*15-28. The winch delivers a horizontal towing force F to

The 0.5-kg ball strikes the rough ground and rebounds with the velocities shown. Determine the magnitude of the impulse the ground exerts on the ball. Assume that the ball does not slip when it strikes the ground, and neglect the size of the ball and the impulse produced by the weight of the ball.

F15-2. If the coefficient of kinetic friction between the 150-lb crate and the ground is fxk = 0.2, determine the speed of the crate when i = 4 s. The crate starts from rest and is towed by the 100-lb force.

F15-3. The motor exerts a force of F = (20r)N on the cable, where / is in seconds. Determine the speed of the 25-kg crate when t = 4 s. The coefficients of static and kinetic friction between the crate and the plane are (jls = 0.3 and fxk = 0.25, respectively.

The wheels of the 1.5-Mg car generate the traction force F described by the graph. If the car starts from rest, determine its speed when t = 6 s.

F15-5. The 2.5-Mg four-wheel-drive SUV tows the 1.5-Mg trailer. The traction force developed at the wheels is Fd - 9 kN. Determine the speed of the truck in 20 s, starting from rest. Also, determine the tension developed in the coupling between the SUV and the trailer. Neglect the mass of the wheels.

FI5-6. The 10-lb block A attains a velocity of 1 ft/s in 5 seconds, starting from rest. Determine the tension in the cord and the coefficient of kinetic friction between block A and the horizontal plane. Neglect the weight of the pulley. Block B has a weight of 8 lb.

F15-7. The freight cars /\ and B have a mass of 20 Mg and 15 Mg. respectively. Determine the velocity of A after collision if the cars collide and rebound, such that B moves to the right with a speed of 2 m/s. If A and B arc in contact for 0.5 s, find the average impulsive force which acts between them.

F15-8. The cart and package have a mass of 20 kg and 5 kg, respectively. If the cart has a smooth surface and it is initially at rest, while the velocity of the package is as shown, determine the final common velocity of the cart and package after the impact.

F15-9. The 5-kg block A has an initial speed of 5 m/s as it slides down the smooth ramp, after which it collides with the stationary block B of mass 8 kg. If the two blocks couple together after collision, determine their common velocity immediately after collision.

The spring is fixed to block A and block B is pressed against the spring. If the spring is compressed s = 200 mm and then the blocks are released, determine their velocity at the instant block B loses contact with the spring. The masses of blocks A and B are 10 kg and 15 kg, respectively.

F15-11. Blocks A and B have a mass of 15 kg and 10 kg. respectively. If A is stationary and B has a velocity of 15 m/s just before collision, and the blocks couple together after impact, determine the maximum compression of the spring.

The cannon and support without a projectile have a mass of 250 kg. If a 20-kg projectile is fired from the cannon with a velocity of 400 m/s, measured relative to the cannon, determine the speed of the projectile as it leaves the barrel of the cannon. Neglect rolling resistance.

Determine the coefficient of restitution e between ball A and ball B.The velocities of A and B before and after the collision are shown.

F15-14. The 15-Mg lank car A and 25-Mg freight car B travel towards each other with the velocities shown. If the coefficient of restitution between the bumpers is e = 0.6, determine the velocity of each car just after the collision.

FI5-15. The 30-lb package A has a speed of 5 ft/s when it enters the smooth ramp. As it slides down the ramp, it strikes the 80-lb package B which is initially at rest. If the coefficient of restitution between A and B is e = 0.6. determine the velocity of B just after the impact.

The ball strikes the smooth wall with a velocity of \(\left(v_{b}\right)_{1}=20 \mathrm{\ m} / \mathrm{s}\). If the coefficient of restitution between the ball and the wall is e = 0.75. determine the velocity of the ball just after the impact.

F15-17. Disk A weighs 2 lb and slides on the smooth horizontal plane with a velocity of 3 fl/s. Disk B weighs 11 lb and is initially at rest. If after impact A has a velocity of 1 ft/s, parallel to the positive .v axis, determine the speed of disk B after impact.

F15-18. Two disks A and B each have a weight of 2 lb and the initial velocities shown just before they collide. If the coefficient of restitution is e - 0.5, determine their speeds just after impact.

The 2-kg particle A has the velocity shown. Determine its angular momentum \(\mathrm{H}_{o}\) about point O.

FI5-20. The 2-kg particle A has the velocity shown. Determine its angular momentum H/, about point P.

F15-21. Initially the 5-kg block is rotating with a constant speed of 2 m/s around the circular path centered at O on the smooth horizontal plane. If a constant tangential force /' = 5 N is applied to the block, determine its speed when t = 3 s. Neglect the size of the block.

F15-22. The 5-kg block is rotating around the circular path centered at O on the smooth horizontal plane when it is subjected to the force F = (10/) N. where / is in seconds. If the block starts from rest, determine its speed when / = 4 s. Neglect the size of the block. The force maintains the same constant angle tangent to the path.

F15-23. The 2-kg sphere is attached to the light rigid rod. which rotates in the horizontal plane centered at O. If the system is subjected to a couple moment M = (0.9r) N m, where t is in seconds, determine the speed of the sphere at the instant / = 5 s starting from rest.

F15-24. Two identical 10-kg spheres are attached to the light rigid rod. which rotates in the horizontal plane centered at pin (>. If the spheres are subjected to tangential forces of P = ION, and the rod is subjected to a couple moment M = (8/) N m. where t is in seconds, determine the speed of the spheres at the instant / = 4 s. The system starts from rest. Neglect the size of the spheres.

A 2-lb ball is thrown in the direction shown with an initial speed \(v_{A}=18 \mathrm{\ ft} / \mathrm{s}\). Determine the time needed for it to reach its highest point B and the speed at which it is traveling at B. Use the principle of impulse and momentum for the solution.

15-2. A 20-lb block slides down a 30 inclined plane with an initial velocity of 2 ft/s. Determine the velocity of the block in 3 s if the coefficient of kinetic friction between the block and the plane is ik = 0.25.

15-3. A 5-lb block is given an initial velocity of 10 ft/s up a 45 smooth slope. Determine the time it will take to travel up the slope before it stops.

*15-4. The 180-lb iron worker is secured by a fall-arrest system consisting of a harness and lanyard AB. which is fixed to the beam. If the lanyard has a slack of 4 ft, determine the average impulsive force developed in the lanyard if he happens to fall 4 feet. Neglect his size in the calculation and assume the impulse takes place in 0.6 seconds.

15-5. A man hits the 50-g golf ball such that it leaves the tee at an angle of 40 with the horizontal and strikes the ground at the same elevation a distance of 20 m away. Determine the impulse of the club C on the ball. Neglect the impulse caused by the ball's weight while the club is striking the ball.

15-6. A train consists of a 50-Mg engine and three cars, each having a mass of 30 Mg. If it takes 80 s for the train to increase its speed uniformly to 40 km/h. starting from rest, determine the force T developed at the coupling between the engine E and the first car A. The wheels of the engine provide a resultant frictional tractive force F which gives the train forward motion, whereas the car wheels roll freely. Also, determine Fading on the engine wheels.

15-7. Crates A and B weigh 100 lb and 50 lb. respectively. If they start from rest, determine their speed when / = 5 s. Also, find the force exerted by crate A on crate B during the motion. The coefficient of kinetic friction between the crates and the ground is fxk = 0.25.

*15-8. If Ihe jets exert a vertical thrust of T = (500f3,2) N. where t is in seconds, determine the mans speed when / = 3 s. The total mass of the man and the jet suit is 100 kg. Neglect the loss of mass due to the fuel consumed during the lift which begins from rest on the ground.

15-9. Under a constant thrust of T = 40 kN. the 1.5-Mg dragster reaches its maximum speed of 125 m/s in 8 s starting from rest. Determine the average drag resistance FL) during this period of time.

15-10. The 50-kg crate is pulled by the constant force P. If the crate starts from rest and achieves a speed of 10 m/s in 5 s. determine the magnitude of P.The coefficient of kinetic friction between the crate and the ground is jik = 0.2.

1511. When the 5-kg block is 6 m from the wall, it is sliding at v, = 14 m/s. If the coefficient of kinetic friction between the block and the horizontal plane is = 0.3. determine the impulse of the wall on the block necessary to stop the block. Neglect the friction impulse acting on the block during the collision.

*15-12. For a short period of time, the frictional driving force acting on the wheels of the 2.5-Mg van is F[} = (60012) N. where t is in seconds. If the van has a speed of 20 km/h when t= 0, determine its speed when / = 5 s.

15-13. The 2.5-Mg van is traveling with a speed of 100 km/h when the brakes are applied and all four wheels lock. If the speed decreases to 40 km/h in 5 s. determine the coefficient of kinetic friction between the tires and the road.

15-14. The force acting on a projectile having a mass m as it passes horizontally through the barrel of the cannon is F = C sin (77///'). Determine the projectiles velocity when / = /'. If the projectile reaches the end of the barrel at this instant, determine the length .v.

During operation the breaker hammer develops on the concrete surface a force which is indicated in the graph. To achieve this the 2-lb spike S is fired from rest into the surface at 200 ft/s. Determine the speed of the spike just after rebounding.

*15-16. The twitch in a muscle of the arm develops a force which can be measured as a function of time as shown in the graph. If the effective contraction of the muscle lasts for a time /0. determine the impulse developed by the muscle.

15-17. A hammer head II having a weight of 0.25 lb is moving vertically downward at 40 ft/s when it strikes the head of a nail of negligible mass and drives it into a block of wood. Find the impulse on the nail if it is assumed that the grip at A is loose, the handle has a negligible mass, and the hammer stays in contact with the nail while it comes to rest. Neglect the impulse caused by the weight of the hammer head during contact with the nail.

15-18. The 40-kg slider block is moving to the right with a speed of 1.5 m/s when it is acted upon by the forces F, and F2. If these loadings vary in the manner shown on the graph, determine the speed of the block at / = 6 s. Neglect friction and the mass of the pulleys and cords. *

15-19. Determine the velocity of each block 2 s after the blocks are released from rest. Neglect the mass of the pulleys and cord.

*15-20. The particle P is acted upon by its weight of 3 lb and forces F( and F:. where t is in seconds. If the particle orginally has a velocity of v, = {3i + 1 j + 6k} ft/s, determine its speed after 2 s.

15-21. If it takes 35 s for the 50-Mg tugboat to increase its speed uniformly to 25 km/h. starling from rest, determine the force of the rope on the tugboat.The propeller provides the propulsion force F which gives the tugboat forward motion, whereas the barge moves freely. Also, determine F acting on the tugboat. The barge has a mass of 75 Mg.

If the force T exerted on the cable by the motor M is indicated by the graph, determine the speed of the 500-lb crate when t = 4 s, starting from rest. The coefficients of static and kinetic friction are \(\mu_{s}=0.3 \text { and } \mu_{k}=0.25 \text {, }\) respectively.

15-23. The 5-kg block is moving downward at = 2 m/s when it is 8 m from the sandy surface. Determine the impulse of the sand on the block necessary to stop its motion. Neglect the distance the block dents into the sand and assume the block does not rebound. Neglect the weight of the block during the impact with the sand.

*15-24. The 5-kg block is falling downward at v, = 2 m/s when it is 8 m from the sandy surface. Determine the average impulsive force acting on the block by the sand if the motion of the block is slopped in 0.9 s once the block strikes the sand. Neglect the distance the block dents into the sand and assume the block docs not rebound. Neglect the weight of the block during the impact with the sand.

15-25. The 0.1 -lb golf ball is struck by the club and then travels along the trajectory shown. Determine the average impulsive force the club imparts on the ball if the club maintains contact with the ball for 0.5 ms.

15-26. As indicated by the derivation, the principle of impulse and momentum is valid for observers in any inertial reference frame. Show that this is So, by Considering the 10-kg block which rests on the smooth surface and is subjected to a horizontal force of 6 N. If observer A is in a fixed frame x, determine the final speed of the block in 4 s if it has an initial speed of 5 m/s measured from the fixed frame. Compare the result with that obtained by an observer 13. attached to the x' axis that moves at a constant velocity of 2 m/s relative to A.

15-27. The winch delivers a horizontal towing force F to its cable at A which varies as shown in the graph. Determine the speed of the 70-kg bucket when / = 18 s. Originally the bucket is moving upward at = 3 m/s.

*15-28. The winch delivers a horizontal towing force F to its cable at A which varies as shown in the graph. Determine the speed of the 80-kg bucket when / = 24 s. Originally the bucket is released from rest.

15-29. The train consists of a 30-Mg engine E. and cars A. B. and C. which have a mass of 15 Mg, 10 Mg. and 8 Mg. respectively. If the tracks provide a traction force of F = 30 kN on the engine wheels, determine the speed of the train when I = 30 s, starting from rest. Also, find the horizontal coupling force at D between the engine E and car A. Neglect rolling resistance.

15-30. The crate B and cylinder A have a mass of 200 kg and 75 kg. respectively. If the system is released from rest, determine the speed of the crate and cylinder when / = 3 s. Neglect the mass of the pulleys.

15-31. Block A weighs 10 lb and block B weighs 3 lb. If B is moving downward with a velocity (vB), = 3 ft/s at t = 0, determine the velocity of A when / = 1 s. Assume that the horizontal plane is smooth. Neglect the mass of the pulleys and cords.

*15-32. Block A weighs 10 lb and block B weighs 3 lb. If B is moving downward with a velocity (%), = 3 ft/s at / = 0, determine the velocity of A when / = 1 s. The coefficient of kinetic friction between the horizontal plane and block A is /x 4 =0.15.

15-33. The log has a mass of 500 kg and rests on the ground for which the coefficients of static and kinetic friction are /xv = 0.5 and /xk = 0.4. respectively. The winch delivers a horizontal towing force T to its cable at A which varies as shown in the graph. Determine the speed of the log when t = 5 s. Originally the tension in the cable is zero. Hint: First determine the force needed to begin moving the log.

15-34. The 50-kg block is hoisted up the incline using the cable and motor arrangement shown. The coefficient of kinetic friction between the block and the surface is /xk = 0.4. If the block is initially moving up the plane at v0 = 2 m/s.and at this instant (t = 0) the motor develops a tension in the cord of T = (300 -I- 120V/) N. where t is in seconds, determine the velocity of the block w hen / = 2 s.

15-35. The bus B has a weight of 15 000 lb and is traveling to the right at 5 ft/s. Meanwhile a 3000-lb car A is traveling at 4 ft/s to the left. If the vehicles crash head-on and become entangled, determine their common velocity just after the collision. Assume that the vehicles are free to roll during collision.

*15-36. The 50-kg boy jumps on the 5-kg skateboard with a horizontal velocity of 5 m/s. Determine the distance .v the boy reaches up the inclined plane before momentarily coming to rest. Neglect the skateboards rolling resistance.

15-37. The 2.5-Mg pickup truck is towing the 1.5-Mg car using a cable as shown. If the car is initially at rest and the truck is coasting with a velocity of 30 km/h when the cable is slack, determine the common velocity of the truck and the car just after the cable becomes taut. Also, find the loss of energy.

15-38. A railroad car having a mass of 15 Mg is coasting at 1.5 m/s on a horizontal track. At the same time another car having a mass of 12 Mg is coasting at 0.75 m/s in the opposite direction. If the cars meet and couple together, determine the speed of both cars just after the coupling. Find the difference between the total kinetic energy before and after coupling has occurred, and explain qualitatively what happened to this energy.

15-39. The car A has a weight of 4500 lb and is traveling to the right at 3 ft/s. Meanwhile a 3000-lb car B is traveling at 6 ft/s to the left. If the cars crash head-on and become entangled, determine their common velocity just after the collision. Assume that the brakes are not applied during collision.

*'15-40. The 200-g projectile is fired with a velocity of 900 m/s towards the center of the 15-kg wooden block, which rests on a rough surface. If the projectile penetrates and emerges from the block with a velocity of 300 m/s. determine the velocity of the block just after the projectile emerges. How long does the block slide on the rough surface, after the projectile emerges, before it comes to rest again? The coefficient of kinetic friction between the surface and the block is fxk = 0.2.

15-41. The block has a mass of 50 kg and rests on the surface of the cart having a mass of 75 kg. If the spring which is attached to the cart and not the block is compressed 0.2 m and the system is released from rest, determine the speed of the block relative to the ground after the spring becomes undeformed. Neglect the mass of the carts wheels and the spring in the calculation. Also neglect friction. Take k= 300 N/m.

15-42. The block has a mass of 50 kg and rests on the surface of the cart having a mass of 75 kg. If the spring which is attached to the cart and not the block is compressed 0.2 m and the system is released from rest, determine the speed of the block with respect to the cart after the spring becomes undeformed. Neglect the mass of the wheels and the spring in the calculation. Also neglect friction.Take k = 300 N/m.

15-43. The three freight cars A. B. and C have masses of 10 Mg, 5 Mg. and 20 Mg, respectively. They arc traveling along the track with the velocities shown. Car A collides with car B first, followed by car C. If the three cars couple together after collision, determine the common velocity of the cars alter the two collisions have taken place.

* 15-44. Two men A and B. each having a weight of 160 lb. stand on the stationary 200-lb. cart. Each then runs with a speed of 3 ft/s measured relative to the cart. Determine the final speed of the cart if (a) A runs and jumps off, then B runs and jumps off the same end. and (b) both run at the same time and jump off at the same time. Neglect the mass of the wheels and assume the jumps are made horizontally.

15-45. The block of mass m travels at vj in the direction 0] shown at the top of the smooth slope. Determine its speed v2 and its direction f)2 when it reaches the bottom.

15-46. The barge B weighs 30 000 lb and supports an automobile weighing 3000 lb. If the barge is not tied to the pier P and someone drives the automobile to the other side of the barge for unloading, determine how far the barge moves away from the pier. Neglect the resistance of the water.

15-47. 'Hie 30-Mg freight car A and 15-Mg freight car B are moving towards each other with the velocities shown. Determine the maximum compression of the spring mounted on car A. Neglect rolling resistance.

*15-48. The barge weighs 45 000 lb and supports two automobiles A and B. which weigh 4000 lb and 3000 lb, respectively. If the automobiles start from rest and drive towards each other, accelerating at ciA = 4 ft/s2 and aR 8 ft/s2 until they reach a constant speed of 6 ft/s relative to the barge, determine the speed of the barge just before the automobiles collide. How much time does this take? Originally the barge is at rest. Neglect water resistance.

15-49. The man M weighs 150 lb and jumps onto the boat B which has a weight of 200 lb. If he has a horizontal component of velocity relative to the boat of 3 ft/s, just before he enters the boat, and the boat is traveling Vu =2 ft/s away from the pier when he makes the jump, determine the resulting velocity of the man and boat.

15-50. The man M weighs 150 lb and jumps onto the boat B which is originally at rest. If he has a horizontal component of velocity of 3 ft/s just before he enters the boat, determine the weight of the boat if it has a velocity of 2 ft/s once the man enters it.

1551. The 20-kg package has a speed of 1.5 m/s when it is delivered to the smooth ramp. After sliding down the ramp it lands onto a 10-kg cart as shown. Determine the speed of the cart and package after the package stops sliding on the cart.

*15-52. The free-rolling ramp has a mass of 40 kg. A 10-kg crate is released from rest at A and slides down 3.5 m to point B. If the surface of the ramp is smooth, determine the ramp's speed when the crate reaches B. Also, what is the velocity of the crate?

15-53. The 80-lb boy and 60-lb girl walk towards each other with a constant speed on the 300-lb cart. If their velocities, measured relative to the cart, arc 3 ft/s to the right and 2 ft/s to the left, respectively, determine the velocities of the boy and girl during the motion. Also, find the distance the cart has traveled at the instant the boy and girl meet.

15-54. The 80-lb boy and 60-lb girl walk towards each other with constant speed on the 300-lb cart. If their velocities measured relative to the cart are 3 ft/s to the right and 2 ft/s to the left, respectively, determine the velocity of the cart while they are walking.

15-55. A tugboat T having a mass of 19 Mg is tied to a barge B having a mass of 75 Mg. If the rope is elastic such that it has a stiffness k = 600kN/m, determine the maximum stretch in the rope during the initial towing. Originally both the tugboat and barge are moving in the same direction with speeds (vr)i = 15 km/h and (u8)i = lOkm/h, respectively. Neglect the resistance of the water.

*15-56. Two boxes A and B, each having a weight of 160 lb, sit on the 500-lb conveyor which is free to roll on the ground. If the belt starts from rest and begins to run with a speed of 3 ft/s, determine the final speed of the conveyor if (a) the boxes are not stacked and A falls off then B falls off. and (b) A is stacked on top of B and both fall off together.

The 10-kg block is held at rest on the smooth inclined plane by the stop block at A. If the 10-g bullet is traveling at 300 m/s when it becomes embedded in the 10-kg block, determine the distance the block will slide up along the plane before momentarily stopping.

A ball having a mass of 200g is released from rest at a height of 400mm above a very large fixed metal surface. If the ball rebounds to a height of 325 mm above the surface, determine the coefficient of restitution between the ball and the surface.

15-59. The 5-Mg truck and 2-Mg car are traveling with the free-rolling velocities shown just before they collide. After the collision, the car moves with a velocity of 15 km/h to the right relative to the truck. Determine the coefficient of restitution between the truck and car and the loss of energy due to the collision.

*15-60. Disk A has a mass of 2 kg and is sliding forward on the smooth surface with a velocity (^)i = 5 m/s when it strikes the 4-kg disk B. which is sliding towards A at (vB), = 2 m/s. with direct central impact. If the coefficient of restitution between the disks is e = 0.4, compute the velocities of A and B just after collision.

15-61. Block A has a mass of 3 kg and is sliding on a rough horizontal surface with a velocity (y^), = 2 m/s when it makes a direct collision with block ft, which has a mass of 2 kg and is originally at rest. If the collision is perfectly elastic (e = 1), determine the velocity of each block just after collision and the distance between the blocks when they stop sliding. The coefficient of kinetic friction between the blocks and the plane is fik 0.3.

15-62. If two disks A and B have the same mass and are subjected to direct central impact such that the collision is perfectly elastic (e = 1), prove that the kinetic energy before collision equals the kinetic energy after collision. The surface upon which they slide is smooth.

Each ball has a mass m and the coefficient of restitution between the balls is e. If they are moving towards one another with a velocity v, determine their speeds after collision. Also, determine their common velocity when they reach the state of maximum deformation. Neglect the size of each ball.

* 15-64. The three balls each have a mass m. If A has a speed v just before a direct collision with B. determine the speed of C after collision. The coefficient of restitution between each pair of balls is e. Neglect the size of each ball.

A 1-lb ball A is traveling horizontally at 20 ft/s when it strikes a 10-lb block B that is at rest. If the coefficient of restitution between A and B is e = 0.6. and the coefficient of kinetic friction between the plane and the block is \(\mu_{k}=0.4\), determine the time for the block B to stop sliding.

If the girl throws the ball with a horizontal velocity of \(v_A=8\mathrm{\ ft}/\mathrm{s}\), determine the distance d so that the ball bounces once on the smooth surface and then lands in the cup at C Take \(e=0.8\).

15-67. The three balls each weigh 0.5 lb and have a coefficient of restitution of e = 0.85. If ball A is released from rest and strikes ball B and then ball B strikes ball C, determine the velocity of each ball after the second collision has occurred. The balls slide without friction.

*15-68. The girl throws the ball with a horizontal velocity of v, = 8 ft/s. If the coefficient of restitution between the ball and the ground is e = 0.8. determine (a) the velocity of the ball just after it rebounds from the ground and (b) the maximum height to which the ball rises after the first bounce.

15-69. A 300-g ball is kicked with a velocity of vA = 25 m/s at point A as shown. If the coefficient of restitution between the ball and the field is e = 0.4. determine the magnitude and direction 0 of the velocity of the rebounding ball at B.

15-70. Two smooth spheres A and B each have a mass m. If A is given a velocity of while sphere B is at rest, determine the velocity of B just after it strikes the wall.The coefficient of restitution for any collision is e.

It was observed that a tennis ball when served horizontally 7.5 ft above the ground strikes the smooth ground at B 20 ft away. Determine the initial velocity \(\mathbf{v}_{A}\) of the ball and the velocity \(\mathbf{v}_{B}(\text { and } \theta)\) of the ball just after it strikes the court at B.Take \(e=0.7\).

*15-72. The tennis ball is struck with a horizontal velocity \A, strikes the smooth ground at B. and bounces upward at 0 = 30. Determine the initial velocity v^, the final velocity vB, and the coefficient of restitution between the ball and the ground.

15-73. The 1 lb ball is dropped from rest and falls a distance of 4 ft before striking the smooth plane at A. If e = 0.8, determine the distance d to where it again strikes the plane at B.

15-74. The I lb ball is dropped from rest and falls a distance of 4 ft before striking the smooth plane at A. If it rebounds and in t = 0.5 s again strikes the plane at B. determine the coefficient of restitution e between the ball and the plane. Also, what is the distance cl?

15-75. The I-kg ball is dropped from rest at point A, 2 m above the smooth plane. If the coefficient of restitution between the ball and the plane is e = 0.6. determine the distance d where the ball again strikes the plane.

*15-76. A ball of mass m is dropped vertically from a height h0 above the ground. If it rebounds to a height of hj. determine the coefficient of restitution between the ball and the ground.

15-77. The cue ball A is given an initial velocity (v4)j = 5 m/s. If it makes a direct collision with ball B (e = 0.8), determine the velocity of B and the angle 0 just after it rebounds from the cushion at C (e' = 0.6). Each ball has a mass of 0.4 kg. Neglect their size.

15-78. Using a slingshot, the boy fires the 0.2-lb marble at the concrete wall, striking it at B. If the coefficient of restitution between the marble and the wall is e = 0.5, determine the speed of the marble after it rebounds from the wall.

15-79. The sphere of mass m falls and strikes the triangular block with a vertical velocity v. If the block rests on a smooth surface and has a mass 3 m. determine its velocity just after the collision.The coefficient of restitution is e.

*15-80. Block A, having a mass m, is released from rest, falls a distance h and strikes the plate B having a mass 2 m. If the coefficient of restitution between A and B is e, determine the velocity of the plate just after collision. The spring has a stiffness k.

15-81. The girl throws the 0.5-kg ball toward the wall with an initial velocity vA = lOm/s. Determine (a) the velocity at which it strikes the wall at B. (b) the velocity at which it rebounds from the wall if the coefficient of restitution e = 0.5, and (c) the distance s from the wall to where it strikes the ground at C.

15-82. The 20-lb box slides on the surface for which ~ 0.3. The box has a velocity v = 15 ft/s when it is 2 ft from the plate. If it strikes the smooth plate, which has a weight of 10 lb and is held in position by an unstretched spring of stiffness k = 400 Ib/ft. determine the maximum compression imparted to the spring. Take e = 0.8 between the box and the plate. Assume that the plate slides smoothly.

Before a cranberry can make it to your dinner plate, it must pass a bouncing test which rates its quality. If cranberries having an \(e \geq 0.8\) are to be accepted, determine the dimensions d and h for the barrier so that when a cranberry falls from rest at A it strikes the incline at B and bounces over the barrier at C.

*15-84. A ball is thrown onto a rough floor at an angle 0. If it rebounds at an angle <t> and the coefficient of kinetic friction is /x, determine the coefficient of restitution e. Neglect the size of the ball. Hint: Show that during impact, the average impulses in the x and y directions are related by /, = Since the time of impact is the same, Fx A/ = iFy A / or Fx = /xFv.

A ball is thrown onto a rough floor at an angle of \(\theta=45^{\circ}\). If it rebounds at the same angle \(\phi=45^{\circ}\), determine the coefficient of kinetic friction between the floor and the ball. The coefficient of restitution is \(e=0.6\). Hint: Show that during impact, the average impulses in the x and y directions are related by \(I_{x}=\mu I_{y}\). Since the time of impact is the same. \(F_x\ \Delta t=\mu F_y\ \Delta t\text{ or }F_x=\mu F_y\).

15-86. The stone A used in the sport of curling slides over the ice track and strikes another stone B as shown. If each stone is smooth and has a weight of 47 lb. and the coefficient of restitution between the stones is e = 0.8. determine their speeds just after collision. Initially A has a velocity of 8 ft/s and B is at rest. Neglect friction.

Two smooth disks A and B each have a mass of 0.5 kg. If both disks are moving with the velocities shown when they collide, determine their final velocities just after collision.The coefficient of restitution is \(e=0.75\).

*15-88. Two smooth disks A and B each have a mass of 0.5 kg. If both disks are moving with the velocities shown when they collide, determine the coefficient of restitution between the disks if after collision B travels along a line. 30 counterclockwise from the y axis.

Two smooth disks A and B have the initial velocities shown just before they collide at O. If they have masses \(m_A=8\mathrm{\ kg}\text{ and }m_B=6\mathrm{\ kg}\), determine their speeds just after impact.The coefficient of restitution is \(e=0.5 \text {. }\)

15-90. If disk A is sliding along the tangent to disk B and strikes B with a velocity v. determine the velocity of B after the collision and compute the loss of kinetic energy during the collision. Neglect friction. Disk B is originally at rest. The coefficient of restitution is e, and each disk has the same size and mass m.

15-91. Two disks A and B weigh 2 lb and 5 lb, respectively. If they are sliding on the smooth horizontal plane with the velocities shown, determine their velocities just after impact. The coefficient of restitution between the disks is e = 0.6.

*15-92. Two smooth coins A and B. each having the same mass, slide on a smooth surface with the motion shown. Determine the speed of each coin after collision if they move off along the dashed paths. Hint: Since the line of impact has not been defined, apply the conservation of momentum along the x and y axes, respectively.

Disks A and B have a mass of \(15 \mathrm{~kg}\) and \(10 \mathrm{~kg}\), respectively. If they are sliding on a smooth horizontal plane with the velocities shown, determine their speeds just after impact. The coefficient of restitution between them is e=0.8.

15-94. Determine the angular momentum H(> of the particle about point O.

15-95. Determine the angular momentum H() of the particle about point O.

*15-96. Determine the angular momentum H,> of the particle about point P

Determine the total angular momentum \(\mathbf{H}_{O}\) for the system of three particles about point O. All the particles are moving in the x-y plane.

15-98. Determine the angular momentum H(, of each of the two particles about point O. Use a scalar solution.

15-99. Determine the angular momentum H/> of each of the two particles about point P. Use a scalar solution.

*15-100. The small cylinder C has a mass of 10 kg and is attached to the end of a rod whose mass may be neglected. If the frame is subjected to a couple M = (8r + 5) N m, where / is in seconds, and the cylinder is subjected to a force of 60 N. which is always directed as shown, determine the speed of the cylinder when / = 2 s. The cylinder has a speed Vq = 2 m/s when t = 0.

15-101. The 10-lb block rests on a surface for which /x< = 0.5. It is acted upon by a radial force of 2 lb and a horizontal force of 7 lb, always directed at 30 from the tangent to the path as shown. If the block is initially moving in a circular path with a speed u, = 2 ft/s at the instant the forces are applied, determine the time required before the tension in cord AB becomes 20 lb. Neglect the size of the block for the calculation.

15-102. The 10-lb block is originally at rest on the smooth surface. It is acted upon by a radial force of 2 lb and a horizontal force of 7 lb. always directed at 30 from the tangent to the path as shown. Determine the time required to break the cord, which requires a tension T = 30 lb. What is the speed of the block when this occurs? Neglect the size of the block for the calculation.

15-103. A 4-lb ball B is traveling around in a circle of radius r, = 3 ft with a speed (ufl), = 6 ft/s. If the attached cord is pulled dowrn through the hole with a constant speed vr = 2 ft/s, determine the ball's speed at the instant r2 = 2 ft. How much work has to be done to pull dowrn the cord? Neglect friction and the size of the ball.

*15-104. A 4-lb ball B is traveling around in a circle of radius /*, = 3 ft with a speed = 6 ft/s. If the attached cord is pulled down through the hole with a constant speed vr = 2 ft/s, determine how much time is required for the ball to reach a speed of 12 ft/s. How far r2 is the ball from the hole when this occurs? Neglect friction and the size of the ball.

15-105. The four 5-lb spheres are rigidly attached to the crossbar frame having a negligible weight. If a couple moment M = (0.5/ + 0.8) lb-ft, where / is in seconds, is applied as shown, determine the speed of each of the spheres in 4 seconds starling from rest. Neglect the size of the spheres.

15-106. A small particle having a mass m is placed inside the semicircular tube. The particle is placed at the position shown and released. Apply the principle of angular momentum about point O CLM0 = I/()), and show that the motion of the particle is governed by the differential equation f) + (g/R)siT\0 = 0.

15-107. The ball B has a weight of 5 lb and is originally rotating in a circle. As shown, the cord AB has a length of 3 ft and passes through the hole A, which is 2 ft above the plane of motion. If 1.5 ft of cord is pulled through the hole, determine the speed of the ball when it moves in a circular path at C.

*15-108. A child having a mass of 50 kg holds her legs up as shown as she swings downward from rest at 01 = 30. Her center of mass is located at point G,. When she is at the bottom position ti = 0, she suddenly lets her legs come down, shifting her center of mass to position G2. Determine her speed in the upswing due to this sudden movement and the angle 02 to which she swings before momentarily coming to rest. Treat the childs body as a particle.

15-109. The 150-lb car of an amusement park ride is connected to a rotating telescopic boom. When r = 15 ft, the car is moving on a horizontal circular path with a speed of 30 ft/s. If the boom is shortened at a rate of 3ft/s, determine the speed of the car when r = 10 ft. Also, find the work done by the axial force F along the boom. Neglect the size of the car and the mass of the boom.

15-110. An amusement park ride consists of a car which is attached to the cable OA.The car rotates in a horizontal circular path and is brought to a speed v, = 4 ft/s when r = 12 ft. The cable is then pulled in at the constant rate of 0.5 ft/s. Determine the speed of the car in 3 s.

15-111. The 800-lb roller-coaster car starts from rest on the track having the shape of a cylindrical helix. If the helix descends 8 ft for every one revolution, determine the speed of the car in t = 4 s. Also, how far has the car descended in this time? Neglect friction and the size of the car.

*15-112. A small block having a mass of 0.1 kg is given a horizontal velocity of t, = 0.4 m/s when /, = 500 mm. It slides along the smooth conical surface. Determine the distance h it must descend for it to reach a speed of ?;2 = 2 m/s. Also, what is the angle of descent ti, that is, the angle measured from the horizontal to the tangent of the path?

15-113. An earth satellite of mass 700 kg is launched into a free-flight trajectory about the earth with an initial speed of vA = lOkm/s when the distance from the center of the earth is rA = 15 Mm. If the launch angle at this position is <f>A 70, determine the speed vR of the satellite and its closest distance rH from the center of the earth. The earth has a mass Me = 5.976(1024) kg. Hint: Under these conditions, the satellite is subjected only to the earth's gravitational force, F = GM/n Jr, Eq. 13-1. For part of the solution, use the conservation of energy.

15-114. The fire boat discharges two streams of seawater, each at a flow of 0.25 m3/s and with a nozzle velocity of 50 m/s. Determine the tension developed in the anchor chain, needed to secure the boat. The density of seawater is pm, = 1020 kg/m3.

A jet of water having a cross-sectional area of \(4 \mathrm{in}^2\) strikes the fixed blade with a speed of \(25 \mathrm{ft} / \mathrm{s}\). Determine the horizontal and vertical components of force which the blade exerts on the water, \(\gamma_w=62.4 \mathrm{lb} / \mathrm{ft}^3\).

*15-116. The 200-kg boat is powered by the fan which develops a slipstream having a diameter of 0.75 m. If the fan ejects air with a speed of 14 m/s, measured relative to the boat, determine the initial acceleration of the boat if it is initially at rest. Assume that air has a constant density of pw = 1.22 kg/m3 and that the entering air is essentially at rest. Neglect the drag resistance of the water.

The chute is used to divert the flow of water, \(Q=0.6 \mathrm{\ m}^{3} / \mathrm{s}\). If the water has a cross-sectional area of \(0.05 \mathrm{\ m}^{2}\). determine the force components at the pin D and roller C necessary for equilibrium. Neglect the weight of the chute and weight of the water on the chute, \(\rho_{w}=1\ \mathrm{Mg} / \mathrm{m}^{3}\).

15-118. The buckets on the Pelton wheel are subjected to a 2-in-diameter jet of water, which has a velocity of 150 ft/s. If each bucket is traveling at 95 ft/s when the water strikes it. determine the power developed by the bucket. yw = 62.4 lb/ft3.

The blade divides the jet of water having a diameter of 3 in. If one-fourth of the water flows downward while the other three-fourths flows upwards, and the total flow is \(Q=0.5\mathrm{\ ft}^3/\mathrm{s}\), determine the horizontal and vertical components of force exerted on the blade by the jet, \(\gamma_{\mathrm{w}}=62.4\mathrm{\ lb}/\mathrm{ft}^3\).

*15-120. The fan draws air through a vent with a speed of 12 ft/s. If the cross-sectional area of the vent is 2 ft2, determine the horizontal thrust on the blade. The specific weight of the air is y(l = 0.076 lb/ft3.

15-121. The gauge pressure of water at C is 40 lb/in2. If water flows out of the pipe at A and B with velocities vA = 12 ft/s and vB = 25 ft/s, determine the horizontal and vertical components of force exerted on the elbow necessary to hold the pipe assembly in equilibrium. Neglect the weight of water within the pipe and the weight of the pipe.The pipe has a diameter of 0.75 in. at C. and at A and B the diameter is 0.5 in. yw = 62.4 lb/ft3.

15-122. A speedboat is powered by the jet drive shown. Seawater is drawn into the pump housing at the rate of 20 ft?/s through a 6-in.-diameter intake A. An impeller accelerates the water flow and forces it out horizontally through a 4*in.-diameter nozzle 13. Determine the horizontal and vertical components of thrust exerted on the speedboat. The specific weight of seawater is ym. = 64.3 lb/ft3.

15-123. A plow located on the front of a locomotive scoops up snow at the rate of 10fi3/s and stores it in the train. If the locomotive is traveling at a constant speed of 12 ft/s, determine the resistance to motion caused by the shoveling.The specific weight of snow is y% = 6 lb/ft3.

* 15-124. The boat has a mass of 180 kg and is traveling forward on a river with a constant velocity of 70 km/h. measured relative to the river. The river is flowing in the opposite direction at 5 km/h. If a tube is placed in the water, as shown, and it collects 40 kg of water in the boat in 80 s. determine the horizontal thrust T on the tube that is required to overcome the resistance due to the water collection and yet maintain the constant speed of the boat. Pk = 1 Mg/m3.

Water is discharged from a nozzle with a velocity of 12 m/s and strikes the blade mounted on the 20-kg cart. Determine the tension developed in the cord, needed to hold the cart stationary, and the normal reaction of the wheels on the cart. The nozzle has a diameter of 50 mm and the density of water is \(\rho_{w}=1000 \mathrm{\ kg} / \mathrm{mg}^{3}\).

15-126. Water is flowing from the 150-mm-diameter fire hydrant with a velocity vB = 15 m/s. Determine the horizontal and vertical components of force and the moment developed at the base joint A. if the static (gauge) pressure at A is 50 kPa.The diameter of the fire hydrant at A is 200 mm. pH, = I Mg/m3.

A coil of heavy open chain is used to reduce the stopping distance of a sled that has a mass M and travels at a speed of \(v_{0}\). Determine the required mass per unit length of the chain needed to slow down the sled to \((1 / 2) v_{0}\) within a distance x = s if the sled is hooked to the chain at x = 0. Neglect friction between the chain and the ground.

The car is used to scoop up water that is lying in a trough at the tracks. Determine the force needed to pull the car forward at constant velocity v for each of the three cases. The scoop has a cross-sectional area A and the density of water is \(\rho_{w}\).

15-129. The water flow enters below the hydrant at C at the rate of 0.75 m3/s. It is then divided equally between the two outlets at A and B. If the gauge pressure at C is 300 kPa. determine the horizontal and vertical force reactions and the moment reaction on the fixed support at ('.The diameter of the two outlets at A and B is 75 mm. and the diameter of the inlet pipe at C is 150 mm. The density of water is pw -1000 kg/m3. Neglect the mass of the contained water and the hydrant.

15-130. The mini hovercraft is designed so that air is drawn in at a constant rale of 20 m3/s by the fan blade and channeled to provide a vertical thrust F. just sufficient to lift the hovercraft off the water, while the remaining air is channeled to produce a horizontal thrust T on the hovercraft. If the air is discharged horizontally at 200 m/s and vertically at 800 m/s, determine the thrust T produced. The hovercraft and its passenger have a total mass of 1.5 Mg. and the density of the air is pa = 1.20 kg/m3.

15-131. Sand is discharged from the silo at A at a rate of 50 kg/s with a vertical velocity of 10 m/s onto the conveyor belt, which is moving with a constant velocity of 1.5 m/s. If the conveyor system and the sand on it have a total mass of 750 kg and center of mass at point (7, determine the horizontal and vertical components of reaction at the pin support B and roller support A. Neglect the thickness of the conveyor.

*15-132. The fan blows air at 6000 ft3/min. If the fan has a weight of 30 lb and a center of gravity at G. determine the smallest diameter d of its base so that it will not tip over. The specific weight of air is y = 0.076 lb/ft3.

15-133. The bend is connected to the pipe at flanges A and B as shown. If the diameter of the pipe is 1 ft and it carries a discharge of 50 ft3/$, determine the horizontal and vertical components of force reaction and the moment reaction exerted at the fixed base D of the support. The total weight of the bend and the water within it is 500 lb. with a mass center at point G. The gauge pressure of the water at the flanges at A and B are 15 psi and 12 psi, respectively. Assume that no force is transferred to the flanges at A and B. The specific weight of water is yw = 62.4 lb/ft3.

15-134. Each of the two stages A and B of the rocket has a mass of 2 Mg when their fuel tanks are empty. They each carry 500 kg of fuel and are capable of consuming it at a rate of 50 kg/s and eject it with a constant velocity of 2500 m/s, measured with respect to the rocket.The rocket is launched vertically from rest by first igniting stage B. Then stage A is ignited immediately after all the fuel in B is consumed and A has separated from B. Determine the maximum velocity of stage A. Neglect drag resistance and the variation of the rocket's weight with altitude.

15-135. A power lawn mower hovers very close over the ground. This is done by drawing air in at a speed of 6 m/s through an intake unit A. which has a cross-sectional area of Aa = 0.25 nr. and then discharging it at the ground. B. where the cross-sectional area is AH = 0.35 nr. If air at A is subjected only to atmospheric pressure, determine the air pressure which the lawn mower exerts on the ground when the weight of the mower is freely supported and no load is placed on the handle. The mower has a mass of 15 kg with center of mass at G. Assume that air has a constant density of pa = 1.22 kg/m3.

*15-136. The 12-ft-long open-link chain has 2 ft of its end suspended from the hole as shown. If a force of P= 10 lb is applied to its end and the chain is released from rest, determine the velocity of the chain at the instant the entire chain has been extended.The chain has a weight of 2 lb/ ft.

15-137. A chain of mass m{) per unit length is loosely coiled on the floor. If one end is subjected to a constant force P when y = 0. determine the velocity of the chain as a function of y.

15-138. The second stage of the two-stage rocket weighs 2000 lb (empty) and is launched from the first stage with a velocity of 3000 mi/h.The fuel in the second stage weighs 1000 lb. If it is consumed at the rate of 50 lb/s and ejected with a relative velocity of 8000 ft/s, determine the acceleration of the second stage just after the engine is fired. What is the rockets acceleration just before all the fuel is consumed? Neglect the effect of gravitation.

15-139. The missile weighs 40 000 lb. The constant thrust provided by the turbojet engine is T = 15 000 lb. Additional thrust is provided by two rocket boosters RThe propellant in each booster is burned at a constant rate of 150 lb/s, with a relative exhaust velocity of 3000 ft/s. If the mass of the propellant lost by the turbojet engine can be neglected, determine the velocity of the missile after the 4-s burn time of the boosters.The initial velocity of the missile is 300 mi/h.

*15-140. The 10-Mg helicopter carries a bucket containing 500 kg of water, which is used to fight fires. If it hovers over the land in a fixed position and then releases 50 kg/s of water at 10 m/s. measured relative to the helicopter, determine the initial upward acceleration the helicopter experiences as the water is being released.

15-141. The earthmovcr initially carries 10 m3 of sand having a density of 1520 kg/m3. The sand is unloaded horizontally through a 2.5 nr dumping port P at a rale of 900 kg/s measured relative to the port. Determine the resultant tractive force F at its front wheels if the acceleration of the earthmover is 0.1 m/s2 when half the sand is dumped. When empty, the earthmover has a mass of 30 Mg. Neglect any resistance to forward motion and the mass of the wheels. The rear wheels are free to roll.

15-142. The 12-Mg jet airplane has a constant speed of 950 km/h when it is flying along a horizontal straight line. Air enters the intake scoops S at the rate of 50 m3/s. If the engine burns fuel at the rate of 0.4 kg/s and the gas (air and fuel) is exhausted relative to the plane with a speed of 450 m/s. determine the resultant drag force exerted on the plane by air resistance. Assume that air has a constant density of 1.22 kg/m3. Hint: Since mass both enters and exits the plane, Eqs. 15-28 and 15-29 must be combined to yield

15-143. The jet is traveling at a speed of 500mi/h. 30 with the horizontal. If the fuel is being spent at 3 Ib/s. and the engine takes in air at 400 Ib/s. whereas the exhaust gas (air and fuel) has a relative speed of 32 800 ft/s. determine the acceleration of the plane at this instant. The drag resistance of the air is PD = (OJv2) lb, where the speed is measured in ft/s. The jet has a weight of 15 000 lb. Hint: See Prob. 15-142.

A four-engine commercial jumbo jet is cruising at a constant speed of 800 km/h in level flight when all four engines are in operation. Each of the engines is capable of discharging combustion gases with a velocity of 775 m/s relative to the plane. If during a test two of the engines, one on each side of the plane, are shut off. determine the new cruising speed of the jet. Assume that air resistance (drag) is proportional to the square of the speed, that is, \(F_{D}=c v^{2}\), where c is a constant to be determined. Neglect the loss of mass due to fuel consumption.

15-145. The car has a mass m0 and is used to tow the smooth chain having a total length / and a mass per unit of length m'. If the chain is originally piled up, determine the tractive force Fthat must be supplied by the rear wheels of the car. necessary to maintain a constant speed v while the chain is being drawn out.

15-146. A rocket has an empty weight of 500 lb and carries 300 lb of fuel. If the fuel is burned at the rate of 1.5 Ib/s and ejected with a velocity of 4400 I't/s relative to the rocket, determine the maximum speed attained by the rocket starting from rest. Neglect the effect of gravitation on the rocket.

Determine the magnitude of force F as a function of time, which must be applied to the end of the cord at A to raise the hook H with a constant speed \(v=0.4 \mathrm{\ m} / \mathrm{s}\). Initially the chain is at rest on the ground. Neglect the mass of the cord and the hook.The chain has a mass of 2 kg/m.

*15-148. The truck has a mass of 50 Mg when empty. When it is unloading 5 m3 of sand at a constant rate of 0.8 m3/s, the sand flows out the back at a speed of 7 m/s, measured relative to the truck, in the direction shown. If the truck is free to roll, determine its initial acceleration just as the load begins to empty. Neglect the mass of the wheels and any frictional resistance to motion. The density of sand is ps = 1520 kg/m3.

15-149. The chain has a total length L < d and a mass per unit length of m'. If a portion h of the chain is suspended over the table and released, determine the velocity of its end A as a function of its position y. Neglect friction.

P15-1. The ball travels to the left when it is struck by the bat. If the ball then moves horizontally to the right, determine which measurements you could make in order to determine the net impulse given to the ball. Use numerical values to give an example of how this can be done.

P15-2. The steel wrecking ball is suspended from the boom using an old rubber tire A. The crane operator lifts the ball then allows it to drop freely to break up the concrete. Explain, using appropriate numerical data, why it is a good idea to use the rubber tire for this work.

P15-3. The train engine on the left, A, is at rest, and the one on the right. B. is coasting to the left. If the engines are identical, use numerical values to show how to determine the maximum compression in each of the spring bumpers that are mounted in the front of the engines. Each engine is free to roll.

P15-4. Three train cars each have the same mass and are rolling freely when they strike the fixed bumper. Legs AB and BC on the bumper are pin connected at their ends and the angle BAC is 30 and BCA is 60. Compare the average impulse in each leg needed to stop the motion if the cars have no bumper and if the cars have a spring bumper. Use appropriate numerical values to explain your answer.