the collision (a) when the second ball is initially at rest, (b) when the second ball is mowing toward the first at a speed of \(1.00 \mathrm{~m} / \mathrm{s}\), and (c) when the second ball is moving away from the first at a speed of \(1.00 \mathrm{~m} / \mathrm{s}\).

A \(90.0-\mathrm{kg}\) fullback running east with a speed of \(5.00 \mathrm{~m} / \mathrm{s}\) is tackled by a \(950-\mathrm{kg}\) opponent running north with a speed of \(3.00 \mathrm{~m} / \mathrm{s}\). (a) Why does the tackle constitute a perfectly inelastic collision? (b) Calculate the velocity of the players immediately after the tackle and (c) determine the mechanical energy that is lost as a result of the collisjon. (d) Where did the lost energy go?

Identical twins, each with mass 55.0 kg, are on ice skates and at rest on a frozen lake, which may be taken as frictionless. Twin A is carrying a backpack of mass ats fractionless. Twin A is carrying a backpack of inass \(12.0 \mathrm{~kg}\). She throws it horizomtally at \(9.00 \mathrm{~m} / \mathrm{s}\) to Twin B. Neglecting any gravity effects, what are the subsequent speeds of Twin \(A\) and Twin \(B\) ?

A 2000 -kg car moving east at \(10.0 \mathrm{~m} / \mathrm{s}\) collides with a \(3000-k g\) car moving nouth. The cars stick together and move as a unit after the collision, at an angle of \(40.0^{\circ}\) north of east and a speed of \(5.22 \mathrm{~m} / \mathrm{s}\). Find the speed of the \(3000-\mathrm{kg}\) car before the collision.

A billiard ball mowing at \(5.00 \mathrm{~m} / \mathrm{s}\) strikes a stationary ball of the same mass. After the collision, the first ball moves at \(4.39 \mathrm{~m} / \mathrm{s}\) at an angle \(0 \mathrm{f} 30^{\circ}\) with respect to the original line of motion. (a) Find the velocity (magnitude and direction) of the second ball after collision. (b) Was the collision inelastic or elastic?

In research in cardiology and exercise physiology, it is often important to know the mass of blood pumped by a person's heart in one stroke. This information can be obtained by means of a ballistocardiograph. The instrument works as follows: The subject lies on a horizontal pallet floating on a film of air. Friction on the pallet is negligible. Initially, the momentum of the system is zero. When the heart beats, it expels a mass \(m\) of blood into the aorta with speed \(v\), and the body and platform move in the opposite direction with speed V. The speed of the blood can be determined independently (e.g., by observing an ultrasound Doppler shift). Assume that the blood's speed is \(50.0 \mathrm{~cm} / \mathrm{s}\) in one typical trial. The mass of the subject plus the pallet is \(54.0 \mathrm{~kg}\). The pallet moves at a speed of \(6.00 \times 10^{-5} \mathrm{~m}\) in \(0.160 \mathrm{~s}\) after one heartbeat. Calculate the mass of blood that leaves the heart. Assume that the mass of blood is negligible compared with the total mass of the person. This simplified example illustrates the principle of ballistocardiography, but in practice a more sophisticated model of heart function is used.

Most of us know intuitively that in a head-on collision between a large dump truck and a subcompact car, you are better off being in the truck than in the car. Why is this? Many people imagine that the collision force exerted on the car is much greater than that exerted on the truck. To substantiate this view, they point out that the car is crushed, whereas the truck is only dented. This idea of unequal forces, of course, is false; Newton's third law tells us that both objects are acted upon by forces of the same magnitude. The truck suffers less damage because it is made of stronger metal. But what about the two drivers? Do they experience the same forces? To answer this question, suppose that each vehicle is initially moving at \(8.00 \mathrm{~m} / \mathrm{s}\) and that they undergo a perfectly inelastic head-on collision. Each driver has mass \(80.0 \mathrm{~kg}\). Inchuding the masses of the drivers, the total masses of the vehicles are \(800 \mathrm{~kg}\) for the car and \(4000 \mathrm{~kg}\) for the truck. If the collision time is \(0.120 \mathrm{~s}\), what force does the seat belt exert on each driver?

A 2.0-g particle moving at 8.0 m/s makes a perfectly elastic head-on collision with a resting 1.0-g object. (a) Find the speed of each particle after the collision. (b) Find the speed of each particle after the collision if the stationary particle has a mass of 10 g. (c) Find the final kinetic energy of the incident 2.0-g particle in the situations described in parts (a) and (b). In which case does the incident particle lose more kinetic energy?

An unstable nucleus of mass 1.7 3 10226 kg, initially at rest at the origin of a coordinate system, disintegrates into three particles. One particle, having a mass of m1 5 5.0 3 10227 kg, moves in the positive y- direction with speed v1 5 6.0 3 106 m/s. Another particle, of mass m2 5 8.4 3 10227 kg, moves in the positive x- direction with speed v2 5 4.0 3 106 m/s. Find the magnitude and direction of the velocity of the third particle.

Two blocks of masses m1 and m2 approach each other on a horizontal table with the same constant speed, v0, as measured by a laboratory observer. The blocks undergo a perfectly elastic collision, and it is observed that m1 stops but m2 moves opposite its original motion with some constant speed, v. (a) Determine the ratio of the two masses, m1/m2. (b) What is the ratio of their speeds, v/v0?

Two objects of masses m and 3m are moving toward each other along the x-axis with the same initial speed v0. The object with mass m is traveling to the left, and the object with mass 3m is traveling to the right. They undergo an elastic glancing collision such that m is moving downward after the collision at right angles from its initial direction. (a) Find the final speeds of the two objects. (b) What is the angle u at which the object with mass 3m is scattered?

A cue ball traveling at 4.00 m/s makes a glancing, elastic collision with a target ball of equal mass that is initially at rest. The cue ball is deflected so that it makes an angle of 30.0° with its original direction of travel. Find (a) the angle between the velocity vectors of the two balls after the collision and (b) the speed of each ball after the collision.

A neutron in a reactor makes an elastic head-on collision with a carbon atom that is initially at rest. (The mass of the carbon nucleus is about 12 times that of the neutron.) (a) What fraction of the neutron’s kinetic energy is transferred to the carbon nucleus? (b) If the neutron’s initial kinetic energy is 1.6 3 10213 J, find its final kinetic energy and the kinetic energy of the carbon nucleus after the collision.

Two blocks collide on a frictionless surface. After the collision, the blocks stick together. Block A has a mass M and is initially moving to the right at speed v. Block B has a mass 2M and is initially at rest. System C is composed of both blocks. (a) Draw a force diagram for each block at an instant during the collision. (b) Rank the magnitudes of the horizontal forces in your diagram. Explain your reasoning. (c) Calculate the change in momentum of block A, block B, and system C. (d) Is kinetic energy conserved in this collision? Explain your answer. (This problem is courtesy of Edward F. Redish. For more such problems, visit http://www.physics.umd.edu/perg.)

(a) A car traveling due east strikes a car traveling due north at an intersection, and the two move together as a unit. A property owner on the southeast corner of the intersection claims that his fence was torn down in the collision. Should he be awarded damages by the insurance company? Defend your answer. (b) Let the eastward-moving car have a mass of 1 300 kg and a speed of 30.0 km/h and the northward-moving car a mass of 1 100 kg and a speed of 20.0 km/h. Find the velocity after the collision. Are the results consistent with your answer to part (a)?