1 A railroad freight car weighing 280 kn and traveling at 1.52 m/s overtakes one weighing 210 kn and traveling at 0.914 m/s in the same direction. (a) If the cars couple together, find the speed of the cars after the collision. (b) If the cars couple together, find the loss of kinetic energy during the collision. (c) If instead, as is very unlikely, the collision is elastic, find the after-collision speed of the lighter car. (d) If instead, as is very unlikely, the collision is elastic, find the after-collision speed of the heavier car. 4 How fast can you set the Earth moving? In particular, when you jump straight up as high as you can, what is the order of magnitude of the maximum recoil speed that you give to the Earth? Model the Earth as a perfectly solid object. In your solution, state the physical quantities you take as data, and the values you measure or estimate for them. (a) 0; (b) 2.25 kj; (b) 2.25 kj; (c) 1.61 m/s; (d) 1.00 m/s 5 A particle of mass m moves with momentum p. (a) Show that the kinetic energy of the particle is K = p 2 /2m. (b) Express the magnitude of the particle s momentum in terms of its kinetic energy and mass. 2 A block of mass m 1 is at rest on a long frictionless table, one end of which is terminated by a wall. Another block of mass m 2 is placed between the first block and the wall and set in motion to the left with a constant speed v 2i as in the figure. Assuming that all collisions are completely elastic, find the value of m 2 for which both blocks move with the same velocity after m 2 has collided once with m 1 and once with the wall. Assume the wall to have infinite mass. 6 High-speed stroboscopic photographs show that the head of a golf club of mass 200 g is traveling at 55.0 m/s just before it strikes a 46.0-g golf ball at rest on a tee. After the collision, the club head travels (in the same direction) at 40.0 m/s. Find the speed of the golf ball just after impact. m 1 /3 7 An archer shoots an arrow towards a target that is sliding towards her with a speed of 2.50 m/s on a smooth, slippery surface. The 22.5-g arrow is shot with a speed of 35.0 m/s and passes through the 300-g target, which is stopped by the impact. What is the speed of the arrow after passing through the target? 3 A 0.100 kg ball is thrown straight up into the air with an initial speed of 15.0 m/s. (a) Find the momentum of the ball at its maximum height (b) Find the momentum of the ball halfway up to its maximum height.
8 A 10.0-g bullet is fired into a stationary block of wood (m = 5.00 kg). The relative motion of the bullet stops inside the block. The speed of the bullet-plus-wood combination immediately after the collision is 0.600 m/s. What was the original speed of the bullet? 12 A neutron in a nuclear reactor makes an elastic head-on collision with the nucleus of a carbon atom initially at rest. (a) What fraction of the neutron's kinetic energy is transferred to the carbon nucleus? (b) If the initial kinetic energy of the neutron is 1.60? 10 13 J, find its final kinetic energy and the kinetic energy of the carbon nucleus after the collision. Note: (The mass of the carbon nucleus is nearly 12.0 times the mass of the neutron.) 9 A railroad car of mass 2.50? 10 4 kg is moving with a speed of 4.00 m/s. It collides and couples with three other coupled railroad cars, each of the same mass as the single car and moving in the same direction with an initial speed of 2.00 m/s. (a) What is the speed of the four cars after the collision? (b) How much mechanical energy is lost in the collision 13 A 7.00-g bullet, when fired from a gun into a 1.00 kg block of wood held in a vise, penetrates the block to a depth of 8.00 cm. What if This block of wood is placed on a frictionless horizontal surface, and a second 7.00-g bullet is fired from the gun into the block. To what depth will the bullet penetrate the block in this case? 10 Four railroad cars, each of mass 2.50 x 10 4 kg, are coupled together and coasting along horizontal tracks at speed vi toward the south. A very strong but foolish movie actor, riding on the second car, uncouples the front car and gives it a big push, increasing its speed to 4.00 m/s southward. The remaining three cars continue moving south, now at 2.00 m/s. (a) Find the initial speed of the cars. (b) How much work did the actor do? (c) State the relationship between the process described here and the process in which the uncoupled car is moving toward the other three cars. 14 Three carts of masses 4.00 kg, 10.0 kg, and 3.00 kg move on a frictionless horizontal track with speeds of 5.00 m/s, 3.00 m/s, and 4.00 m/s, as shown in the figure. Velcro couplers make the carts stick together after colliding. (a) Find the final velocity of the train of three carts. (b) Does your answer require that all the carts collide and stick together at the same time? (c) What if they collide in a different order? 11 A 45.0 kg girl is standing on a plank that has a mass of 150 kg. The plank, originally at rest, is free to slide on a frozen lake, which is a flat, frictionless supporting surface. The girl begins to walk along the plank at a constant speed of 1.50 m/s relative to the plank. (a) What is her speed relative to the ice surface? (b) What is the speed of the plank relative to the ice surface? 15 A proton, moving with a velocity of v i, collides elastically with another proton that is initially at rest. The two protons have equal speeds after the collision (a) Find the speed of each proton after the collision in terms of v i (b) Find the direction of the velocity vectors after the collision.
16 17 An object of mass 3.00 kg, with an initial velocity of 5i m/s, collides with and sticks to an object of mass 2.00 kg, with an initial velocity of -3i m/s. Find the final velocity of the composite object. Two particles with masses m and 3m are moving toward each other along the x axis with the same initial speeds v i. Particle m is traveling to the left, while particle 3m is traveling to the right. They undergo an elastic glancing collision such that particle m is moving downward after the collision at right angles from its initial direction. (a) Find the final speeds of the two particles. (b) What is the angle at which the particle 3 m is scattered? 19 the following (a) What is the final linear momentum of the target in the figure if the initial linear momentum of the projectile is 6 kg. m/s and the final linear momentum of the projectile is 2 kg. m/s? (b) What is the final linear momentum of the target in the figure if the initial linear momentum of the projectile is 6 kg. m/s and the final linear momentum of the projectile is -2 kg. m/s? (c) What is the final kinetic energy of the target if the initial and final kinetic energies of the projectile are, respectively, 5 J and 2 J? 18 An 80.0 kg astronaut is working on the engines of his ship, which is drifting through space with a constant velocity. The astronaut, wishing to get a better view of the Universe, pushes against the ship and much later finds himself 30.0 m behind the ship. Without a thruster, the only way to return to the ship is to throw his 0.500 kg wrench directly away from the ship. If he throws the wrench with a speed of 20.0 m/s relative to the ship, how long does it take the astronaut to reach the ship? 20 (a) 4 kg.m/s; (b) 8 kg.m/s; (c) 3 J A block slides along a frictionless floor and into a stationary second block with the same mass. The figure shows four choices for a graph of the kinetic energies K of the blocks, before and after the collision. (a) Determine which choices represent physically impossible situations; explain why. (b) Of the possible choices, which best represents an elastic collision? (c) Of the possible choices, which best represents an inelastic collision? (a) c, kinetic energy cannot be negative; d, total energy cannot increase; (b) a; (c) b
21 In the figure, particle 1 of mass m 1 = 0.30 kg slides rightward along an x axis on a frictionless floor with a speed of 2.0 m/s. When it reaches x = 0, it undergoes a one-dimensional elastic collision with stationary particle 2 of mass m 2 = 0.40 kg. When particle 2 then reaches a wall at x w = 70 cm, it bounces from the wall with no loss of speed. At what position on the x axis does particle 2 then collide with particle 1? 24 A small ball of mass m is aligned above a larger ball of mass M = 0.63 kg (with a slight separation, as with the baseball and basketball of figure a), and the two are dropped simultaneously from height h = 1.8 m. (Assume the radius of each ball is negligible relative to h.) (a) If the larger ball rebounds elastically from the floor and then the small ball rebounds elastically from the larger ball, what value of m results in the larger ball stopping when it collides with the small ball? (b) What height does the small ball then reach (figure b)? -28 cm 22 A body of mass 2.0 kg makes an elastic collision with another body at rest and continues to move in the original direction but with one-fourth of its original speed. (a) What is the mass of the other body? (b) What is the speed of the two-body center of mass if the initial speed of the 2.0 kg body was 4.0 m/s? (a) 0.21 kg; (b) 7.2 m (a) 1.2 kg; (b) 2.5 m/s 25 In the figure, block 1 of mass m 1 slides along an x axis on a frictionless floor with a speed of vj 1i = 4.00 m/s. Then it undergoes a one-dimensional elastic collision with stationary block 2 of mass m 2 = 0.500 ml. Next, block 2 undergoes a one-dimensional elastic collision with stationary block 3 of mass m 3 = 0.50m 2 (a) What then is the speed of block 3? (b) Is the speed of block 3 greater than, less than, or the same as the initial values for block 1? (c) Is the kinetic energy of block 3 greater than, less than, or the same as the initial values for block 1? (d) Is the momentum of block 3 greater than, less than, or the same as the initial values for block 1? 23 Block 1 of mass m 1 slides along a frictionless floor and into a one-dimensional elastic collision with stationary block 2 of mass m 2 = 3m l. Prior to the collision, the center of mass of the two-block system had a speed of 3.00 m/s. Afterward, what is the speed of (a) Afterward, what is the speed of the center of mass? (b) Afterward, what is the speed of block 2? (a) 3.00 m/s; (b) 6.00 m/s (a) 7.11 m/s; (b) greater; (c) less; (d) less
26 In the figure, block 1 of mass m 1 slides along an x axis on a frictionless floor at speed 4.00 m/s. Then it undergoes a onedimensional elastic collision with stationary block 2 of mass m 2 = 2.00 ml. Next, block 2 undergoes a one-dimensional elastic collision with stationary block 3 of mass m 3 = 2.00m 2. (a) What then is the speed of block 3? (b) Is the speed of block 3 greater than, less than, or the same as the initial values for block 1? (c) Is the kinetic energy of block 3 greater than, less than, or the same as the initial values for block 1? (d) Is the momentum of block 3 greater than, less than, or the same as the initial values for block 1? 28 A pellet gun fires ten 2.0 g pellets per second with a speed of 500 m/s. The pellets are stopped by a rigid wall. (a) What is the magnitude of the momentum of each pellet? (b) What is the kinetic energy of each pellet? (c) What is the magnitude of the average force on the wall from the stream of pellets? (d) If each pellet is in contact with the wall for 0.60 ms, what is the magnitude of the average force on the wall from each pellet during contact? (e) Why is this average force so different from the average force calculated in (c)? 27 (a) 1.78 m/s; (b) less; (c) less; (d) greater Block 1, with mass m 1 and speed 4.0 m/s, slides along an x axis on a frictionless floor and then undergoes a one-dimensional elastic collision with stationary block 2, with mass m 2 = 0.40 m l. The two blocks then slide into a region where the coefficient of kinetic friction is 0.50; there they stop. (a) How far into that region does block 1 slide? (b) How far into that region does block 2 slide? 29 (a) 1.0 kg.m/s; (b) 2.5 x 10 2 J; (c) 10 N; (d) 1.7 kn; (e) answer for (c) includes time between pellet collisions A railroad freight car of mass 3.18 x 10 4 kg collides with a stationary caboose car. They couple together, and 27.0% of the initial kinetic energy is transferred to thermal energy, sound, vibrations, and so on. Find the mass of the caboose. 1.18 x 10 4 kg (a) 30 cm; (b) 3.3 m 30 In the figure, a ball of mass m 60 g is shot with speed v i = 22 m/s into the barrel of a spring gun of mass M = 240 g initially at rest on a frictionless surface. The ball sticks in the barrel at the point of maximum compression of the spring. Assume that the increase in thermal energy due to friction between the ball and the barrel is negligible. (a) What is the speed of the spring gun after the ball stops in the barrel? (b) What fraction of the initial kinetic energy of the ball is stored in the spring? (a) 4.4 m/s; (b) 0.80
31 An electron undergoes a one-dimensional elastic collision with an initially stationary hydrogen atom. What percentage of the electron's initial kinetic energy is transferred to kinetic energy of the hydrogen atom? (The mass of the hydrogen atom is 1840 times the mass of the electron.) 0.22% 34 The ballistic pendulum was used to measure the speeds of bullets before electronic timing devices were developed. The version shown in the figure consists of a large block of wood of mass M = 5.4 kg, hanging from two long cords. A bullet of mass m = 9.5 g is fired into, the block, coming quickly to rest, The block + bullet then swing upward, their center of mass rising a vertical distance h = 6.3 cm before the pendulum comes momentarily to rest at the end of its arc. What is the speed of the bullet just prior to the collision? 32 In the figure, block 1 of mass m 1 = 6.6 kg is at rest on a long frictionless table that is up against a wall. Block 2 of mass m 2 is placed between block 1 and the wall and sent sliding to the left, toward block 1, with constant speed v 2i. Find the value of m 2 for which both blocks move with the same velocity after block 2 has collided once with block 1 and once with the wall. Assume all collisions are elastic (the collision with the wall does not change the speed of block 2). 35 A bullet of mass 10 g strikes a ballistic pendulum of mass 2.0 kg. The center of mass of the pendulum rises a vertical distance of 12 cm. Assuming that the bullet remains embedded in the pendulum, calculate the bullet's initial speed. 2.2 kg 33 In a railroad accident, a boxcar weighing 200 kn and traveling at 3.00 m/s on horizontal track slams into a stationary caboose weighing 400 kn. The collision connects the caboose to the boxcar. How much energy is transferred from kinetic energy to other forms of energy in the collision? 61.2 kj 36 3.1 x 10 2 m/s A steel ball of mass 0.500 kg is fastened to a cord that is 70.0 cm long and fixed at the far end. The ball is then released when the cord is horizontal (see figure). At the bottom of its path, the ball strikes a 2.50 kg steel block initially at rest on a frictionless surface. The collision is elastic. (a) Find the speed of the ball just after the collision. (b) Find the speed of the block just after the collision. (a) 2.47 m/s; (b) 1.23m/s
37 In the figure, block 1 of mass m 1 slides from rest along a frictionless ramp from height h = 2.50 m and then collides with stationary block 2, which has mass m 2 = 2.00 ml. After the collision, block 2 slides into a region where the coefficient of kinetic friction? k is 0.500 and comes to a stop in distance d within that region. (a) What is the value of distance d if the collision is elastic? (b) What is the value of distance d if the collision is completely inelastic? 39 In the ballistic pendulum of the figure, assume the bullet's mass m is 8.00 g, the block's mass M is 7.00 kg, and the vertical distance h the block rises is 5.00 cm. (a) When the bullet is fired into the block, what fraction of the bullet's initial kinetic energy remains as mechanical energy of the bullet-block pendulum after the collision? (b) If we increase the initial speed of the bullet, does that fraction increase, decrease, or remain the same? Why? (a) 2.22 m; (b) 0.556 m (a) 1.14 x 10-3 ; (b) same 38 In the two-sphere arrangement of the figure, assume that sphere 1 has a mass of 50 g and an initial height of h 1 = 9.0 cm, and that sphere 2 has a mass of 85 g. (a) After sphere 1 is released and collides elastically with sphere 2, what height is reached by sphere 1? (b) After sphere 1 is released and collides elastically with sphere 2, what height is reached by sphere 2? (c) After the next (elastic) collision, what height is reached by sphere 1? (d) After the next (elastic) collision, what height is reached by sphere 2? (Hint: Do not use rounded-off values.) 40 Particle A and particle B are held together with a compressed spring between them. When they are released, the spring pushes them apart, and they then fly off in opposite directions, free of the spring. The mass of A is 2.00 times the mass of B, and the energy stored in the spring was 60 J. Assume that the spring has negligible mass and that all its stored energy is transferred to the particles. Once that transfer is complete, what is the kinetic energies of (a) Once that transfer is complete, what is the kinetic energies of particle A? (b) Once that transfer is complete, what is the kinetic energies of particle B? (a) 20 J; (b) 40 J (a) 0.60 cm; (b) 4.9 cm; (c) 9.0 cm; (d) 0
41 In the figure, block 1 (mass 2.0 kg) is moving rightward at 10 m/s and block 2 (mass 5.0 kg) is moving rightward at 3.0 m/s. The surface is frictionless, and a spring with a spring constant of 1120 N/m is fixed to block 2. When the blocks collide, the compression of the spring is maximum at the instant the blocks have the same velocity. Find the maximum compression. 44 In the figure block 1 (mass 6.0 kg) is moving rightward at 8.0 m/s and block 2 (mass 4.0 kg) is moving rightward at 2.0 m/s. The surface is frictionless, and a spring with a spring constant of 8000 N/m is fixed to block 2. Eventually block 1 overtakes block 2. At the instant block 1 is moving rightward at 6.4 m/s, what is (a) At the instant block 1 is moving rightward at 6.4 m/s, what is the speed of block 2? (b) At the instant block 1 is moving rightward at 6.4 m/s, what is the elastic potential energy of the spring? 25 cm 42 In the figure, block 2 (mass 1.0 kg) is at rest on a frictionless surface and touching the end of an unstretched spring of spring constant 200 N/m. The other end of the spring is fixed to a wall. Block 1 (mass 2.0 kg), traveling at speed v 1 = 4.0 m/s, collides with block 2, and the two blocks stick together. When the blocks momentarily stop, by what distance is the spring compressed? (a) 4.4 m/s; (b) 38 J 33 cm 45 Two blocks of masses M and 3M are placed on a horizontal, frictionless surface. A light spring is attached to one of them, and the blocks are pushed together with the spring between them. A cord initially holding the blocks together is burned; after this, the block of mass 3 M moves to the right with a speed of 2.00 m/s. (a) What is the speed of the block of mass M? (b) Find the original elastic potential energy in the spring if M = 0.350 kg. 43 In the figure, block L of mass m L = 1.00 kg and block R of mass m R = 0.500 kg are held in place with a compressed spring between them. When the blocks are released, the spring sends them sliding across a frictionless floor. (The spring has negligible mass and falls to the floor after the blocks leave it.) (a) If the spring gives block L a release speed of 1.20 m/s relative to the floor, how far does block R travel in the next 0.800 s? (b) If, instead, the spring gives block L a release speed of 1.20 m/s relative to the velocity that the spring gives block R, how far does block R travel in the next 0.800 s? (a) 1.92 m; (b) 0.640 m
46 A glider of mass m is free to slide along a horizontal air track. It is pushed against a launcher at one end of the track. Model the launcher as a light spring of force constant k, compressed by a distance x. The glider is released from rest. (a) Show that the glider attains a speed v = x (k/m) 1/2. (b) Does a glider of large or of small mass attain a greater speed? (c) Show that the impulse imparted to the glider is given by the expression x(k m) 1/2. (d) Is a greater impulse injected into a large or a small mass? (e) Is more work done on a large or a small mass? 48 A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large spring, initially unstretched and with force constant k = 2.00? 10 4 N/m, as in the figure. The cannon fires a 200-kg projectile at a velocity of 125 m/s directed 45.0 above the horizontal. (a) If the mass of the cannon and its carriage is 5 000 kg, find the recoil speed of the cannon. (b) Determine the maximum extension of the spring. (c) Find the maximum force the spring exerts on the carriage. (d) Consider the system consisting of the cannon, carriage, and shell. Is the momentum of this system conserved during the firing? Why or why not? 47 Two gliders are set in motion on an air track. A spring of force constant k is attached to the near side of one glider. The first glider of mass m 1 has velocity v 1 and the second glider of mass m 2 moves more slowly, with velocity v 2, as in the figure. When m 1 collides with the spring attached to m 2 and compresses the spring to its maximum compression x max, the velocity of the gliders is v. (a) In terms of v 1, v 2, m 1, m 2, and k, find the velocity v at maximum compression, (b) In terms of v 1, v 2, m 1, m 2, and k, find the maximum compression x max, and (c) In terms of v 1, v 2, m 1, m 2, and k, find the velocity of each glider after m 1 has lost contact with the spring. 49 An object is tracked by a radar station and found to have a position vector given by r = (3500-160t)i + 2700j + 300k, with y in meters and t in seconds. The radar station's x axis points east, its y axis north, and its z axis vertically up. (a) If the object is a 250 kg meteorological missile, what is its linear momentum? (b) If the object is a 250 kg meteorological missile, what is its direction of motion? (c) If the object is a 250 kg meteorological missile, what is the net force on it? (a) -4.0 x 10 4 kg.m/s; (b) west; (c) 0
50 A 2100 kg truck traveling north at 41 km/h turns east and accelerates to 51 km/h. (a) What is the change in the truck's kinetic energy? (b) What is the magnitude of the change in its momentum? (c) What is the direction of the change in its momentum? (a) 7.5 x 10 4 J; (b) 3.8 x 10 4 kg m/s; (c) 39 0 south of east 53 A 20.0 kg body is moving through space in the positive direction of an x axis with a speed of 200 m/s when, due to an internal explosion, it breaks into three parts. One part, with a mass of 10.0 kg, moves away from the point of explosion with a speed of 100 m/s in the positive y direction. A second part, with a mass of 4.00 kg, moves in the negative x direction with a speed of 500 m/s. (a) In unit-vector notation, what is the velocity of the third part? (b) How much energy is released in the explosion? Ignore effects due to the gravitational force. 51 An object, with mass m and speed v relative to an observer, explodes into two pieces, one three times as massive as the other; the explosion takes place in deep space. The less massive piece stops relative to the observer, How much kinetic energy is added to the system during the explosion, as measured in the observer's reference frame? mv 2 /6 54 (a) (1.00 km/s) i - (0.167 km/s) j; (b) 3.23 MJ A body is traveling at 2.0 m/s along the positive direction of an x axis; no net force acts on the body. An internal explosion separates the body into two parts, each of 4.0 kg, and increases the total kinetic energy by 16 J. The forward part continues to move in the original direction of motion. (a) What is the speed of the rear part? (b) What is the speed of the forward part? 52 In the figure, a stationary block explodes into two pieces L and R that slide across a frictionless floor and then into regions with friction, where they stop. Piece L, with a mass of 2.0 kg, encounters a coefficient of kinetic friction? L = 0.40 and slides to a stop in distance d L = 0.15 m. Piece R encounters a coefficient of kinetic friction? R = 0.50 and slides to a stop in distance d R = 0.25 m. What was the mass of the block? (a) 0; (b) 4.0 m/s 55 At time t = 0, force F 1 = (-4.00i + 5.00j) N acts on an initially stationary particle of mass 2.00 x 10-3 kg and force F 2 = (2.00i - 4.00j) N acts on an initially stationary particle of mass 4.00 x 10-3 kg. From time t = 0 to t = 2.00 ms, what is the (a) From time t = 0 to t = 2.00 ms, what is the magnitude of the displacement of the center of mass of the two-particle system? (b) From time t = 0 to t = 2.00 ms, what is the angle (relative to the positive direction of the x axis) of the displacement of the center of mass of the two-particle system? (c) What is the kinetic energy of the center of mass at t = 2.00 ms? 3.4 kg (a) 0.745 mm; (b) 153 0 ; (c) 1.67 mj
56 An atomic nucleus at rest at the origin of an xy coordinate system transforms into three particles. Particle 1, mass 16.7 x 10-27 kg, moves away from the origin at velocity (6.00 x 10 6 m/s)i; particle 2, mass 8.35 x 10-27 kg, moves away at velocity (-8.00 x 10 6 m/s)j. (a) In unit-vector notation, what is the linear momentum of the third particle, mass 11.7 x 10-27 kg? (b) How much kinetic energy appears in this transformation? (a) (-1.00 x 10-19 kg.m/s) i + (0.67 x 10-19 kg m/s) j; (b) 1.19 x 10-12 J 58 The three balls in the overhead view of the figure are identical. Balls 2 and 3 touch each other and are aligned perpendicular to the path of ball 1. The velocity of ball 1 has magnitude v 0 = 10 m/s and is directed at the contact point of balls 1 and 2. (a) After the collision, what is the speed of ball 2? (b) After the collision, what is the direction of the velocity of ball 2? (c) After the collision, what is the speed of ball 3? (d) After the collision,what is the direction of the velocity of ball 3? (e) After the collision, what is the speed of ball 1? (f) After the collision,what is the direction of the velocity of ball 1? (Hint: With friction absent, each impulse is directed along the line connecting the centers of the colliding balls, normal to the colliding surfaces.) 57 A certain radioactive (parent) nucleus transforms to a different (daughter) nucleus by emitting an electron and a neutrino. The parent nucleus was at rest at the origin of an xy coordinate system. The electron moves away from the origin with linear momentum (- 1.2 x 10-22 kg.m/s)i; the neutrino moves away from the origin with linear momentum (-6.4 x 10-23 kg.m/s)". (a) What is the magnitude of the linear momentum of the daughter nucleus? (b) What is the direction of the linear momentum of the daughter nucleus? (c) If the daughter nucleus has a mass of 5.8 x 10-26 kg, what is its kinetic energy? (a) 6.9 m/s; (b) 30 0 ; (c) 6.9 m/s; (d) -30 0 ; (e) 2.0 m/s; (f) -180 0 (a) 1.4 x 10-22 kg.m/s; (b) 28 0 ; (c) 1.6 x 10-19 J 59 In a game of pool, the cue ball strikes another ball of the same mass and initially at rest. After the collision, the cue ball moves at 3.50 m/s along a line making an angle of 22.0 0 with the cue ball's original direction of motion, and the second ball has a speed of 2.00 m/s. (a) Find the angle between the direction of motion of the second ball and the original direction of motion of the cue ball. (b) Find the original speed of the cue ball. (c) Is kinetic energy (of the centers of mass, don't consider the rotation) conserved? (a) 41.0 0 ; (b) 4.75 m/s; (c) no
60 A ball with an initial speed of 10 m/s collides elastically with two identical balls whose centers are on a line perpendicular to the initial velocity and that are initially in contact with each other as shown in the figure to the left. The first ball is aimed directly at the contact point, and all the balls are frictionless. Find the velocities of all three balls after the collision? 63 A 90.0 kg fullback running east with a speed of 5.00 m/s is tackled by a 95.0 kg opponent running north with a speed of 3.00 m/s. The collision is perfectly inelastic, (a) Calculate the speed and direction of the players just after the tackle (b) Determine the mechanical energy lost as a result of the collision. (c) Account for the missing energy. 61 ball 2 = 6.93 ball 3 = 6.93 original Ball = 2 m/s The figure to the left, which is drawn to scale, shows two balls during an elastic collision. The balls enter from the left of the page, collide, and bounce away. The heavier ball at the top of the diagram has a mass of 600 g, while the ball on the bottom has a mass of 400 g. Using a vector diagram, determine if momentum is conserved in this collision. 64 Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37.0 with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk. 65 Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed v i. After the collision, the orange disk moves along a direction that makes an angle? with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk. 62 A 3.00 kg particle has a velocity of (3,00i - 4.00j ) m/s. (a) Find its x and y components of momentum. (b) Find the magnitude and direction of its momentum.
66 The mass of the blue puck in the figure is 20.0% greater than the mass of the green one. Before colliding, the pucks approach each other with momenta of equal magnitude and opposite directions, and the green puck has an initial speed of 10.0 m/s. Find the speeds of the pucks after the collision if half the kinetic energy is lost during the collision. 69 An unstable atomic nucleus of mass 17.0? 10 27 kg initially at rest disintegrates into three particles. One of the particles, of mass 5.00? 10 27 kg, moves along the y axis with a speed of 6.00? 10 6 m/s. Another particle, of mass 8.40? 10 27 kg, moves along the x axis with a speed of 4.00? 10 6 m/s. (a) Find the velocity of the third particle (b) Find the total kinetic energy increase in the process. 70 Block 1 with mass m 1 slides along an x axis across a frictionless floor and then undergoes an elastic collision with a stationary block 2 with mass m 2. The figure shows a plot of position x versus time t of block 1 until the collision occurs at position x c and time t c. (a) In which of the lettered regions on the graph will the plot be continued (after the collision) if m 1 < m 2? (b) In which of the lettered regions on the graph will the plot be continued (after the collision) if m 1 > m 2? (c) Along which of the numbered dashed lines will the plot be continued if m 1 = m 2? 67 Two automobiles of equal mass approach an intersection. One vehicle is traveling with velocity 13.0 m/s toward the east and the other is traveling north with speed v 2i. Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of 55.0 o north of east. The speed limit for both roads is 35 mi/h and the driver of the northward-moving vehicle claims he was within the speed limit when the collision occurred. Is he telling the truth? (a) C; (b) B; (c) 3 68 A billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.33 m/s, at an angle of 30.0 with respect to the original line of motion. Assuming an elastic collision (and ignoring friction and rotational motion), find the struck ball's velocity after the collision.
71 In the "before" part of the figure, car A (mass 1100 kg) is stopped at a traffic light when it is rear-ended by car B (mass 1400 kg). Both cars then slide with locked wheels until the frictional force from the slick road (with a low? k of 0.13) stops them, at distances d A = 8.2 m and d B = 6.1 m. (a) What is the speed of car A at the start of the sliding, just after the collision? (b) What is the speed of car B at the start of the sliding, just after the collision? (c) Assuming that linear momentum is conserved during the collision, find the speed of car B just before the collision. (d) Explain why this assumption may be invalid. 73 In the figure, block A (mass 1.6 kg) slides into block B (mass 2.4 kg), along a frictionless surface. The directions of three velocities before (i) and after (f) the collision are indicated; the corresponding speeds are v Ai = 5.5 m/s, v Bi = 2.5 m/s, and v Bf = 4.9 m/s. (a) What is the speed of velocity v Af? (b) What is the direction (left or right) of velocity v Af? (c) Is the collision elastic? (a) 1.9 m/s; (b) right; (c) yes 72 (a) 4.6m/s; (b) 3.9m/s; (c) 7.5m/s A cart with mass 340 g moving on a frictionless linear air track at an initial speed of 1.2 m/s undergoes an elastic collision with an initially stationary cart of unknown mass. After the collision, the first cart continues in its original direction at 0.66 m/s. (a) What is the mass of the second cart? (b) What is its speed after impact? (c) What is the speed of the two-cart center of mass? 74 Ball B, moving in the positive direction of an x axis at speed v, collides with stationary ball A at the origin, A and B have different masses. After the collision, B moves in the negative direction of the y axis at speed v/2. (a) In what direction does A move? (b) Show that the speed of A cannot be determined from the given information. (a) 27 0 (a) 99 g; (b) 1.9m/s; (c) 0.93m/s 75 Particle 1 with mass 3.0 kg and velocity (5.0 m/s)i undergoes a one-dimensional elastic collision with particle 2 with mass 2.0 kg and velocity (-6.0 m/s)i. (a) After the collision, what is the velocitiy of particle 1? (b) After the collision, what is the velocitiy of particle 2? (a) (-3.8 m/s) i; (b) (7.2 m/s) i
76 A 2.00 kg " particle" traveling with velocity v = (4.0 m/s)i collides with a 4.00 kg "particle" traveling with velocity v = (2.0 m/s)j. The collision connects the two particles (a) What then is their velocity in unit-vector notation? (b) What then is their velocity as a magnitude? (c) What then is their velocity as an angle? 80 A 500.0 kg module is attached to a 400.0 kg shuttle craft, which moves at 1000 m/s relative to the stationary main spaceship. Then a small explosion sends the module backward with speed 100.0 m/s relative to the new speed of the shuttle craft. As measured by someone on the main spaceship, by what fraction did the kinetic energy of the module and shuttle craft increase because of the explosion? (a) (1.3 m/s) i + (1.3 m/s) j; (b) 1.9 m/s; (c) 45 0 2.5 x 10-3 77 A 3.0 kg object moving at 8.0 m/s in the positive direction of an x axis has a one-dimensional elastic collision with an object of mass M, initially at rest. After the collision the object of mass M has a velocity of 6.0 m/s in the positive direction of the axis. What is mass M? 81 A 6.00 kg model rocket is traveling horizontally and due south with a speed of 20.0 m/s when it explodes into two pieces. The velocity of one piece, with a mass of 2.00 kg, is v1 = (-12.0 m/s)i + (30.0 m/s)j - (15.0 m/s)k, with i pointing due east, j pointing due north, and k pointing vertically upward. (a) What is the linear momentum of the other piece, in unitvector notation? (b) What is the kinetic energy of the other piece? (c) How much kinetic energy is produced by the explosion? 5.0 kg 78 Ball A, rolling west at 3.0 m/s, has a mass of 1.0 kg. Ball B has a mass of 2.0 kg and is stationary. After colliding with ball B, ball A moves south at 2.0 m/s. Calculate the momentum and velocity of ball B after the collision. (a) (24.0 kg.m/s) i - (180 kg.m/s) j +(30.0 kg.m/s) k; (b) 4.23 kj; (c) 4.30 kj 79 3.6 kg.m/s at 34 degrees north of west 1.8 m/s at 34 degrees N of west A glass ball of mass 5.0 g moves with a velocity of 20 m/s. The ball collides with a second glass ball of mass 10.0 g, which is moving along the same line with a velocity of 10.0 m/s. After the collision, the 5.0 g ball is still moving at a velocity of 8.0 m/s. (a) What is the change of momentum of the 5.0 g ball? (b) What is the change of momentum of the 10.0 g ball? 82 A 6100 kg rocket is set for vertical firing from the ground. (a) If the exhaust speed is 1200 m/s, how much gas must be ejected each second if the thrust is to equal the magnitude of the gravitational force on the rocket? (b) If the exhaust speed is 1200 m/s, how much gas must be ejected each second if the thrust is to give the rocket an initial upward acceleration of 21 m/s 2? (a) 50 kg/s; (b) 1.6 x 10 2 kg/s (a) -.06 kg.m/s (b) +.06 kg.m/s
83 The first stage of a Saturn V space vehicle consumed fuel and oxidizer at the rate of 1.50? 10 4 kg/s, with an exhaust speed of 2.60? 10 3 m/s. (a) Calculate the thrust produced by these engines. (b) Find the acceleration of the vehicle just as it lifted off the launch pad on the Earth if the vehicle s initial mass was 3.00? 10 6 kg. Note: You must include the gravitational force to solve part (b). 86 Rocket Science. A rocket has total mass M i = 360 kg, including 330 kg of fuel and oxidizer. In interstellar space it starts from rest, turns on its engine at time t = 0, and puts out exhaust with relative speed v e = 1 500 m/s at the constant rate k = 2.50 kg/s. The fuel will last for an actual burn time of 330 kg/(2.5 kg/s) = 132 s, but define a projected depletion time as T p = M i/ k = 144 s. (This would be the burn time if the rocket could use its payload and fuel tanks as fuel, and even the walls of the combustion chamber.) (a) Show that during the burn the velocity of the rocket is given as a function of time by v(t) =?v e ln [1? (t/t p )] 84 Model rocket engines are sized by thrust, thrust duration, and total impulse, among other characteristics. A size C5 model rocket engine has an average thrust of 5.26 N, a fuel mass of 12.7 grams, and an initial mass of 25.5 grams. The duration of its burn is 1.90 s. (a) What is the average exhaust speed of the engine? (b) If this engine is placed in a rocket body of mass 53.5 grams, what is the final velocity of the rocket if it is fired in outer space? Assume the fuel burns at a constant rate. (b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to 132 s. (c) Show that the acceleration of the rocket is a(t) = v e /(T p? t) (d) Graph the acceleration as a function of time. (e) Show that the displacement of the rocket is x(t) = v e (T p? t) ln [1? (t/t p )] + v e t 85 A rocket for use in deep space is to be capable of boosting a total load (payload plus rocket frame and engine) of 3.00 metric tons to a speed of 10 000 m/s. (a) It has an engine and fuel designed to produce an exhaust speed of 2 000 m/s. How much fuel plus oxidizer is required? (b) If a different fuel and engine design could give an exhaust speed of 5 000 m/s, what amount of fuel and oxidizer would be required for the same task? 87 An orbiting spacecraft is described not as a zero- g, but rather as a micro gravity environment for its occupants and for onboard experiments. Astronauts experience slight lurches due to the motions of equipment and other astronauts, and due to venting of materials from the craft. Assume that a 3500-kg spacecraft undergoes an acceleration of 2.50 g = 2.45? 10 5 m/s 2 due to a leak from one of its hydraulic control systems. The fluid is known to escape with a speed of 70.0 m/s into the vacuum of space. How much fluid will be lost in one hour if the leak is not stopped?
88 In the figure, a 10 g bullet moving directly upward at 1000 m/s strikes and passes through the center of mass of a 5.0 kg block initially at rest. The bullet emerges from the block moving directly upward at 400 m/s. To what maximum height does the block then rise above its initial position? 91 Two titanium spheres approach each other head-on with the same speed and collide elastically. After the collision, one of the spheres, whose mass is 300 g, remains at rest. (a) What is the mass of the other sphere? (b) What is the speed of the two-sphere center of mass if the initial speed of each sphere is 2.00 m/s? (a) 100 g; (b) 1.0 m/s 89 7.3 cm A 5.0 kg block with a speed of 3.0 m/s collides with a 10 kg block that has a speed of 2.0 m/s in the same direction. After the collision, the 10 kg block is observed to be traveling in the original direction with a speed of 2.5 m/s. (a) What is the velocity of the 5.0 kg block immediately after the collision? (b) By how much does the total kinetic energy of the system of two blocks change because of the collision? (c) Suppose, instead, that the 10 kg block ends up with a speed of 4.0 m/s. What then is the change in the total kinetic energy? (d) Account for the result you obtained in (c). 92 In the figure, puck 1 of mass m 1 = 0.20 kg is sent sliding across a frictionless lab bench, to undergo a one-dimensional elastic collision with stationary puck 2. Puck 2 then slides off the bench and lands a distance d from the base of the bench. Puck 1 rebounds from the collision and slides off the opposite edge of the bench, landing a distance 2d from the base of the bench. What is the mass of puck 2? (Hint: Be careful with signs.) 1.0 kg (a) +2.0 m/s; (b) -1.3 J; (c) +40 J; (d) energy entered system from some source such as a small explosion 90 A completely inelastic collision occurs between two balls of wet putty that move directly toward each other along a vertical axis. Just before the collision, one ball, of mass 3.0 kg, is moving upward at 20 m/s and the other ball, of mass 2.0 kg, is moving downward at 12 m/s. How high do the combined two balls of putty rise above the collision point? (Neglect air drag.) 2.6 m
93 Tyrannosaurus Rex may have known from experience not to run particularly fast because of the danger of tripping, in which case its short forearms would have been no help in cushioning the fall. Suppose a T. Rex of mass m trips while walking, toppling over, with its center of mass falling freely a distance of 1.5 m. Then its center of mass descends an additional 0.30 m due to compression of its body and the ground. (a) In multiples of the dinosaur's weight, what is the approximate magnitude of the average vertical force on the dinosaur during its collision with the ground (during the descent of 0.30 m)? Now assume that the dinosaur is running at a speed of 19 m/s (fast) when it trips, falls to the ground, and then slides to a stop with a coefficient of kinetic friction of 0.6. Assume also that the average vertical force during the collision and sliding is that in (a). (b) What, approximately, is the magnitude of the average total force on the dinosaur from the ground (again in multiples of its weight)? (c) What, approximately, is the sliding distance? The force magnitudes of (a) and (b) strongly suggest that the collision would injure the torso of the dinosaur. The head, which would fall farther, would suffer even greater injury. 95 96 A 3000 kg block falls vertically through 6.0 m and then collides with a 500 kg pile, driving it 3.0 cm into bedrock Assuming that the block-pile collision is completely inelastic, find the magnitude of the average force on the pile from the bedrock during the 3.0 cm descent. 5.0 x 10 6 N A brick that weighs 30.0 N is released from rest on a 1.00 m long, frictionless plane inclined at an angle of 20.0 degrees. The brick slides down the incline and strikes a second brick that weighs 36.8 N. (a) What is the initial velocity of the first brick at the bottom of the incline before it strikes the second brick (b) If the two bricks stick together, with what initial speed will they move along the table? (c) If the force of friction acting on the two bricks is 5.0 N, what time will elapse before the bricks come to rest? (d) How far will the two bricks slide before coming to rest. (a) 0.54 m/s; (b) 0; (c) 1.1 m/s (a) 2.59 m/s (b) 1.16 m/s (c) 1.58 s (d) 0.92 m 94 In the figure, a 3.2 kg box of running shoes slides on a horizontal frictionless table and collides with a 2.0 kg box of ballet slippers initially at rest on the edge of the table, at height h = 0.40 m. The speed of the 3.2 kg box is 3.0 m/s just before the collision. If the two boxes stick together because of packing tape on their sides, what is their kinetic energy just before they strike the floor? 97 A brick that weighs 50.0 N is released from rest on a 3.00 m long, frictionless plane inclined at an angle of 30.0 degrees. The brick slides down the incline and strikes a second brick that weighs 40.0 N. (a) What is the initial velocity of the first brick at the bottom of the incline before it strikes the second brick (b) If the two bricks stick together, with what initial speed will they move along the table? (c) If the force of friction acting on the two bricks is 5.0 N, what time will elapse before the bricks come to rest? (d) How far will the two bricks slide before coming to rest. (a) 5 mg; (b) 7 mg; (c) 5 m (a) 5.42 m/s (b) 3.01 m/s (c) 5.53 m/s (d) 8.37 m
98 A brick that weighs 24.5 N is released from rest on a 1.00 m long, frictionless plane inclined at an angle of 30.0 degrees. The brick slides down the incline and strikes a second brick that weighs 36.8 N. (a) What is the initial velocity of the first brick at the bottom of the incline before it strikes the second brick (b) If the two bricks stick together, with what initial speed will they move along the table? (c) If the force of friction acting on the two bricks is 5.0 N, what time will elapse before the bricks come to rest? (d) How far will the two bricks slide before coming to rest. 100 As shown in the figure, a bullet of mass m and speed v passes completely through a pendulum bob of mass M. The bullet emerges with a speed of v/2. The pendulum bob is suspended by a stiff rod of length and negligible mass. What is the minimum value of v such that the pendulum bob will barely swing through a complete vertical circle? (a) 3.14 m/sec (b) 1.25 m/s (c) 1.6 s (d) 1.0 m 99 Two blocks are free to slide along the frictionless wooden track ABC shown in the figure. A block of mass m1 = 5.00 kg is released from A. Protruding from its front end is the north pole of a strong magnet, repelling the north pole of an identical magnet embedded in the back end of the block of mass m 2 = 10.0 kg, initially at rest. The two blocks never touch. Calculate the maximum height to which m 1 rises after the elastic collision. 101 A 12.0-g wad of sticky clay is hurled horizontally at a 100-g wooden block initially at rest on a horizontal surface. The clay sticks to the block. After impact, the block slides 7.50 m before coming to rest. If the coefficient of friction between the block and the surface is 0.650, what was the speed of the clay immediately before impact?
102 A 60.0 kg person running at an initial speed of 4.00 m/s jumps onto a 120-kg cart initially at rest (see figure). The person slides on the cart s top surface and finally comes to rest relative to the cart. The coefficient of kinetic friction between the person and the cart is 0.400. Friction between the cart and ground can be neglected. (a) Find the final velocity of the person and cart relative to the ground. (b) Find the friction force acting on the person while he is sliding across the top surface of the cart. (c) How long does the friction force act on the person? (d) Find the change in momentum of the person and the change in momentum of the cart. (e) Determine the displacement of the person relative to the ground while he is sliding on the cart. (f) Determine the displacement of the cart relative to the ground while the person is sliding. (g) Find the change in kinetic energy of the person. (h) Find the change in kinetic energy of the cart. (i) Explain why the answers to (g) and (h) differ. (What kind of collision is this, and what accounts for the loss of mechanical energy?) 104 105 A 0.500 kg sphere moving with a velocity (2.00i - 3.00j + 1.00k)m/s strikes another sphere of mass 1.50 kg moving with a velocity (-1.00i + 2.00j - 3.00k) m/s. (a) If the velocity of the 0.500 kg sphere after the collision is (-1.00i + 3.00j - 800k) m/s, find the final velocity of the 1.50 kg sphere and identify the kind of collision (elastic, inelastic, or perfectly inelastic). (b) If the velocity of the 0.500 kg sphere after the collision is (-0.250i + 0.750J - 2.00k) m/s, find the final velocity of the 1.50 kg sphere and identify the kind of collision. (c) If the velocity of the 0.500 kg sphere after the collision is (-1.00i + 3.00j + ak) m/s, find the value of a and the velocity of the 1.50 kg sphere after an elastic collision. A small block of mass m 1 = 0.500 kg is released from rest at the top of a curve-shaped frictionless wedge of mass m 2 = 3.00 kg, which sits on a frictionless horizontal surface as in figure a. When the block leaves the wedge, its velocity is measured to be 4.00 m/s to the right, as in figure b. (a) What is the velocity of the wedge after the block reaches the horizontal surface? (b) What is the height h of the wedge? 103 A bullet of mass m is fired into a block of mass M initially at rest at the edge of a frictionless table of height h (see figure). The bullet remains in the block, and after impact the block lands a distance d from the bottom of the table. Determine the initial speed of the bullet. 106 A bucket of mass m and volume V is attached to a light cart, completely covering its top surface. The cart is given a quick push along a straight, horizontal, smooth road. It is raining, so that as the cart cruises along without friction, the bucket gradually fills with water. By the time the bucket is full, its speed is v. (a) What was the initial speed vi of the cart? Let represent the density of water. (b) Assume that when the bucket is half full, it develops a slow leak at the bottom, so that the level of the water remains constant thereafter. Describe qualitatively what happens to the speed of the cart after the leak develops.