Solved: When an object breaks into two pieces (explosion,

(a) What is the magnitude of the momentum of a 10,000-kg truck whose speed is 12.0 m/s? (b) What speed would a 2000-kg SUV have to attain in order to have (i) the same momentum? (ii) the same kinetic energy?

In a certain mens track and field event, the shotput has a mass of 7.30 kg and is released with a speed of at 40.0 above the horizontal over a mans straight left leg. What are the initial horizontal and vertical components of the momentum of this shotput?

(a) Show that the kinetic energy K and the momentum magnitude p of a particle with mass m are related by (b) A 0.040-kg cardinal (Richmondena cardinalis) and a 0.145-kg baseball have the same kinetic energy. Which has the greater magnitude of momentum? What is the ratio of the cardinals magnitude of momentum to the baseballs? (c) 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 mans kinetic energy to that of the woman?

Two vehicles are approaching an intersection. One is a 2500-kg pickup traveling at from east to west (the x-direction), and the other is a 1500-kg sedan going from south to north (the (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?

One 110-kg football lineman is running to the right at while another 125-kg lineman is running directly toward him at What are (a) the magnitude and direction of the net momentum of these two athletes, and (b) their total kinetic energy?

Biomechanics. The mass of a regulation tennis ball is 57 g (although it can vary slightly), and tests have shown that the ball is in contact with the tennis racket for 30 ms. (This number can also vary, depending on the racket and swing.) We shall assume a 30.0-ms contact time for this exercise. The fastest-known served tennis ball was served by “Big Bill” Tilden in 1931, and its speed was measured to be 73.14 m/s. (a) What impulse and what force did Big Bill exert on the tennis ball in his record serve? (b) If Big Bill’s opponent returned his serve with a speed of 55 m/s, what force and what impulse did he exert on the ball, assuming only horizontal motion?

Force of a Golf Swing. A 0.0450-kg golf ball initially at rest is given a speed of 25.0 m/s when a club strikes. If the club and ball are in contact for 2.00 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?

Force of a Baseball Swing. A baseball has a mass 0.145 kg. (a) If the velocity of a pitched ball has a magnitude of 45.0 m/s and the batted ball’s velocity is 55.0 m/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 ms, find the magnitude of the average force applied by the bat.

A 0.160-kg hockey puck is moving on an icy, frictionless, horizontal surface. At the puck is moving to the right at (a) Calculate the velocity of the puck (magnitude and direction) after a force of 25.0 N directed to the right has been applied for 0.050 s. (b) If, instead, a force of 12.0 N directed to the left is applied from to what is the final velocity of the puck?

An engine of the orbital maneuvering system (OMS) on a space shuttle exerts a force of for 3.90 s, exhausting a negligible mass of fuel relative to the 95,000-kg mass of the shuttle. (a) What is the impulse of the force for this 3.90 s? (b) What is the shuttles change in momentum from this impulse? (c) What is the shuttles change in velocity from this impulse? (d) Why cant we find the resulting change in the kinetic energy of the shuttle?

At time t = 0, a 2150-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 N when t = 1.25 s. (a) Find the SI value of the constant A, including its units. (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?

A bat strikes a 0.145-kg baseball. Just before impact, the ball is traveling horizontally to the right at and it leaves the bat traveling to the left at an angle of above horizontal with a speed of If the ball and bat are in contact for 1.75 ms, find the horizontal and vertical components of the average force on the ball

A 2.00-kg stone is sliding to the right on a frictionless horizontal surface at 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. E8.13 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 stones velocity if the force acts (i) to the right or (ii) to the left.

BIO Bone Fracture. Experimental tests have shown that bone will rupture if it is subjected to a force density of . Suppose a 70.0-kg person carelessly rollerskates into an overhead metal beam that hits his forehead and completely stops his forward motion. If the area of contact with the persons forehead is what is the greatest speed with which he can hit the wall without breaking any bone if his head is in contact with the beam for 10.0 ms?

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

Starting at \(t=0\), a horizontal net force \(\overrightarrow{\boldsymbol{F}}= (0.280 \mathrm{~N} / \mathrm{s}) t \hat{\boldsymbol{\imath}}+\left(-0.450 \mathrm{~N} / \mathrm{s}^2\right) t^2 \hat{\boldsymbol{\jmath}}\) is applied to a box that has an initial momentum \(\vec{p}=(-3.00 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}) \hat{\imath}+(4.00 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}) \hat{\boldsymbol{j}}\). What is the momentum of the box at \(t=2.00\mathrm{\ s}\)?

The expanding gases that leave the muzzle of a rifle also contribute to the recoil. A .30-caliber bullet has mass 0.00720 kg and a speed of relative to the muzzle when fired from a rifle that has mass 2.80 kg. The loosely held rifle recoils at a speed of 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.

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

Animal Propulsion. 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-kg squid (including the water in the cavity) at rest suddenly sees a dangerous predator. (a) If the squid has 1.75 kg of water in its cavity, at what speed must it expel this water to suddenly achieve a speed of 2.50 m / 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?

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-kg ball that is traveling horizontally at Your mass is 70.0 kg. (a) If you catch the ball, with what speed do you and the ball move afterward? (b) If the ball hits you and bounces off your chest, so afterward it is moving horizontally at in the opposite direction, what is your speed after the collision?

On a frictionless, horizontal air table, puck A (with mass 0.250 kg) is moving toward puck B (with mass 0.350 kg), which is initially at rest. After the collision, puck A has a velocity of 0.120 m/s to the left, and puck B has a velocity of 0.650 m/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.

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-kg car traveling to the right at collides with a 1450-kg car going to the left at Measurements show that the heavier cars speed just after the collision was 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.

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

Block A in Fig. E8.24 has mass 1.00 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 (a) What is the final speed of block A? (b) How much potential energy was stored in the compressed spring?

A hunter on a frozen, essentially frictionless pond uses a rifle that shoots 4.20-g bullets at The mass of the hunter (including his gun) is 72.5 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 above the horizontal.

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 momentum 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 \(K_{A}/L_{B} = m_{B}/m_{A}\), where \(K_{A}\) and \(K_{B}\) are the kinetic energies of the two pieces. Text Transcription: m_A v_a m_B v_b m_A, m_B K_A/K_B = m_B/m_A K_A K_B

Two ice skaters, Daniel (mass 65.0 kg) and Rebecca (mass 45.0 kg), are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at before she collides with him. After the collision, Rebecca has a velocity of magnitude at an angle of from her initial direction. Both skaters move on the frictionless, horizontal surface of the rink. (a) What are the magnitude and direction of Daniels velocity after the collision? (b) What is the change in total kinetic energy of the two skaters as a result of the collision?

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 relative to the earth at an angle of above the horizontal. If your mass is 70.0 kg and the rocks mass is 15.0 kg, what is your speed after you throw the rock? (See Discussion Question Q8.7.)

Changing Mass. An open-topped freight car with mass 24,000 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 (a) What is the speed of the car after it has collected 3000 kg of rainwater? (b) Since the rain is falling downward, how is it able to affect the horizontal motion of the car?

An astronaut in space cannot use a conventional means, such as a scale or balance, to determine the mass of an object. But she does have devices to measure distance and time accurately. She knows her own mass is 78.4 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 m/s, she pushes against it, which slows it down to 1.20 m/s (but does not reverse it) and gives her a speed of 2.40 m/s. What is the mass of this canister?

Asteroid Collision. 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 is deflected 30.0 from its original direction, while asteroid B, which was initially at rest, travels at 45.0 to the original direction of A (Fig. E8.31). (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?

Two skaters collide and grab on to each other on frictionless ice. One of them, of mass 70.0 kg, is moving to the right at while the other, of mass 65.0 kg, is moving to the left at What are the magnitude and direction of the velocity of these skaters just after they collide?

A 15.0-kg fish swimming at 1.10 m/s suddenly gobbles up a 4.50-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?

Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 kg, is sliding to the left at while the other, of mass 5.75 kg, is slipping to the right at 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?

Deep Impact Mission. In July 2005, NASAs Deep Impact mission crashed a 372-kg probe directly onto the surface of the comet Tempel 1, hitting the surface at The original speed of the comet at that time was about and its mass was estimated to be in the range Use the smallest value of the estimated mass. (a) What change in the comets velocity did this collision produce? Would this change be noticeable? (b) Suppose this comet were to hit the earth and fuse with it. By how much would it change our planets velocity? Would this change be noticeable? (The mass of the earth is )

A 1050-kg sports car is moving westbound at 15.0 m/s on a level road when it collides with a 6320-kg truck driving east on the same road at 10.0 m/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 the 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?

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

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

Two cars, one a compact with mass 1200 kg and the other a large gas-guzzler with mass 3000 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 \(\Delta v\). (c) Which car’s occupants would you expect to sustain greater injuries? Explain. Text Transcription: Delta v

BIO Bird Defense. To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a 600-g falcon flying at hit a 1.50-kg raven flying at The falcon hit the raven at right angles to its original path and bounced back at (These figures were estimated by the author as he watched this attack occur in northern New Mexico.) (a) By what angle did the falcon change the ravens direction of motion? (b) What was the ravens speed right after the collision?

At the intersection of Texas Avenue and University Drive, a yellow subcompact car with mass 950 kg traveling east on University collides with a red pickup truck with mass 1900 kg that is traveling north on Texas and has run a red light (Fig. E8.41). The two vehicles stick together as a result of the collision, and the wreckage slides at in the direction 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.

A 5.00-g bullet is fired horizontally into a 1.20-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 m along the surface before stopping. What was the initial speed of the bullet?

A Ballistic Pendulum. A 12.0-g rifle bullet is fired with a speed of into a ballistic pendulum with mass 6.00 kg, suspended from a cord 70.0 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 pendulum.

Combining Conservation Laws. A 15.0-kg block is attached to a very light horizontal spring of force constant 500.0 N/m and is resting on a frictionless horizontal table. (Fig. E8.44). Suddenly it is struck by a 3.00-kg stone traveling horizontally at 8.00 m/s to the right, whereupon the stone rebounds at 2.00 m/s horizontally to the left. Find the maximum distance that the block will compress the spring after the collision.

A 5.00-kg ornament is hanging by a 1.50-m wire when it is suddenly hit by a 3.00-kg missile traveling horizontally at The missile embeds itself in the ornament during the collision. What is the tension in the wire immediately after the collision?

A 0.150-kg glider is moving to the right on a frictionless, horizontal air track with a speed of It has a head-on collision with a 0.300-kg glider that is moving to the left with a speed of Find the final velocity (magnitude and direction) of each glider if the collision is elastic

Blocks A (mass 2.00 kg) and B (mass 10.00 kg) move on a frictionless, horizontal surface. Initially, block B is at rest and block A is moving toward it at The blocks are equipped with ideal spring bumpers, as in Example 8.10 (Section 8.4). 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.

A 10.0-g marble slides to the left with a velocity of magnitude on the frictionless, horizontal surface of an icy New York sidewalk and has a head-on, elastic collision with a larger 30.0-g marble sliding to the right with a velocity of magnitude (Fig. E8.48). (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.

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 of its original value?

You are at the controls of a particle accelerator, sending a beam of 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 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?

Three odd-shaped blocks of chocolate have the following masses and center-of-mass coordinates: (1) 0.300 kg, 10.200 m, Find the coordinates of the center of mass of the system of three chocolate blocks.

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 outside the sun? Use the data in Appendix F

Pluto and Charon. Plutos diameter is approximately 2370 km, and the diameter of its satellite Charon is 1250 km. Although the distance varies, they are often about 19,700 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.

A 1200-kg station wagon is moving along a straight highway at Another car, with mass 1800 kg and speed has its center of mass 40.0 m ahead of the center of mass of the station wagon (Fig. E8.54). (a) Find the position of the center of mass of the system consisting of the two automobiles. (b) Find the magnitude of the total momentum of the system from the given data. (c) Find the speed of the center of mass of the system. (d) Find the total momentum of the system, using the speed of the center of mass. Compare your result with that of part (b).

A machine part consists of a thin, uniform 4.00-kg bar that is 1.50 m long, hinged perpendicular to a similar vertical bar of mass 3.00 kg and length 1.80 m. The longer bar has a small but dense 2.00-kg ball at one end (Fig. E8.55). By what distance will the center of mass of this part move horizontally and vertically if the vertical bar is pivoted counterclockwise through 90 to make the entire part horizontal?

At one instant, the center of mass of a system of two particles is located on the x-axis at and has a velocity of . One of the particles is at the origin. The other particle has a mass of 0.10 kg and is at rest on the x-axis at (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?

In Example 8.14 (Section 8.5), Ramon pulls on the rope to give himself a speed of What is Jamess speed?

CALC A system consists of two particles. At one particle is at the origin; the other, which has a mass of 0.50 kg, is on the y-axis at At the center of mass of the system is on the y-axis at The velocity of the center of mass is given by . (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 s.

CALC A radio-controlled model airplane has a momentum given by . What are the x-, y-, and z- components of the net force on the airplane?

BIO Changing Your Center of Mass. To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 92.0 cm long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a 70.0-kg person, the mass of the upper leg would be 8.60 kg, while that of the lower leg (including the foot) would be 5.25 kg. Find the location of the center of mass of this leg, relative to the hip joint, if it is (a) stretched out horizontally and (b) bent at the knee to form a right angle with the upper leg remaining horizontal.

A 70-kg astronaut floating in space in a 110-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 m/s, how much gas is used by the thruster in 5.0 s? (b) What is the thrust of the thruster?

A small rocket burns 0.0500 kg of fuel per second, ejecting it as a gas with a velocity relative to the rocket of magnitude 1600 m/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?

A C6-5 model rocket engine has an impulse of while burning 0.0125 kg of propellant in 1.70 s. It has a maximum thrust of 13.3 N. The initial mass of the engine plus propellant is 0.0258 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 gravityfree outer space.

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 and you want the rockets speed eventually to be 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 m>s?

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 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 final speed v of 8.00 km/s (about equal to the orbital speed of an earth satellite)?

A young girl with mass 40.0 kg is sliding on a horizontal, frictionless surface with an initial momentum that is due east and that has magnitude \(90.0 \mathrm {\ kg} \cdot \mathrm{m}/ \mathrm{s}\). Starting at t = 0, a net force with magnitude F = (8.20 N/s)t and direction due west is applied to the girl. (a) At what value of t does the girl have a westward momentum of magnitude \(60.0 \mathrm{\ kg} \cdot \mathrm{m}/ \mathrm{s}\)? (b) How much work has been done on the girl by the force in the time interval from t = 0 to the time calculated in part (a)? (c) What is the magnitude of the acceleration of the girl at the time calculated in part (a)?

A steel ball with mass 40.0 g is dropped from a height of 2.00 m onto a horizontal steel slab. The ball rebounds to a height of 1.60 m. (a) Calculate the impulse delivered to the ball during impact. (b) If the ball is in contact with the slab for 2.00 ms, find the average force on the ball during impact.

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 318 m directly north of the point of the explosion. Where will the other fragment land? Neglect any air resistance.

Just before it is struck by a racket, a tennis ball weighing 0.560 N has a velocity of \((20.0 \mathrm{~m} / \mathrm{s}) \hat{\boldsymbol{\imath}}-(4.0 \mathrm{~m} / \mathrm{s}) \hat{\boldsymbol{\jmath}}\). During the 3.00 ms that the racket and ball are in contact, the net force on the ball is constant and equal to \(-(380 \mathrm{~N}) \hat{\boldsymbol{\imath}}+(110 \mathrm{~N}) \hat{\boldsymbol{\jmath}}\). (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?

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?

A 1500-kg blue convertible is traveling south, and a 2000-kg red SUV is traveling west. If the total momentum of the system consisting of the two cars is directed at west of south, what is the speed of each vehicle?

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 kg and is traveling east with a velocity of magnitude Find the final velocity of the car in each case, assuming that the handcar does not leave the tracks. (a) A 25.0-kg mass is thrown sideways out of the car with a velocity of magnitude relative to the cars initial velocity. (b) A 25.0-kg mass is thrown backward out of the car with a velocity of relative to the initial motion of the car. (c) A 25.0-kg mass is thrown into the car with a velocity of relative to the ground and opposite in direction to the initial velocity of the car

Spheres A B and C are approaching the origin as they slide on a frictionless air table (Fig. P8.73). 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 in the 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?

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

The nucleus of decays radioactively by emitting an alpha particle (mass kg) with kinetic energy as measured in the laboratory reference frame. Assuming that the Po was initially at rest in this frame, find the recoil velocity of the nucleus that remains after the decay

At a classic auto show, a 840-kg 1955 Nash Metropolitan motors by at 9.0 m/s, followed by a 1620-kg 1957 Packard Clipper purring past at 5.0 m/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 \(F_{\mathrm{N}}\) 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 \(F_{\mathrm{P}}\)?or What is the ratio \(F_{\mathrm{N}} / F_{\mathrm{P}}\)?

CP An 8.00-kg block of wood sits at the edge of a frictionless table, 2.20 m above the floor. A 0.500- kg blob of clay slides along the length of the table with a speed of , strikes the block of wood, and sticks to it. The combined object leaves the edge of the table and travels to the floor. What horizontal distance has the combined object traveled when it reaches the floor?

CP A small wooden block with mass 0.800 kg is suspended from the lower end of a light cord that is 1.60 m long. The block is initially at rest. A bullet with mass 12.0 g is fired at the block with a horizontal velocity . The bullet strikes the block and becomes embedded in it. After the collision the combined object swings on the end of the cord. When the block has risen a vertical height of 0.800 m, the tension in the cord is 4.80 N. What was the initial speed of the bullet?

Combining Conservation Laws. A 5.00-kg chunk of ice is sliding at on the floor of an ice-covered valley when it collides with and sticks to another 5.00-kg chunk of ice that is initially at rest. (Fig. P8.79). Since the valley is icy, there is no friction. After the collision, how high above the valley floor will the combined chunks go?

Automobile Accident Analysis. You are called as an expert witness to analyze the following auto accident: Car B, of mass 1900 kg, was stopped at a red light when it was hit from behind by car A, of mass 1500 kg. The cars locked bumpers during the collision and slid to a stop with brakes locked on all wheels. Measurements of the skid marks left by the tires showed them to be 7.15 m long. The coefficient of kinetic friction between the tires and the road was 0.65. (a) What was the speed of car A just before the collision? (b) If the speed limit was 35 mph, was car A speeding, and if so, by how many miles per hour was it exceeding the speed limit?

Accident Analysis. A 1500-kg sedan goes through a wide intersection traveling from north to south when it is hit by a 2200-kg SUV traveling from east to west. The two cars become enmeshed due to the impact and slide as one thereafter. On-thescene measurements show that the coefficient of kinetic friction between the tires of these cars and the pavement is 0.75, and the cars slide to a halt at a point 5.39 m west and 6.43 m south of the impact point. How fast was each car traveling just before the collision?

CP A 0.150-kg frame, when suspended from a coil spring, stretches the spring 0.070 m. A 0.200-kg lump of putty is dropped from rest onto the frame from a height of 30.0 cm (Fig. P8.82). Find the maximum distance the frame moves downward from its initial position.

A rifle bullet with mass 8.00 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. P8.83). The impact compresses the spring 15.0 cm. Calibration of the spring shows that a force of 0.750 N is required to compress the spring 0.250 cm. (a) Find the magnitude of the blocks velocity just after impact. (b) What was the initial speed of the bullet?

A Ricocheting Bullet. 0.100-kg stone rests on a frictionless, horizontal surface. A bullet of mass 6.00 g, traveling horizontally at strikes the stone and rebounds horizontally at right angles to its original direction with a speed of (a) Compute the magnitude and direction of the velocity of the stone after it is struck. (b) Is the collision perfectly elastic?

A movie stuntman (mass 80.0 kg) stands on a window ledge 5.0 m above the floor (Fig. P8.85). Grabbing a rope attached to a chandelier, he swings down to grapple with the movies villain (mass 70.0 kg), who is standing directly under the chandelier. (Assume that the stuntmans center of mass moves downward 5.0 m. He releases the rope just as he reaches the villain.) (a) With what speed do the entwined foes start to slide across the floor? (b) If the coefficient of kinetic friction of their bodies with the floor is how far do they slide?

CP Two identical masses are released from rest in a smooth hemispherical bowl of radius R from the positions shown in Fig. P8.86. 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?

A ball with mass M, moving horizontally at 4.00 m/s, collides elastically with a block with mass 3M that is initially hanging at rest from the ceiling on the end of a 50.0-cm wire. Find the maximum angle through which the block swings after it is hit.

CP A 20.00-kg lead sphere is hanging from a hook by a thin wire 3.50 m long and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00-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?

CP An 8.00-kg ball, hanging from the ceiling by a light wire 135 cm long, is struck in an elastic collision by a 2.00-kg ball moving horizontally at just before the collision. Find the tension in the wire just after the collision.

A 7.0-kg shell at rest explodes into two fragments, one with a mass of 2.0 kg and the other with a mass of 5.0 kg. If the heavier fragment gains 100 J of kinetic energy from the explosion, how much kinetic energy does the lighter one gain?

A 4.00-g bullet, traveling horizontally with a velocity of magnitude is fired into a wooden block with mass 0.800 kg, initially at rest on a level surface. The bullet passes through the block and emerges with its speed reduced to The block slides a distance of 45.0 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?

A 5.00-g bullet is shot through a 1.00-kg wood block suspended on a string 2.00 m long. The center of mass of the block rises a distance of 0.38 cm. Find the speed of the bullet as it emerges from the block if its initial speed is 450 m>s

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 \(K_{0}\) 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?

Energy Sharing in Elastic Collisions. A stationary object with mass is struck head-on by an object with mass that is moving initially at speed (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 (i) and (ii) (c) For what values, if any, of the mass ratio is the original kinetic energy shared equally by the two objects after the collision?

In a shipping company distribution center, an open cart of mass 50.0 kg is rolling to the left at a speed of 5.00 m/s (Fig. P8.95). You can ignore friction between the cart and the floor. A 15.0-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 m/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 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?

A blue puck with mass 0.0400 kg, sliding with a velocity of magnitude 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 in the same direction as its initial velocity. Find (a) the velocity (magnitude and direction) of the red puck after the collision and (b) the mass m of the red puck.

Jack and Jill are standing on a crate at rest on the frictionless, horizontal surface of a frozen pond. Jack has mass 75.0 kg, Jill has mass 45.0 kg, and the crate has mass 15.0 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 relative to the crate. (a) What is the final speed of the crate if both Jack and Jill jump simultaneously and 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 Jill jumps first and then Jack, again in the same direction?

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 elastic 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 and the small ball has velocity (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 ball? (b) From the answer to part (a), what is the ratio of the small balls rebound distance to the distance it fell before the collision?

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 and is deflected from its initial direction. Assume that the collision is perfectly elastic. Find the final speed of each puck and the direction of Bs velocity after the collision.

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

Neutron Decay. 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?

A (thorium) nucleus at rest decays to a (radium) nucleus with the emission of an alpha particle. The total kinetic energy of the decay fragments is An alpha particle has 1.76% of the mass of a nucleus. Calculate the kinetic energy of (a) the recoiling nucleus and (b) the alpha particle.

Antineutrino. In beta decay, a nucleus emits an electron. A (bismuth) nucleus at rest undergoes beta decay to (polonium). Suppose the emitted electron moves to the right with a momentum of The nucleus, with mass recoils to the left at a speed of 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.

Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathans weight is 800 N, Janes weight is 600 N, and that of the sleigh is 1000 N. They see a poisonous spider on the floor of the sleigh and immediately jump off. Jonathan jumps to the left with a velocity of at above the horizontal (relative to the ice), and Jane jumps to the right at at above the horizontal (relative to the ice). Calculate the sleighs horizontal velocity (magnitude and direction) after they jump out.

Two friends, Burt and Ernie, are standing at opposite ends of a uniform log that is floating in a lake. The log is 3.0 m long and has mass 20.0 kg. Burt has mass 30.0 kg and Ernie has mass 40.0 kg. Initially the log and the two friends are at rest relative to the shore. Burt then offers Ernie a cookie, and Ernie walks to Burts end of the log to get it. Relative to the shore, what distance has the log moved by the time Ernie reaches Burt? Neglect any horizontal force that the water exerts on the log and assume that neither Burt nor Ernie falls off the log.

A 45.0-kg woman stands up in a 60.0-kg canoe 5.00 m long. She walks from a point 1.00 m from one end to a point 1.00 m from the other end (Fig. P8.106). If you ignore resistance to motion of the canoe in the water, how far does the canoe move during this process?

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 m/s relative to the ice, with what speed, relative to the ice, does the slab move?

A 20.0-kg projectile is fired at an angle of \(60.0^{\circ}\) above the horizontal with a speed of 80.0 m/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?

CP A fireworks rocket is fired vertically upward. At its maximum height of 80.0 m, it explodes and breaks into two pieces: one with mass 1.40 kg and the other with mass 0.28 kg. In the explosion, 860 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.

A 12.0-kg shell is launched at an angle of above the horizontal with an initial speed of When it is at its highest point, the shell explodes 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, and how much energy was released in the explosion?

A wagon with two boxes of gold, having total mass 300 kg, is cut loose from the horses by an outlaw when the wagon is at rest 50 m up a slope (Fig. P8.111). 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 m from the canyon edge wait the Lone Ranger (mass 75.0 kg) and Tonto (mass 60.0 kg). They drop vertically into the wagon as it passes beneath them. (a) If they require 5.0 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?

CALC 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 earths 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 8.15 (Section 8.6) if it is near the earths 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 earths 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?

A Multistage Rocket. Suppose the first stage of a two-stage rocket has total mass 12,000 kg, of which 9000 kg is fuel. The total mass of the second stage is 1000 kg, of which 700 kg is fuel. Assume that the relative speed \(v_{\mathrm{ex}}\) 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 kg. In terms of \(v_{\mathrm{ex}}\), 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 km/s?

CALC A Variable-Mass Raindrop. 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 Suppose the mass of the raindrop depends on the distance x that it has fallen. Then where k is a constant, and This gives, since Or, dividing by k, This is a differential equation that has a solution of the form where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for find the acceleration a. (b) Find the distance the raindrop has fallen in (c) Given that find the mass of the raindrop at (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics, Vol. 49 (1981), pp. 113117.)

. CALC 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 where x and y are the coordinates of the small piece of the object that has mass dm. The integration is over the whole of the object. xcm = 1 M Lxdm ycm = 1 M Lydm Consider a thin rod of length L, mass M, and cross-sectional area A. Figure P8.116 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 of the object is uniform, perform the integration described above to 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 xthat is, where is a positive constantcalculate the x-coordinate of the rods center of mass.

CALC Use the methods of Challenge Problem 8.115 to calculate the x- and y-coordinates of the center of mass of a semicircular metal plate with uniform density and thickness t. Let the radius of the plate be a. The mass of the plate is thus Use the coordinate system indicated in Fig. P8.116. M = 1 2rpa2 t.

Problem 1DQ In splitting logs with a hammer and wedge, is a heavy hammer more effective than a lighter hammer? Why?

Problem 1E (a) What is the magnitude of the momentum of a 10,000-kg truck whose speed is 12.0 m/s? (b) What speed would a 2000-kg SUV have to attain in order to have (i) the same momentum? (ii) the same kinetic energy?

Problem 2DQ Suppose you catch a baseball and then someone invites you to catch a bowling ball with either the same momentum or the same kinetic energy as the baseball. Which would you choose? Explain.

Problem 2E In a certain track and field event, the shotput has a mass of 7.30 kg and is released with a speed of 15.0 m/s at 40.0° above the horizontal over a competitor’s straight left leg. What are the initial horizontal and vertical components of the momentum of this shotput?

Problem 3DQ When rain falls from the sky, what happens to its momentum as it hits the ground? Is your answer also valid for Newton’s famous apple?

Problem 3E (a) Show that the kinetic energy K and the momentum magnitude p of a particle with mass m are related by K = p2/2m. (b) A 0.040-kg cardinal (Richmondera cardinalis) and a 0.145-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 baseball’s? (c) 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?

Problem 4DQ A car has the same kinetic energy when it is traveling south at 30 m/s as when it is traveling northwest at 30 m/s. Is the momentum of the car the same in both cases? Explain.

Problem 4E Two vehicles are approaching an intersection. One is a 2500-kg pickup traveling at 14.0 m/s from east to west (the -x-direction), and the other is a 1500-kg sedan going from south to north (the +y-direction) at 23.0 m/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?

A truck is accelerating as it speeds down the highway. One inertial frame of reference is attached to the ground with its origin at a fence post. A second frame of reference is attached to a police car that is traveling down the highway at constant velocity. Is the momentum of the truck the same in these two reference frames? Explain. Is the rate of change of the truck’s momentum the same in these two frames? Explain.

Problem 5E One 110-kg football lineman is running to the right at 2.75 m/s while another 125-kg lineman is running directly toward him at 2.60 m/s. What are (a) the magnitude and direction of the net momentum of these two athletes, and (b) their total kinetic energy?

Problem 6DQ (a) When a large car collides with a small car, which one undergoes the greater change in momentum: the large one or the small one? Or is it the same for both? (b) In light of your answer to part (a), why are the occupants of the small car more likely to be hurt than those of the large car, assuming that both cars are equally sturdy?

Problem 6E Biomechanics. The mass of a regulation tennis ball is 57 g (although it can vary slightly), and tests have shown that the ball is in contact with the tennis racket for 30 ms. (This number can also vary, depending on the racket and swing.) We shall assume a 30.0-ms contact time for this exercise. The fastest-known served tennis ball was served by “Big Bill” Tilden in 1931, and its speed was measured to be 73.14 m/s. (a) What impulse and what force did Big Bill exert on the tennis ball in his record serve? (b) If Big Bill’s opponent returned his serve with a speed of 55 m/s, what force and what impulse did he exert on the ball, assuming only horizontal motion?

Problem 7DQ A woman holding a large rock stands on a frictionless, horizontal sheet of ice. She throws the rock with speed v0 at an angle ? above the horizontal. Consider the system consisting of the woman plus the rock. Is the momentum of the system conserved? Why or why not? Is any component of the momentum of the system conserved? Again, why or why not?

Problem 7E Force of a Golf Swing. A 0.0450-kg golf ball initially at rest is given a speed of 25.0 m/s when a club strikes it. If the club and ball are in contact for 2.00 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?

In Example 8.7 (Section 8.3), where the two gliders in Fig. 8.15 stick together after the collision, the collision is inelastic because \(K_{2}<K_{1}\) In Example 8.5 (Section 8.2), is the collision inelastic? Explain. Equation Transcription: Text Transcription: K_2<K_1

Problem 8E Force of a Baseball Swing. A baseball has mass 0.145 kg. (a) If the velocity of a pitched ball has a magnitude of 45.0 m/s and the batted ball’s velocity is 55.0 m/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 ms, find the magnitude of the average force applied by the bat.

In a completely inelastic collision between two objects, where the objects stick together after the collision, is it possible for the final kinetic energy of the system to be zero? If so, give an example in which this would occur. If the final kinetic energy is zero, what must the initial momentum of the system be? Is the initial kinetic energy of the system zero? Explain.

A 0.160-kg hockey puck is moving on an icy, frictionless, horizontal surface. At t = 0, the puck is moving to the right at 3.00 m/s. (a) Calculate the velocity of the puck (magnitude and direction) after a force of 25.0 N directed to the right has been applied for 0.050 s. (b) If, instead, a force of 12.0 N directed to the left is applied from t = 0 to t = 0.050 s, what is the final velocity of the puck?

Since for a particle the kinetic energy is given by \(K=\frac{1}{2} m v^{2}\) and the momentum by \(\vec{p}=m \vec{v}\), it is easy to show that \(K=p^{2} / 2 m\) How, then, is it possible to have an event during which the total momentum of the system is constant but the total kinetic energy changes? Equation Transcription: = Text Transcription: K=\frac{1}{2} m v^{2} \vec{p}=m \vec{v} K=p^{2} / 2 m

An engine of the orbital maneuvering system (OMS) on a space shuttle exerts a force of \((26,700\mathrm{\ N})\hat{j}\) for 3.90 s, exhausting a negligible mass of fuel relative to the 95,000-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 shuttle’s change in velocity from this impulse? (d) Why can’t we find the resulting change in the kinetic energy of the shuttle?

In each of Examples 8.10, 8.11, and 8.12 (Section 8.4), verify that the relative velocity vector of the two bodies has the same magnitude before and after the collision. In each case what happens to the direction of the relative velocity vector?

Problem 11E CALC At time t = 0 a 2150-kg rocket in outer space fires an engine that exerts an increasing force on it in the +x-direction. This force obeys the equation Fx = At2, where t is time, and has a magnitude of 781.25 N when t = 1.25 s. (a) Find the SI value of the constant A , including its units. (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? Assume constant mass.

A glass dropped on the floor is more likely to break if the floor is concrete than if it is wood. Why? (Refer to Fig. 8.3b.)

Problem 12E A bat strikes a 0.145-kg baseball. Just before impact, the ball is traveling horizontally to the right at 50.0 m/s, and it leaves the bat traveling to the left at an angle of 30° above horizontal with a speed of 65.0 m/s. If the ball and bat are in contact for 1.75 ms, find the horizontal and vertical components of the average force on the ball.

In Fig. 8.22b, the kinetic energy of the Ping-Pong ball is larger after its interaction with the bowling ball than before. From where does the extra energy come? Describe the event in terms of conservation of energy.

A 2.00-kg stone is sliding to the right on a frictionless horizontal surface at 5.00 m/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. E8.13 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.

A machine gun is fired at a steel plate. Is the average force on the plate from the bullet impact greater if the bullets bounce off or if they are squashed and stick to the plate? Explain.

Problem 14E BIO Bone Fracture. Experimental tests have shown that bone will rupture if it is subjected to a force density of 1.03 X 108 N/m2. Suppose a 70.0-kg person carelessly roller-skates into an overhead metal beam that hits his forehead and completely stops his forward motion. If the area of contact with the person’s forehead is 1.5 cm2, what is the greatest speed with which he can hit the wall without breaking any bone if his head is in contact with the beam for 10.0 ms?

A net force of 4 N acts on an object initially at rest for 0.25 s and gives it a final speed of 5 m/s. How could a net force of 2 N produce the same final speed?

Problem 15E To warm up for a match, a tennis player hits the 57.0-g ball vertically with her racket. If the ball is stationary just before it is hit and goes 5.50 m high, what impulse did she impart to it?

A net force with x-component \(\Sigma F_{x}\) acts on an object from time \(t_{1}\) to time\(t_{2}\) The x-component of the momentum of the object is the same at \(t_{1}\) as it is at \(t_{2}\) but is not zero at all times between \(t_{1}\) and \(t_{2}\). What can you say about the graph of \(\Sigma F_{x}\) versus? Equation Transcription: Text Transcription: x-component \Sigma F_x t_1 t_2 x-component t_1 t_2 \Sigma F_x t_1 t_2 F_x

Starting at , a horizontal net force \(\vec{F}=(0.280 \mathrm{~N} / \mathrm{s}) t \hat{1}+\left(-0.450 \mathrm{~N} / \mathrm{s}^{2}\right) t^{2} \hat{j}\)is applied to a box that has an initial momentum \(p \overrightarrow{=}=(-3.00 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}) \hat{\mathrm{i}}+(4.00 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}) \hat{\mathrm{j}}\). What is the momentum of the box at \(t=2.00 \mathrm{~s}^{\mathrm{s}}\)? Equation Transcription: Text Transcription: t=0 \vec{F}=(0.280{~N}{s}) t \hat{1}+\left(-0.450{~N}{s}^{2}\right) t^{2} \hat{j} p \overrightarrow{=}=(-3.00{~kg} \cdot{m}{s}) \hat{{i}}+(4.00{~kg} \cdot{m}{s}) \hat{{j} t=2.00 s

Problem 17DQ A tennis player hits a tennis ball with a racket. Consider the system made up of the ball and the racket. Is the total momentum of the system the same just before and just after the hit? Is the total momentum just after the hit the same as 2 s later, when the ball is in midair at the high point of its trajectory? Explain any differences between the two cases.

Problem 17E The expanding gases that leave the muzzle of a rifle also contribute to the recoil. A .30-caliber bullet has mass 0.00720 kg and a speed of 601 m/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 m/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.

In Example 8.4 (Section 8.2), consider the system consisting of the rifle plus the bullet. What is the speed of the system’s center of mass after the rifle is fired? Explain.

Problem 18E A 68.5-kg astronaut is doing a repair in space on the orbiting space station. She throws a 2.25-kg tool away from her at 3.20 m/s relative to the space station. With what speed and in what direction will she begin to move?

Problem 19DQ An egg is released from rest from the roof of a building and falls to the ground. As the egg falls, what happens to the momentum of the system of the egg plus the earth?

Problem 19E BIO Animal Propulsion. 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-kg squid (including the water in the cavity) at rest suddenly sees a dangerous predator. (a) If the squid has 1.75 kg of water in its cavity, at what speed must it expel this water suddenly to achieve a speed of 2.50 m/s to escape the predator? Ignore any drag effects of the surrounding water. (b) How much kinetic energy does the squid create by this maneuver?

Problem 20DQ A woman stands in the middle of a perfectly smooth, frictionless, frozen lake. She can set herself in motion by throwing things, but suppose she has nothing to throw. Can she propel herself to shore without throwing anything?

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-kg ball that is traveling horizontally at 10.0 m/s. Your mass is 70.0 kg. (a) If you catch the ball, with what speed do you and the ball move afterward? (b) If the ball hits you and bounces off your chest, so afterward it is moving horizontally at 8.0 m/s in the opposite direction, what is your speed after the collision?

Problem 21E On a frictionless, horizontal air table, puck A (with mass 0.250 kg) is moving toward puck B (with mass 0.350 kg), which is initially at rest. After the collision, puck A has a velocity of 0.120 m/s to the left, and puck B has a velocity of 0.650 m/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.

Problem 22DQ When an object breaks into two pieces (explosion, radio-active decay, recoil, etc.), the lighter fragment gets more kinetic energy than the heavier one. This is a consequence of momentum conservation, but can you also explain it by using Newton’s laws of motion?

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-kg car traveling to the right at 1.50 m/s collides with a 1450-kg car going to the left at 1.10 m/s. Measurements show that the heavier car’s speed just after the collision was 0.250 m/s in its original direction. 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

Problem 23DQ An apple falls from a tree and feels no air resistance. As it is falling, which of these statements about it are true? (a) Only its momentum is conserved; (b) only its mechanical energy is conserved; (c) both its momentum and its mechanical energy are conserved; (d) its kinetic energy is conserved.

Problem 23E Two identical 1.50-kg masses are pressed against opposite ends of a light spring of force constant 1.75 N/cm, compressing the spring by 20.0 cm from its normal length. Find the speed of each mass when it has moved free of the spring on a frictionless horizontal table.

Problem 24DQ Two pieces of clay collide and stick together. During the collision, which of these statements are true? (a) Only the momentum of the clay is conserved; (b) only the mechanical energy of the clay is conserved; (c) both the momentum and the mechanical energy of the clay are conserved; (d) the kinetic energy of the clay is conserved.

Block A in Fig. E8.24 has mass \(1.00 \mathrm{~kg}\), and block B has mass \(3.00 \mathrm{~kg}\). 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? (b) How much potential energy was stored in the compressed spring? Equation Transcription: Text Transcription: 1.00 kg 3.00 kg 1.20 m/s

Problem 25DQ Two marbles are pressed together with a light ideal spring between them, but they are not attached to the spring in any way. They are then released on a frictionless horizontal table and soon move free of the spring. As the marbles are moving away from each other, which of these statements about them are true? (a) Only the momentum of the marbles is conserved, (b) only the mechanical energy of the marbles is conserved, (c) both the momentum and the mechanical energy of the marbles are conserved, (d) the kinetic energy of the marbles is conserved.

Problem 25E A hunter on a frozen, essentially frictionless pond uses a rifle that shoots 4.20-g bullets at 965 m/s. The mass of the hunter (including his gun) is 72.5 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° above the horizontal.

Problem 26DQ A very heavy SUV collides head-on with a very light compact car. Which of these statements about the collision are correct? (a) The amount of kinetic energy lost by the SUV is equal to the amount of kinetic energy gained by the compact; (b) the amount of momentum lost by the SUV is equal to the amount of momentum gained by the compact; (c) the compact feels a considerably greater force during the collision than the SUV does; (d) both cars lose the same amount of kinetic energy.

Problem 26E An atomic nucleus suddenly bursts apart (fissions) into two pieces. Piece A, of mass mA, travels off to the left with speed vA. Piece B, of mass mB, travels off to the right with speed vB. (a) Use conservation of momentum to solve for v B in terms of mA, mB, and vA. (b) Use the results of part (a) to show that KA/KB = mB/mA, where KA and KB are the kinetic energies of the two pieces.

Problem 27E Two ice skaters, Daniel (mass 65.0 kg) and Rebecca (mass 45.0 kg), are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 m/s before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 m/s at an angle of 53.1o 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?

Problem 28E 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 m/s relative to the earth at an angle of 35.0o above the horizontal. If your mass is 70.0 kg and the rock’s mass is 3.00 kg, what is your speed after you throw the rock? (See Discussion Question Q8.7.) Q8.7 A woman holding a large rock stands on a frictionless, horizontal sheet of ice. She throws the rock with speed v0 at an angle ? above the horizontal. Consider the system consisting of the woman plus the rock. Is the momentum of the system conserved? Why or why not? Is any component of the momentum of the system conserved? Again, why or why not?

Problem 29E Changing Mass. An open-topped freight car with mass 24,000 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 m/s. (a) What is the speed of the car after it has collected 3000 kg of rainwater? (b) Since the rain is falling downward, how is it able to affect the horizontal motion of the car?

An astronaut in space cannot use a conventional means, such as a scale or balance, to determine the mass of an object. But she does have devices to measure distance and time accurately. She knows her own mass is 78.4 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 m/s, she pushes against it, which slows it down to 1.20 m/s (but does not reverse it) and gives her a speed of 2.40 m/s. What is the mass of this canister?

Asteroid Collision. 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, which was initially at rest, travels at \(45.0^{\circ}\) to the original direction of A (Fig. E8.31). (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? Equation Transcription: ° ° Text Transcription: 40.o m/s 30.0^° 45.0^°

Problem 32E Two skaters collide and grab on to each other on frictionless ice. One of them, of mass 70.0 kg, is moving to the right at 4.00 m/s, while the other, of mass 65.0 kg, is moving to the left at 2.50 m/s. What are the magnitude and direction of the velocity of these skaters just after they collide?

Problem 33E A 15.0-kg fish swimming at 1.10 m/s suddenly gobbles up a 4.50-kg fish that is initially stationary. Ignore 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?

Problem 34E Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 kg, is sliding to the left at 5.00 m/s, while the other, of mass 5.75 kg, is slipping to the right at 6.00 m/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?

Problem 35E Deep Impact Mission. In July 2005, NASA’s “Deep Impact” mission crashed a 372-kg probe directly onto the surface of the comet Tempel 1, hitting the surface at 37,000 km/h. The original speed of the comet at that time was about 40,000 km/h, and its mass was estimated to be in the range (0.10 - 2.5) X 1014 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 noticeable? (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 X 1024 kg.)

Problem 36E A 1050-kg sports car is moving westbound at 15.0 m/s on a level road when it collides with a 6320-kg truck driving east on the same road at 10.0 m/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 both it and the car are 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?

Problem 37E On a very muddy football field, a 110-kg linebacker tackles an 85-kg halfback. Immediately before the collision, the linebacker is slipping with a velocity of 8.8 m/s north and the half-back is sliding with a velocity of 7.2 m/s east. What is the velocity (magnitude and direction) at which the two players move together immediately after the collision?

Problem 38E Accident Analysis. Two cars collide at an intersection. Car A , with a mass of 2000 kg, is going from west to east, while car B , of mass 1500 kg, is going from north to south at 15 m/s. As a result, the two cars become enmeshed and move as one. As an expert witness, you inspect the scene and determine that, after the collision, the enmeshed cars moved at an angle of 65° south of east from the point of impact. (a) How fast were the enmeshed cars moving just after the collision? (b) How fast was car A going just before the collision?

Problem 39E Two cars, one a compact with mass 1200kg and the other a large gas-guzzler with mass 3000 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 target car changes its velocity by v, calculate the change in the velocity of the small car in terms of v. (c) Which car’s occupants would you expect to sustain greater injuries? Explain.

Problem 40E BIO Bird Defense. To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a 600-g falcon flying at 20.0 m/s hit a 1.50-kg raven flying at 9.0 m/s. The falcon hit the raven at right angles to its original path and bounced back at 5.0 m/s. (These figures were estimated by the author as he watched this attack occur in northern New Mexico.) (a) By what angle did the falcon change the raven’s direction of motion? (b) What was the raven’s speed right after the collision?

At the intersection 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 has run a red light (Fig. E8.41). The two vehicles stick together as a result of the collision, 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. Equation Transcription: ° Text Transcription: 950 kg 1900 kg 16.0 m/s 24.0^°

Problem 42E A 5.00-g bullet is fired horizontally into a 1.20-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 0.310 m along the surface before stopping. What was the initial speed of the bullet?

A Ballistic Pendulum. A \(text { 12.0-g }\) rifle bullet is fired with a speed of \(380 \mathrm{~m} / \mathrm{s}\) into a ballistic pendulum with mass \(6.00 \mathrm{~kg}\), 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 pendulum. Equation Transcription: Text Transcription: 12.0-g 380 m/s 6.00 kg 70.0 cm

Combining Conservation Laws. A \(15.0-\mathrm{kg}\) block is attached to a very light horizontal spring of force constant \(500.0 \mathrm{~N} / \mathrm{m}\) and is resting on a frictionless horizontal table. (Fig. E8.44). Suddenly it is struck by a \(\text { 3.00-kg }\) stone traveling horizontally at \(8.00 \mathrm{~m} / \mathrm{s}\) to the right, whereupon the stone rebounds at \(2.00 \mathrm{~m} / \mathrm{s}\)horizontally to the left. Find the maximum distance that the block will compress the spring after the collision. Equation Transcription: Text Transcription: 15.0-kg 500.0 N/m 3.00-kg 8.00 m/s 2.00 m/s

Problem 45E A 5.00-kg ornament is hanging by a 1.50-m wire when it is suddenly hit by a 3.00-kg missile traveling horizontally at 12.0 m/s. The missile embeds itself in the ornament during the collision. What is the tension in the wire immediately after the collision?

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

Blocks A (mass \(2.00 \mathrm{~kg} \text {. }\)) and B (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 (Section 8.4). 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. Equation Transcription: Text Transcription: 2.00 kg 10.00 kg 2.00 m/s

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-g marble sliding to the right with a velocity of magnitude 0.200 \(\mathrm{~m} / \mathrm{s}\) (Fig. E8.48). (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. Equation Transcription: Text Transcription: 10.0-g m/s 30.0-g m/s

Moderators. Canadian nuclear reactors use heavy water moderators in which elastic collisions occur between the neutrons and deuterons of mass \(2.0 \mathrm{u}\) (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? Equation Transcription: Text Transcription: 2.0 u

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?

Problem 51E Three odd-shaped blocks of chocolate have the following masses and center-of-mass coordinates: (1) 0.300 kg, (0.200 m,0.300 m); (2) 0.400 kg, (0.100 m, - 0.400 m); (3) 0.200 kg, (-0.300 m, 0.600 m). Find the coordinates of the center of mass of the system of three chocolate blocks.

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 outside the sun? Use the data in Appendix F.

Problem 53E Pluto and Charon. Pluto’s diameter is approximately 2370 km, and the diameter of its satellite Charon is 1250 km. Although the distance varies, they are often about 19,700 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.

A \(1200-\mathrm{kg}\) station wagon is moving along a straight highway at 12.0\(\mathrm{m} / \mathrm{s}\). Another car, with mass 1800 \(\mathrm{kg}\) and speed 20.0 \(\mathrm{m} / \mathrm{s}\), has its center of mass 40.0 m ahead of the center of mass of the station wagon (Fig. E8.54). (a) Find the position of the center of mass of the system consisting of the two automobiles. (b) Find the magnitude of the total momentum of the system from the given data. (c) Find the speed of the center of mass of the system. (d) Find the total momentum of the system, using the speed of the center of mass. Compare your result with that of part (b). Equation Transcription: Text Transcription: 1200-kg m/s kg m/s

A machine part consists of a thin, uniform \(\text { 4.00-kg }\) 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 m. The longer bar has a small but dense \(2.00-\mathrm{kg}\) ball at one end (Fig. E8.55). By what distance will the center of mass of this part move horizontally and vertically if the vertical bar is pivoted counterclockwise through \(90^{\circ}\) to make the entire part horizontal? Equation Transcription: ° Text Transcription: 4.00-kg 1.50 m 3.00 kg 1.80 m 2.00 kg 90Degree°

At one instant, the center of mass of a system of two particles is located on the x-axis at x = 2.0 m and has a velocity of \((5.0\mathrm{\ m}/\mathrm{s})\hat{i}\). One of the particles is at the origin. The other particle has a mass of 0.10 kg and is at rest on the x-axis at x = 8.0 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 par-ticle at the origin?

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? Equation Transcription: Text Transcription: 0.70 m/s

A system consists of two particles. At t = 0 one particle is at the origin; the other, which has a mass of 0.50 kg, is on the y-axis at y = 6.0 m. At t = 0 the center of mass of the system is on the y-axis at y = 2.4 m. The velocity of the center of mass is given by \(\left(0.75 \mathrm{\ m} / \mathrm{s}^{3}\right) t^{2} \hat{\imath}\). (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 s.

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

Changing Your Center of Mass. To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 92.0 cm long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a 70.0-kg person, the mass of the upper leg would be 8.60 kg, while that of the lower leg (including the foot) would be 5.25 kg. Find the location of the center of mass of this leg, relative to the hip joint, if it is (a) stretched out horizontally and (b) bent at the knee to form a right angle with the upper leg remaining horizontal.

Problem 61E A 70-kg astronaut floating in space in a 110-kg MMU (manned maneuvering unit) experiences an acceleration of 0.029 m/s2 when he fires one of the MMU’s thrusters. (a) If the speed of the escaping N2 gas relative to the astronaut is 490 m/s, how much gas is used by the thruster in 5.0 s? (b) What is the thrust of the thruster?

Problem 62E A small rocket burns 0.0500 kg of fuel per second, ejecting it as a gas with a velocity relative to the rocket of magnitude 1600 m/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?

Problem 64E 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 m/s and you want the rocket’s speed eventually to be 1.00 X 10-3c, where c is the speed of light in vacuum, 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 m/s?

Problem 65E 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 s and the relative speed of the exhaust gas is vex = 2100 m/s, what must the mass ratio m0/m be for a final speed v of 8.00 km/s (about equal to the orbital speed of an earth satellite)?

A young girl with mass 40.0 kg is sliding on a horizontal, frictionless surface with an initial momentum that is due east and that has magnitude \(90.0 \mathrm{\ kg} \cdot \mathrm{m} / \mathrm{s}\). Starting at t = 0, a net force with magnitude F = (8.20 N/s)t and direction due west is applied to the girl. (a) At what value of t does the girl have a westward momentum of magnitude \(60.0 \mathrm{\ kg} \cdot \mathrm{m} / \mathrm{s}\)? (b) How much work has been done on the girl by the force in the time interval from t = 0 to the time calculated in part (a)? (c) What is the magnitude of the acceleration of the girl at the time calculated in part (a)?

A steel ball with mass 40.0 g is dropped from a height of 2.00 m onto a horizontal steel slab. The ball rebounds to a height of 1.60 m. (a) Calculate the impulse delivered to the ball during impact. (b) If the ball is in contact with the slab for 2.00 ms, find the average force on the ball during impact.

Problem 68P 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 318 m directly north of the point of the explosion. Where will the other fragment land? Neglect any air resistance.

Just before it is struck by a racket, a tennis ball weighing 0.560 N has a velocity of \((20.0 \mathrm{~m} / \mathrm{s}) \hat{\boldsymbol{\imath}}-(4.0 \mathrm{~m} / \mathrm{s}) \hat{\boldsymbol{J}}\). During the 3.00 ms that the racket and ball are in contact, the net force on the ball is constant and equal to \(-(380 \mathrm{~N}) \hat{\boldsymbol{\imath}}+(110 \mathrm{~N}) \hat{\boldsymbol{\jmath}}\). (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?

Problem 70P 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?

Problem 71P A 1500-kg blue convertible is traveling south, and a 2000-kg red SUV is traveling west. If the total momentum of the system consisting of the two cars is 7200 kg ? m/s directed at 60.0° west of south, what is the speed of each vehicle?

Problem 72P 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 kg and is traveling east with a velocity of magnitude 5.00 m/s. Find the final velocity of the car in each case, assuming that the handcar does not leave the tracks. (a) A 25.0-kg mass is thrown sideways out of the car with a velocity of magnitude 2.00 m/s relative to the car’s initial velocity. (b) A 25.0-kg mass is thrown backward out of the car with a velocity of 5.00 m/s relative to the initial motion of the car. (c) A 25.0-kg mass is thrown into the car with a velocity of 6.00 m/s relative to the ground and opposite in direction to the initial velocity of the car.

Spheres \(A\) (mass \(0.020 \mathrm{~kg}\) ), \(B\) (mass \(0.030 \mathrm{~kg}\) ), and \(C\) (mass \(0.050 \mathrm{~kg}\) ) are approaching the origin as they slide on a frictionless air table (Fig. P8.73). 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 in the 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?

Problem 74P You and your friends are doing physics experiments on a frozen pond that serves as a frictionless, horizontal surface. Sam, with mass 80.0 kg, is given a push and slides eastward. Abigail, with mass 50.0 kg, is sent sliding northward. They collide, and after the collision Sam is moving at 37.0o north of east with a speed of 6.00 m/s and Abigail is moving at 23.0o south of east with a speed of 9.00 m/s. (a) What was the speed of each per-son before the collision? (b) By how much did the total kinetic energy of the two people decrease during the collision?

Problem 75P The nucleus of 214Po decays radioactively by emitting an alpha particle (mass 6.65 × 10?27 kg) with kinetic energy 1.23 × 10?12 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 nucleus that remains after the decay.

At a classic auto show, a 840-kg 1955 Nash Metropolitan motors by 9.0 m/s at followed by a 1620-kg 1957 Packard Clipper purring past at 5.0 m/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 \(F_{\mathrm{N}}\) 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 \(F_{\mathrm{N}} / F_{\mathrm{p}}\)?

Problem 77P CP An 8.00-kg block of wood sits at the edge of a frictionless table, 2.20 m above the floor. A 0.500-kg blob of clay slides along the length of the table with a speed of 24.0 m/s, strikes the block of wood, and sticks to it. The combined object leaves the edge of the table and travels to the floor. What horizontal distance has the combined object traveled when it reaches the floor?

Combining Conservation Laws. A 5.00-kg chunk of ice is sliding at 12.0 m/s on the floor of an ice-covered valley when it collides with and sticks to another 5.00-kg chunk of ice that is initially at rest. (Fig. P8.79). Since the valley is icy, there is no friction. After the collision, how high above the valley floor will the combined chunks go?

Problem 80P Automobile Accident Analysis. You are called as an expert witness to analyze the following auto accident: Car B, of mass 1900 kg, was stopped at a red light when it was hit from behind by car A, of mass 1500 kg. The cars locked bumpers during the collision and slid to a stop with brakes locked on all wheels. Measurements of the skid marks left by the tires showed them to be 7.15 m long. The coefficient of kinetic friction between the tires and the road was 0.65. (a) What was the speed of car A just before the collision? (b) If the speed limit was 35 mph, was car A speeding, and if so, by how many miles per hour was it exceeding the speed limit?

Problem 81P Accident Analysis. A 1500-kg sedan goes through a wide intersection traveling from north to south when it is hit by a 2200-kg SUV traveling from east to west. The two cars become enmeshed due to the impact and slide as one thereafter. On-the-scene measurements show that the coefficient of kinetic friction between the tires of these cars and the pavement is 0.75, and the cars slide to a halt at a point 5.39 m west and 6.43 m south of the impact point. How fast was each car traveling just before the collision?

A 0.150-\(\mathrm{kg}\) frame, when suspended from a coil spring, stretches the spring 0.070 m. A 0.200-kg lump of putty is dropped from rest onto the frame from a height of 30.0 \(\mathrm{cm}\) (Fig. P8.82). Find the maximum distance the frame moves downward from its initial position. Equation Transcription: Text Transcription: kg cm 30.0 cm

A rifle bullet with mass \(8.00 \mathrm{~g}\) strikes and embeds itself in a block with mass \(0.992 \mathrm{~kg}\) that rests on a frictionless, horizontal surface and is attached to a coil spring (Fig. P8.83). The impact compresses the spring \(15.0 \mathrm{~cm}\). Calibration of the spring shows that a force of \(0.750 \mathrm{~N}\) is required to compress the spring \(0.250 \mathrm{~cm}\). (a) Find the magnitude of the block’s velocity just after impact. (b) What was the initial speed of the bullet? Equation Transcription: Text Transcription: 8.00 g 0.992 kg 15.0 cm 0.750 N 0.250 cm

Problem 84P A Ricocheting Bullet. A 0.100-kg stone rests on a frictionless, horizontal surface. A bullet of mass 6.00 g, traveling horizontally at 350 m/s, strikes the stone and rebounds horizontally at right angles to its original direction with a speed of 250 m/s. (a) Compute the magnitude and direction of the velocity of the stone after it is struck. (b) Is the collision perfectly elastic?

A movie stuntman (mass \(80.0 \mathrm{~kg}\)) stands on a window ledge \(5.0 \mathrm{~m}\) above the floor (Fig. P8.85). 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 \(5.0 \mathrm{~m} \text {. }\). He releases the rope just as he reaches the villain.) (a) With what speed do the entwined foes start to slide across the floor? (b) If the coefficient of kinetic friction of their bodies with the floor is \(\mu_{\mathrm{k}}\) how far do they slide? Equation Transcription: Text Transcription: 80.0 kg 5.0 m 70.0 kg 5.0 m \mu k

Two identical masses are released from rest in a smooth hemispherical bowl of radius R from the positions shown in Fig. P8.86. 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?

Problem 87P A ball with mass M, moving horizontally at 4.00 m/s, collides elastically with a block with mass 3 M that is initially hanging at rest from the ceiling on the end of a 50.0-cm wire. Find the maximum angle through which the block swings after it is hit.

Problem 88P CP A 20.00-kg lead sphere is hanging from a hook by a thin wire 2.80 m long and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00-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?

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

Problem 90P A 7.0-kg shell at rest explodes into two fragments, one with a mass of 2.0 kg and the other with a mass of 5.0 kg. If the heavier fragment gains 100 J of kinetic energy from the explosion, how much kinetic energy does the lighter one gain?

Problem 91P A 4.00-g bullet, traveling horizontally with a velocity of magnitude 400 m/s, is fired into a wooden block with mass 0.800 kg, initially at rest on a level surface. The bullet passes through the block and emerges with its speed reduced to 190 m/s. The block slides a distance of 45.0 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 life block at the instant after the bullet passes through it?

Problem 92P A 5.00-g bullet is shot through a 1.00-kg wood block suspended on a string 2.00 m long. The center of mass of the block rises a distance of 0.38 cm. Find the speed of the bullet as it emerges from the block if its initial speed is 450 m/s.

Problem 93P 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 K0, the kinetic energy that it loses during the collision is 4mMK0/(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?

In a shipping company distribution center, an open cart of mass \(50.0 \mathrm{~kg}\) is rolling to the left at a speed of \(5.00 \mathrm{~m} / \mathrm{s}\) (Fig. P8.95). You can ignore friction between the cart and the floor. A \(\text { 15.0-kg }\) 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? Equation Transcription: ° Text Transcription: 50.0 kg 5.00 m/s 15.0-kg 37 degree 3.00m/s 4.00 m

A blue puck with mass 0.0400 kg, sliding with a velocity of magnitude 0.200 m/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 m/s in the same direction as its initial velocity. Find (a) the velocity (magnitude and direction) of the red puck after the collision and (b) the mass m of the red puck.

Problem 97P Jack and Jill are standing on a crate at rest on the frictionless; horizontal surface of a frozen pond. Jack has mass 75.0 kg. Jill has mass 45.0 kg, and the crate has mass 15.0 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 m/s relative to the crate. (a) What is the final speed of the crate if both Jack and Jill jump simultaneously and 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 Jill jumps first and then Jack, again in the same direction?

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 elastic 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 \(\vec v\) and the small ball has velocity - \(\vec v\) (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 ball? (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? Equation Transcription: Text Transcription: vector v

Problem 99P 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 m/s and is deflected 25.0° from its initial direction. Assume that the collision is perfectly elastic. Find the final speed of each puck and the direction of B’s velocity after the collision.

Problem 100P Energy Sharing. An object with mass m, initially at rest, explodes into two fragments, one with mass mA and the other with mass mB, where mA + mB = m. (a) If energy Q is released in the explosion, how much kinetic energy does each fragment have immediately after the explosion? (b) What percentage of the total energy released does each fragment get when one fragment has four times the mass of the other?

Problem 101P Neutron Decay. 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?

Problem 102P A 232Th (thorium) nucleus at rest decays to a 228Ra (radium) nucleus with the emission of an alpha particle. The total kinetic energy of the decay fragments is 6.54 × 10?13 J. An alpha particle has 1.76% of the mass of a 228Ra nucleus. Calculate the kinetic energy of (a) the recoiling 228Ra nucleus and (b) the alpha particle.

Antineutrino. In beta decay, a nucleus emits an electron. \(A{ }^{210} \mathrm{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 \(\text { 210 Po }\) 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. Equation Transcription: Text Transcription: A ^210Bi 210^Po 5.60 times 10-22kg dot m/s 240^Po 3.50 times 10^-25kg 1.14 times 10^3 m/s

Problem 104P Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathan’s weight is 800 N, Jane’s weight is 600 N, and that of the sleigh is 1000 N. They 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 m/s at 30.0o above the horizontal (relative to the ice), and Jane jumps to the right at 7.00 m/s at 36.9o above the horizontal (relative to the ice). Calculate the sleigh’s horizontal velocity (magnitude and direction) after they jump out.

Problem 105P Friends Burt and Ernie stand at opposite ends of a uniform log that is floating in a lake. The log is 3.0 m long and has mass 20.0 kg. Burt has mass 30.0 kg; Ernie has mass 40.0 kg. Initially, the log and the two friends are at rest relative to the shore. Burt then offers Ernie a cookie, and Ernie walks to Burt’s end of the log to get it. Relative to the shore, what distance has the log moved by the time Ernie reaches Burt? Ignore any horizontal force that the water exerts on the log, and assume that neither friend falls off the log.

A \(\text { 45.0-kg }\) 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. P8.106). If you ignore resistance to motion of the canoe in the water, how far does the canoe move during this process? Equation Transcription: Text Transcription: 45.0-kg 60.0-kg 5.00 m 1.00 m 1.00 m

Problem 107P 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 m/s relative to the ice, with what speed, relative to the ice, does the slab move?

Problem 110P A 12.0-kg shell is launched at an angle of 55.0o above the horizontal with an initial speed of 150 m/s. At its highest point, the shell explodes into two fragments, one three times heavier than the other. The two fragments reach the ground at the same time. Ignore air resistance. If the heavier fragment lands back at the point from which the shell was launched, where will the lighter fragment land, and how much energy was released in the explosion?

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. P8.111). 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? Equation Transcription: ° Text Transcription: 300 kg 50 m 6.0 degree 40 m 75.0 kg 60.0 kg 5.0 s

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 of the rocket, analogous to Eq. (8.39). (c) What is the acceleration of the rocket in Example 8.15 (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?

Problem 113P A Multistage Rocket. Suppose the first stage of a two stage rocket has total mass 12,000 kg, of which 9000 kg is fuel. The total mass of the second stage is 1000 kg, of which 700 kg is fuel. Assume that the relative speed vex 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 kg. In terms of vex, 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 vex is required to give the second stage of the rocket a speed of 7.00 km/s?

A Variable-Mass Raindrop. 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 that it has fallen. Then (m=k x\) where is a constant, and \(d m / d t=k v\) This gives, since \(F_{e x t}=m g\) m g=m \frac{d v}{d t}+v(k v) Or, dividing by , This is a differential equation that has a solution of the form where is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for , find the acceleration . (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 K. S. Krane, American Journal of Physics, Vol. 49 (1981), pp. 113–117.) Equation Transcription: Text Transcription: F_\ext=\frac d p d t= m \frac d vd t+v \frac d m d t M = kx d m / d t=k v \(F_e x t=m g\) m g=m \frac{d v}{d t}+v(k v) v=at t=3.00 s k=2.00 g/m t=3.00 s

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 \(x_{\mathrm{cm}}=\frac{1}{M} \int x d m \quad y_{\mathrm{cm}}=\frac{1}{M} \int y d m\) where x and y are the coordinates of the small piece of the object that has mass dm. 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 the object is uniform, perform the integration described above to 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 is a positive constant—calculate the x-coordinate of the rod’s center of mass. Equation Transcription: Text Transcription: x_cm=1M \quad xdm y_cm=1M \quad ydm \rho =M/V \rho =ax

Use the methods of Challenge Problem 8.115 to calculate the x- and y-coordinates of the center of mass of a semicircular metal plate with uniform density \(\rho\) and thickness . Let the radius of the plate be . The mass of the plate is thus \(M=\frac{1}{2} \rho \pi a^{2} t\) Use the coordinate system indicated in Fig. P8.116. Equation Transcription: Text Transcription: \rho t a M=\frac{1}{2} \rho \pi a^{2} t

Problem 21DQ In a zero-gravity environment, can a rocket-propelled spaceship ever attain a speed greater than the relative speed with which the burnt fuel is exhausted?

Problem 63E A C6-5 model rocket engine has an impulse of 10.0 N·s while burning 0.0125 kg of propellant in 1.70 s. It has a maximum thrust of 13.3 N. The initial mass of the engine plus propellant is 0.0258 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.

Problem 78P CP A small wooden block with mass 0.800 kg is suspended from the lower end of a light cord that is 1.60 m long. The block is initially at rest. A bullet with mass 12.0 g is fired at the block with a horizontal velocity v0. The bullet strikes the block and becomes embedded in it. After the collision the combined object swings on the end of the cord. When the block has risen a vertical height of 0.800 m, the tension in the cord is 4.80 N. What was the initial speed v0 of the bullet?

Problem 94P Energy Sharing in Elastic Collisions. A stationary object with mass mB is struck head-on by an object with mass mA that is moving initially at speed v0. (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 (i) mA = mB and (ii) mA = 5mB? (c) For what values, if any, of the mass ratio mA/mB is the original kinetic energy shared equally by the two objects after the collision?

Problem 108P CP A 20.0-kg projectile is fired at an angle of 60.0o above the horizontal with a speed of 80.0 m/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. 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?

Problem 109P CP A fireworks rocket is fired vertically upward. At its maximum height of 80.0 m, it explodes and breaks into two pieces: one with mass 1.40 kg and the other with mass 0.28 kg. In the explosion, 860 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.