The cue ball A is given an initial velocity (vA)1 = 5 m>s. If it makes a direct

Determine the impulse of the force for t = 2 s. a) 100 N 30 b) 200 N c) F (6t) N 3 4 5 d) F 30 t (s) F (N) 20 1 3 e) k 10 N/m 80 N f) 60 N 5 3

Determine the linear momentum of the 10-kg block. a) 10 m/s 6 m b) 30 2 m/s c) 100 N 60 N 45 3 m/s P

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.

If the coefficient of kinetic friction between the 150-lb crate and the ground is \(\mu_{k} = 0.2\), determine the speed of the crate when t = 4 s. The crate starts from rest and is towed by the 100-lb force.

The motor exerts a force of \(F = (20t^{2}) \ N\) on the cable, where t 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 \(\mu_{s} = 0.3\) and \(\mu_{k} = 0.25\), 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.

The 2.5-Mg four-wheel-drive SUV tows the 1.5-Mg trailer. The traction force developed at the wheels is \(F_{D} = 9 \ kN\). Determine the speed of the truck in 20 s, starting from rest. Also, determine the tension developed in the coupling, A, between the SUV and the trailer. Neglect the mass of the wheels.

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.

The freight cars A 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 are in contact for 0.5 s, find the average impulsive force which acts between them.

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.

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.

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.

The 15-Mg tank car A and 25-Mg freight car B travel toward 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.

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 \((v_{b})_{1} = 20 \ m/s\). 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.

Disk A has a mass of 2 kg and slides on the smooth horizontal plane with a velocity of 3 m/s. Disk B has a mass of 11 kg and is initially at rest. If after impact A has a velocity of 1 m/s, parallel to the positive x axis, determine the speed of disk B after impact.

Two disks A and B each have a mass of 1 kg 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 \(\mathbf{H}_{O}\) about point O.

The 2-kg particle A has the velocity shown. Determine its angular momentum \mathbf{H}_{P} about point P.

Initially the 5-kg block is moving 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 F = 5 N is applied to the block, determine its speed when t = 3 s. Neglect the size of the block.

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

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 \(M = (0.9t^{2}) \ N \cdot m\), where t is in seconds, determine the speed of the sphere at the instant t = 5 s starting from rest.

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

A man kicks the 150-g ball such that it leaves the ground at an angle of 60 and strikes the ground at the same elevation a distance of 12 m away. Determine the impulse of his foot on the ball at A. Neglect the impulse caused by the balls weight while its being kicked.

A 20-lb block slides down a \(30^{\circ}\) 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 \(\mu_k = 0.25\).

The uniform beam has a weight of 5000 lb. Determine the average tension in each of the two cables AB and AC if the beam is given an upward speed of 8 ft/s in 1.5 s starting from rest. Neglect the mass of the cables.

Each of the cables can sustain a maximum tension of 5000 lb. If the uniform beam has a weight of 5000 lb, determine the shortest time possible to lift the beam with a speed of 10 ft/s starting from rest.

A hockey puck is traveling to the left with a velocity of \(v_{1} = 10 \ m/s\) when it is struck by a hockey stick and given a velocity of \(v_{2} = 20 \ m/s\) as shown. Determine the magnitude of the net impulse exerted by the hockey stick on the puck. The puck has a mass of 0.2 kg.

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 F acting on the engine wheels.

Crates A and B weigh 100 lb and 50 lb, respectively. If they start from rest, determine their speed when t = 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 \(\mu_{k} = 0.25\).

The automobile has a weight of 2700 lb and is traveling forward at 4 ft/s when it crashes into the wall. If the impact occurs in 0.06 s, determine the average impulsive force acting on the car. Assume the brakes are not applied. If the coefficient of kinetic friction between the wheels and the pavement is \(\mu_{k} = 0.3\), calculate the impulsive force on the wall if the brakes were applied during the crash.The brakes are applied to all four wheels so that all the wheels slip.

The 200-kg crate rests on the ground for which the coefficients of static and kinetic friction are \(\mu_{s} = 0.5\) and \(\mu_{k} = 0.4\), 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 crate when t = 4 s. Originally the tension in the cable is zero. Hint: First determine the force needed to begin moving the crate.

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 \(\mu_{k} = 0.2\).

During operation the jack hammer strikes the concrete surface with a force which is indicated in the graph. To achieve this the 2-kg spike S is fired into the surface at 90 m/s. Determine the speed of the spike just after rebounding.

For a short period of time, the frictional driving force acting on the wheels of the 2.5-Mg van is \(F_{D} = (600t^{2}) \ N\), where t is in seconds. If the van has a speed of 20 km/h when t = 0, determine its speed when t = 5 s.

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.

A tankcar has a mass of 20 Mg and is freely rolling to the right with a speed of 0.75 m/s. If it strikes the barrier, determine the horizontal impulse needed to stop the car if the spring in the bumper B has a stiffness (a) \(k \rightarrow \infty\) (bumper is rigid), and (b) k = 15 kN/m.

The motor, M, pulls on the cable with a force \(F = (10t^{2} + 300) \ N\), where t is in seconds. If the 100 kg crate is originally at rest at t = 0, determine its speed when t = 4 s. Neglect the mass of the cable and pulleys. Hint: First find the time needed to begin lifting the crate.

The choice of a seating material for moving vehicles depends upon its ability to resist shock and vibration. From the data shown in the graphs, determine the impulses created by a falling weight onto a sample of urethane foam and CONFOR foam.

The towing force acting on the 400-kg safe varies as shown on the graph. Determine its speed, starting from rest, when t = 8 s. How far has it traveled during this time?

The motor exerts a force F on the 40-kg crate as shown in the graph. Determine the speed of the crate when t = 3 s and when t = 6 s. When t = 0, the crate is moving downward at 10 m/s.

The 30-kg slider block is moving to the left with a speed of 5 m/s when it is acted upon by the forces \(\mathbf{F}_{1}\) and \mathbf{F}_{2}. If these loadings vary in the manner shown on the graph, determine the speed of the block at t = 6 s. Neglect friction and the mass of the pulleys and cords.

The 200-lb cabinet is subjected to the force F = 20(t + 1) lb where t is in seconds. If the cabinet is initially moving to the left with a velocity of 20 ft/s, determine its speed when t = 5 s. Neglect the size of the rollers.

If it takes 35 s for the 50-Mg tugboat to increase its speed uniformly to 25 km/h, starting 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.

The thrust on the 4-Mg rocket sled is shown in the graph. Determine the sleds maximum velocity and the distance the sled travels when t = 35 s. Neglect friction.

The motor pulls on the cable at A with a force \(F = (30 + t^{2}) \ lb\), where t is in seconds. If the 34-lb crate is originally on the ground at t = 0, determine its speed in t = 4 s. Neglect the mass of the cable and pulleys. Hint: First find the time needed to begin lifting the crate.

The motor pulls on the cable at A with a force \(F = (e^{2t}) \ lb\), where t is in seconds. If the 34-lb crate is originally at rest on the ground at t = 0, determine the crate’s velocity when t = 2 s. Neglect the mass of the cable and pulleys. Hint: First find the time needed to begin lifting the crate.

The balloon has a total mass of 400 kg including the passengers and ballast. The balloon is rising at a constant velocity of 18 km/h when h = 10 m. If the man drops the 40-kg sand bag, determine the velocity of the balloon when the bag strikes the ground. Neglect air resistance.

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 slides along 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 B, attached to the \(x^{\prime}\) axis that moves at a constant velocity of 2 m/s relative to A.

The 20-kg crate is lifted by a force of \(F = (100 + 5t^{2}) \ N\), where t is in seconds. Determine the speed of the crate when t = 3 s, starting from rest.

The 20-kg crate is lifted by a force of \(F = (100 + 5t^{2}) \ N\), where t is in seconds. Determine how high the crate has moved upward when t = 3 s, starting from rest.

In case of emergency, the gas actuator is used to move a 75-kg block B by exploding a charge C near a pressurized cylinder of negligible mass. As a result of the explosion, the cylinder fractures and the released gas forces the front part of the cylinder, A, to move B forward, giving it a speed of 200 mm/s in 0.4 s. If the coefficient of kinetic friction between B and the floor is \(\mu_{k} = 0.5\), determine the impulse that the actuator imparts to B.

A jet plane having a mass of 7 Mg takes off from an aircraft carrier such that the engine thrust varies as shown by the graph. If the carrier is traveling forward with a speed of 40 km>h, determine the plane’s airspeed after 5 s.

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

Block A weighs 10 lb and block B weighs 3 lb. If B is moving downward with a velocity \((v_{B})_{1} = 3 \ ft/s\) at t = 0, determine the velocity of A when t = 1 s. The coefficient of kinetic friction between the horizontal plane and block A is \(\mu_{A} = 0.15\).

The log has a mass of 500 kg and rests on the ground for which the coefficients of static and kinetic friction are \(\mu_{s} = 0.5\) and \(\mu_{k} = 0.4\), 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.

The 0.15-kg baseball has a speed of v = 30 m/s just before it is struck by the bat. It then travels along the trajectory shown before the outfielder catches it. Determine the magnitude of the average impulsive force imparted to the ball if it is in contact with the bat for 0.75 ms.

The 5-Mg bus B is traveling to the right at 20 m/s. Meanwhile a 2-Mg car A is traveling at 15 m/s to the right. If the vehicles crash and become entangled, determine their common velocity just after the collision. Assume that the vehicles are free to roll during collision.

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

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.

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.

A ballistic pendulum consists of a 4-kg wooden block originally at rest, \(\theta = 0^{\circ}\). When a 2-g bullet strikes and becomes embedded in it, it is observed that the block swings upward to a maximum angle of \(\theta = 6^{\circ}\). Estimate the initial speed of the bullet.

The boy jumps off the flat car at A with a velocity of \(v = 4 \ ft/s\) relative to the car as shown. If he lands on the second flat car B, determine the final speed of both cars after the motion. Each car has a weight of 80 lb. The boy’s weight is 60 lb. Both cars are originally at rest. Neglect the mass of the car’s wheels.

A 0.03-lb bullet traveling at 1300 ft/s strikes the 10-lb wooden block and exits the other side at 50 ft/s as shown. Determine the speed of the block just after the bullet exits the block, and also determine how far the block slides before it stops. The coefficient of kinetic friction between the block and the surface is \(\mu_{k} = 0.5\).

A 0.03-lb bullet traveling at 1300 ft/s strikes the 10-lb wooden block and exits the other side at 50 ft/s as shown. Determine the speed of the block just after the bullet exits the block. Also, determine the average normal force on the block if the bullet passes through it in 1 ms, and the time the block slides before it stops. The coefficient of kinetic friction between the block and the surface is \(\mu_{k} = 0.5\).

The 20-g bullet is traveling at 400 m/s when it becomes embedded in the 2-kg stationary block. Determine the distance the block will slide before it stops. The coefficient of kinetic friction between the block and the plane is \(\mu_{k} = 0.2\).

A toboggan having a mass of 10 kg starts from rest at A and carries a girl and boy having a mass of 40 kg and 45 kg, respectively. When the toboggan reaches the bottom of the slope at B, the boy is pushed off from the back with a horizontal velocity of \(v_{b/t} = 2 \ m/s\), measured relative to the toboggan. Determine the velocity of the toboggan afterwards. Neglect friction in the calculation.

The block of mass m travels at \(v_{1}\) in the direction \(\theta_{1}\) shown at the top of the smooth slope. Determine its speed \(v_{2}\) and its direction \(\theta_{2}\) when it reaches the bottom.

The two blocks A and B each have a mass of 5 kg and are suspended from parallel cords. A spring, having a stiffness of k = 60 N/m, is attached to B and is compressed 0.3 m against A and B as shown. Determine the maximum angles \(\theta\) and \(\phi\) of the cords when the blocks are released from rest and the spring becomes unstretched.

The 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.

Blocks A and B have masses of 40 kg and 60 kg, respectively. They are placed on a smooth surface and the spring connected between them is stretched 2 m. If they are released from rest, determine the speeds of both blocks the instant the spring becomes unstretched.

A boy A having a weight of 80 lb and a girl B having a weight of 65 lb stand motionless at the ends of the toboggan, which has a weight of 20 lb. If they exchange positions, A going to B and then B going to A’s original position, determine the final position of the toboggan just after the motion. Neglect friction between the toboggan and the snow.

A boy A having a weight of 80 lb and a girl B having a weight of 65 lb stand motionless at the ends of the toboggan, which has a weight of 20 lb. If A walks to B and stops, and both walk back together to the original position of A, determine the final position of the toboggan just after the motion stops. Neglect friction between the toboggan and the snow.

The 10-Mg barge B supports a 2-Mg automobile A. If someone drives the automobile to the other side of the barge, determine how far the barge moves. Neglect the resistance of the water.

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?

Block A has a mass of 5 kg and is placed on the smooth triangular block B having a mass of 30 kg. If the system is released from rest, determine the distance B moves from point O when A reaches the bottom. Neglect the size of block A.

Solve Prob. 15–53 if the coefficient of kinetic friction between A and B is \(\mu_{k} = 0.3\). Neglect friction between block B and the horizontal plane.

The cart has a mass of 3 kg and rolls freely from A down the slope. When it reaches the bottom, a spring loaded gun fires a 0.5-kg ball out the back with a horizontal velocity of \(v_{b/c} = 0.6 \ m/s\), measured relative to the cart. Determine the final velocity of the cart.

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.

Disk A has a mass of 250 g and is sliding on a smooth horizontal surface with an initial velocity \((v_{A})_1 = 2 \ m/s\). It makes a direct collision with disk B, which has a mass of 175 g and is originally at rest. If both disks are of the same size and the collision is perfectly elastic (e = 1), determine the velocity of each disk just after collision. Show that the kinetic energy of the disks before and after collision is the same.

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.

Disk A has a mass of 2 kg and is sliding forward on the smooth surface with a velocity \((v_{A})_{1} = 5 \ m/s\) when it strikes the 4-kg disk B, which is sliding towards A at \((v_{B})_{1} = 2 \ m/s\), 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.

The 15-kg block A slides on the surface for which \(\mu_{k} = 0.3\). The block has a velocity v = 10 m/s when it is s = 4 m from the 10-kg block B. If the unstretched spring has a stiffness k = 1000 N/m, determine the maximum compression of the spring due to the collision. Take \(e = 0.6\).

The four smooth balls each have the same mass m. If A and B are rolling forward with velocity v and strike C, explain why after collision C and D each move off with velocity v. Why doesn’t D move off with velocity 2v? The collision is elastic, e = 1. Neglect the size of each ball.

The four balls each have the same mass m. If A and B are rolling forward with velocity v and strike C, determine the velocity of each ball after the first three collisions. Take e = 0.5 between each ball.

Ball A has a mass of 3 kg and is moving with a velocity of 8 m/s when it makes a direct collision with ball B, which has a mass of 2 kg and is moving with a velocity of 4 m/s. If e = 0.7, determine the velocity of each ball just after the collision. Neglect the size of the balls.

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.

Block A, having a mass m, is released from rest, falls a distance h and strikes the plate B having a mass 2m. 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.

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.

A pitching machine throws the 0.5-kg ball toward the wall with an initial velocity \(v_{A} = 10 \ m/s\) as shown. Determine (a) the velocity at which it strikes the wall at B, (b) the velocity at which it rebounds from the wall if e = 0.5, and (c) the distance s from the wall to where it strikes the ground at C.

A 300-g ball is kicked with a velocity of \(v_{A} = 25 \ m/s\) 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 \(\theta\) of the velocity of the rebounding ball at B.

Two smooth spheres A and B each have a mass m. If A is given a velocity of \(v_{0}\), 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 \(\mathbf{v}_{B}\) (and \(\theta\)) of the ball just after it strikes the court at B. Take e = 0.7.

The tennis ball is struck with a horizontal velocity \(\mathbf{v}_{A}\), strikes the smooth ground at B, and bounces upward at \(\theta = 30^{\circ}\). Determine the initial velocity \(\mathbf{v}_{A}\), the final velocity \(\mathbf{v}_{B}\), and the coefficient of restitution between the ball and the ground.

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.

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, \(30^{\circ}\) counterclockwise from the y axis.

The 0.5-kg ball is fired from the tube at A with a velocity of \(v = 6 \ m/s\). If the coefficient of restitution between the ball and the surface is e = 0.8, determine the height h after it bounces off the surface.

A ball of mass m is dropped vertically from a height \(h_{0}\) above the ground. If it rebounds to a height of \(h_{1}\), determine the coefficient of restitution between the ball and the ground.

The cue ball A is given an initial velocity \((v_{A})_{1} = 5 \ m/s\). If it makes a direct collision with ball B (e = 0.8), determine the velocity of B and the angle \(\theta\) just after it rebounds from the cushion at \(C \ (e^{\prime} = 0.6)\). Each ball has a mass of 0.4 kg. Neglect their size.

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.

The two disks A and B have a mass of 3 kg and 5 kg, respectively. If they collide with the initial velocities shown, determine their velocities just after impact. The coefficient of restitution is e = 0.65.

A ball of negligible size and mass m is given a velocity of \(\mathbf{v}_{0}\) on the center of the cart which has a mass M and is originally at rest. If the coefficient of restitution between the ball and walls A and B is e, determine the velocity of the ball and the cart just after the ball strikes A. Also, determine the total time needed for the ball to strike A, rebound, then strike B, and rebound and then return to the center of the cart. Neglect friction.

The girl throws the 0.5-kg ball toward the wall with an initial velocity \(v_{A} = 10 m/s\). 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.

The 20-lb box slides on the surface for which \(\mu_ {k} = 0.3\). 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 lb/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.

The 10-lb collar B is at rest, and when it is in the position shown the spring is unstretched. If another 1-lb collar A strikes it so that B slides 4 ft on the smooth rod before momentarily stopping, determine the velocity of A just after impact, and the average force exerted between A and B during the impact if the impact occurs in 0.002 s. The coefficient of restitution between A and B is e = 0.5.

A ball is thrown onto a rough floor at an angle \(\theta\). If it rebounds at an angle \(\phi\) and the coefficient of kinetic friction is \(\mu\), 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 \(I_{x} = \mu I_{y}\). Since the time of impact is the same, \(F_{x} \ \Delta t = \mu F_{y} \ \Delta t\) or \(F_{x} = \mu F_{y}\).

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, \(F_{x} \ \Delta t = \mu F_{y} \ \Delta t\) or \(F_{x} = \mu F_{y}\).

Two smooth billiard balls A and B each have a mass of 200 g. If A strikes B with a velocity \((v_{A})_{1} = 1.5 \ m/s\) as shown, determine their final velocities just after collision. Ball B is originally at rest and the coefficient of restitution is e = 0.85. Neglect the size of each ball.

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.

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 “stone” is e = 0.8, determine the time required just after collision for B to slide off the runway. This requires the horizontal component of displacement to be 3 ft.

Two smooth disks A and B have the initial velocities shown just before they collide. If they have masses \(m_{A} = 4 \ kg\) and \(m_{B} = 2 \ kg\), determine their speeds just after impact. The coefficient of restitution is e = 0.8.

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.

The 200-g billiard ball is moving with a speed of 2.5 m/s when it strikes the side of the pool table at A. If the coefficient of restitution between the ball and the side of the table is e = 0.6, determine the speed of the ball just after striking the table twice, i.e., at A, then at B. Neglect the size of the ball.

The two billiard balls A and B are originally in contact with one another when a third ball C strikes each of them at the same time as shown. If ball C remains at rest after the collision, determine the coefficient of restitution. All the balls have the same mass. Neglect the size of each ball.

Disks A and B have a mass of 15 kg and 10 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.

Determine the angular momentum \(\mathbf{H}_{O}\) of the 6-lb particle about point O.

Determine the angular momentum \(\mathbf{H}_{p}\) of the 6-lb particle about point P.

Determine the angular momentum \(\mathbf{H}_{o}\) of each of the two particles about point O.

Determine the angular momentum \(\mathbf{H}_{p}\) of each of the two particles about point P.

Determine the angular momentum \(\mathbf{H}_{O}\) of the 3-kg particle about point O.

Determine the angular momentum \(\mathbf{H}_{P}\) of the 3-kg particle about point P.

Each ball has a negligible size and a mass of 10 kg and is attached to the end of a rod whose mass may be neglected. If the rod is subjected to a torque \(M = (t^{2} + 2) \ N \cdot m\), where t is in seconds, determine the speed of each ball when t = 3 s. Each ball has a speed v = 2 m/s when t = 0.

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 when t = 4 s. Also, how far has the car descended in this time? Neglect friction and the size of the car.

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 time required for the car to attain a speed of 60 ft/s. Neglect friction and the size of the car.

A 4-lb ball B is traveling around in a circle of radius \(r_{1} = 3 \ ft\) with a speed \((v_{B})_{1} = 6 \ ft/s\). If the attached cord is pulled down through the hole with a constant speed \(v_{r} = 2 \ ft/s\), determine the ball’s speed at the instant \(r_{2} = 2 \ ft\). How much work has to be done to pull down the cord? Neglect friction and the size of the ball.

A 4-lb ball B is traveling around in a circle of radius \(r_{1} = 3 \ ft\) with a speed \((v_{B})_{1} = 6 \ ft/s\). If the attached cord is pulled down through the hole with a constant speed \(v_{r} = 2 \ ft/s\), determine how much time is required for the ball to reach a speed of 12 ft/s. How far \(r_{2}\) is the ball from the hole when this occurs? Neglect friction and the size of the ball.

The two blocks A and B each have a mass of 400 g. The blocks are fixed to the horizontal rods, and their initial velocity along the circular path is 2 m/s. If a couple moment of \(M = (0.6) \ N \cdot m\) is applied about CD of the frame, determine the speed of the blocks when t = 3 s. The mass of the frame is negligible, and it is free to rotate about CD. Neglect the size of the blocks.

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 \(O \ (\Sigma M_{O} = H_O)\), and show that the motion of the particle is governed by the differential equation \(\ddot{\theta}+ (g/R) sin \ \theta = 0\).

If the rod of negligible mass is subjected to a couple moment of \(M = (30t^{2}) \ N \cdot m\), and the engine of the car supplies a traction force of F = (15t) N to the wheels, where t is in seconds, determine the speed of the car at the instant t = 5 s. The car starts from rest. The total mass of the car and rider is 150 kg. Neglect the size of the car.

When the 2-kg bob is given a horizontal speed of 1.5 m/s, it begins to move around the horizontal circular path A. If the force F on the cord is increased, the bob rises and then moves around the horizontal circular path B. Determine the speed of the bob around path B. Also, find the work done by force F.

The elastic cord has an unstretched length \(l_{0} = 1.5 \ ft\) and a stiffness k = 12 lb/ft. It is attached to a fixed point at A and a block at B, which has a weight of 2 lb. If the block is released from rest from the position shown, determine its speed when it reaches point C after it slides along the smooth guide. After leaving the guide, it is launched onto the smooth horizontal plane. Determine if the cord becomes unstretched. Also, calculate the angular momentum of the block about point A, at any instant after it passes point C.

The amusement park ride consists of a 200-kg car and passenger that are traveling at 3 m/s along a circular path having a radius of 8 m. If at t = 0, the cable OA is pulled in toward O at 0.5 m/s, determine the speed of the car when t = 4 s. Also, determine the work done to pull in the cable.

A box having a weight of 8 lb is moving around in a circle of radius \(r_{A} = 2 \ ft\) with a speed of \((v_{A})_{1} = 5 \ ft/s\) while connected to the end of a rope. If the rope is pulled inward with a constant speed of \(v_{r} = 4 \ ft/s\), determine the speed of the box at the instant \(r_{B} = 1 \ ft\). How much work is done after pulling in the rope from A to B? Neglect friction and the size of the box.

A toboggan and rider, having a total mass of 150 kg, enter horizontally tangent to a \(90^{\circ}\) circular curve with a velocity of \(v_{A} = 70 \ km/h\). If the track is flat and banked at an angle of \(60^{\circ}\), determine the speed \(v_{B}\) and the angle \(\theta\) of “descent,” measured from the horizontal in a vertical x–z plane, at which the toboggan exists at B. Neglect friction in the calculation.

An earth satellite of mass 700 kg is launched into a free-flight trajectory about the earth with an initial speed of \(v_{A} = 10 \ km/s\) when the distance from the center of the earth is \(r_{A} = 15 \ Mm\). If the launch angle at this position is \(\phi_{A} = 70^{\circ}\), determine the speed \(v_{B}\) of the satellite and its closest distance \(r_{B}\) from the center of the earth. The earth has a mass \(M_{e} = 5.976(10^{24}) \ kg\). Hint: Under these conditions, the satellite is subjected only to the earth’s gravitational force, \(F = GM_{e} m_{s} \ / \ r^{2}\), Eq. 13–1. For part of the solution, use the conservation of energy.

The fire boat discharges two streams of seawater, each at a flow of \(0.25 \ m^{3} /s\) 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 \(\rho_{s \omega} = 1020 \ kg/m^{3}\).

The chute is used to divert the flow of water, \(Q = 0.6 \ m^{3} \ /s\). If the water has a cross-sectional area of \(0.05 \ m^{2}\), 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 \ Mg/m^{3}.

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 \(\rho_{w} = 1.22 \ kg/m^{3}\) and that the entering air is essentially at rest. Neglect the drag resistance of the water.

The nozzle discharges water at a constant rate of \(2 \ ft^{3} /s\). The cross-sectional area of the nozzle at A is \(4 \ in^{2}\), and at B the cross-sectional area is \(12 \ in^{2}\). If the static gauge pressure due to the water at B is \(2 \ lb/in^{2}\), determine the magnitude of force which must be applied by the coupling at B to hold the nozzle in place. Neglect the weight of the nozzle and the water within it. \(\gamma_{w} = 62.4 \ lb/ft^{3}\).

The blade divides the jet of water having a diameter of 4 in. If one-half of the water flows to the right while the other half flows to the left, and the total flow is \(Q = 1.5 \ ft^{3} /s\), determine the vertical force exerted on the blade by the jet, \(\gamma_{\omega} = 62.4 \ lb/ft^{3}\).

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 upward, and the total flow is \(Q = 0.5 \ ft^{3}/s\), determine the horizontal and vertical components of force exerted on the blade by the jet, \(\gamma_{w} = 62.4 \ lb/ft^{3}\).

The gauge pressure of water at A is 150.5 kPa. Water flows through the pipe at A with a velocity of 18 m/s, and out the pipe at B and C with the same velocity v. 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 50 mm at A, and at B and C the diameter is 30 mm. \(\rho_{w} = 1000 \ kg/m^{3}\).

The gauge pressure of water at C is \(40 \ lb/in^{2}\). If water flows out of the pipe at A and B with velocities \(v_{A} = 12 \ ft/s\) and \(v_{B} = 25 \ ft/s\), 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. \(\gamma_{w} = 62.4 \ lb/ft^{3}\).

The fountain shoots water in the direction shown. If the water is discharged at \(30^{\circ}\) from the horizontal, and the cross-sectional area of the water stream is approximately \(2 \ in^{2}\), determine the force it exerts on the concrete wall at B. \(\gamma_{w} = 62.4 \ lb/ft^{3}\).

A plow located on the front of a locomotive scoops up snow at the rate of \(10 \ ft^{3}/s\) 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 \(\gamma_{s} = 6 \ lb/ft^{3}\).

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. \(\rho_{w} = 1 \ Mg/m^{3}\).

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 \(\rho_{w} = 1000 \ kg/m^{3}\).

A snowblower having a scoop S with a cross-sectional area of \(A_{s} = 0.12 \ m^{3}\) is pushed into snow with a speed of \(v_{s} = 0.5 \ m/s\). The machine discharges the snow through a tube T that has a cross-sectional area of \(A_{T} = 0.03 \ m^{2}\) and is directed \(60^{\circ}\) from the horizontal. If the density of snow is \(\rho_{s} = 104 \ kg/m^{3}\) , determine the horizontal force P required to push the blower forward, and the resultant frictional force F of the wheels on the ground, necessary to prevent the blower from moving sideways. The wheels roll freely.

The fan blows air at \(6000 \ ft^{3} /min\). 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 \(\gamma = 0.076 \ lb/ft^{3}\).

The nozzle has a diameter of 40 mm. If it discharges water uniformly with a downward velocity of 20 m/s against the fixed blade, determine the vertical force exerted by the water on the blade. \(\rho_{w} = 1 \ Mg/m^{3}\).

The water flow enters below the hydrant at C at the rate of \(0.75 \ m^{3} /s\). 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 C. 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 \(\rho_{w} = 1000 \ kg/m^{3}\). Neglect the mass of the contained water and the hydrant.

Sand drops onto the 2-Mg empty rail car at 50 kg/s from a conveyor belt. If the car is initially coasting at 4 m/s, determine the speed of the car as a function of time.

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 G, determine the horizontal and vertical components of reaction at the pin support B and roller support A. Neglect the thickness of the conveyor.

Sand is deposited from a chute onto a conveyor belt which is moving at 0.5 m/s. If the sand is assumed to fall vertically onto the belt at A at the rate of 4 kg/s, determine the belt tension \(F_{B}\) to the right of A. The belt is free to move over the conveyor rollers and its tension to the left of A is \(F_{C} = 400 \ N\).

The tractor together with the empty tank has a total mass of 4 Mg. The tank is filled with 2 Mg of water. The water is discharged at a constant rate of 50 kg/s with a constant velocity of 5 m/s, measured relative to the tractor. If the tractor starts from rest, and the rear wheels provide a resultant traction force of 250 N, determine the velocity and acceleration of the tractor at the instant the tank becomes empty.

A rocket has an empty weight of 500 lb and carries 300 lb of fuel. If the fuel is burned at the rate of 15 lb/s and ejected with a relative velocity of 4400 ft/s, determine the maximum speed attained by the rocket starting from rest. Neglect the effect of gravitation on the rocket.

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 \(A_{A} = 0.25 \ m^{2}\), and then discharging it at the ground, B, where the cross-sectional area is \(A_{B} = 0.35 \ m^{2}\). 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 \(\rho_{a} = 1.22 \ kg/m^{3}\).

The rocket car has a mass of 2 Mg (empty) and carries 120 kg of fuel. If the fuel is consumed at a constant rate of 6 kg/s and ejected from the car with a relative velocity of 800 m/s, determine the maximum speed attained by the car starting from rest. The drag resistance due to the atmosphere is \(F_{D} = (6.8v^{2}) \ N\), where v is the speed in m/s.

If the chain is lowered at a constant speed v = 4 ft/s, determine the normal reaction exerted on the floor as a function of time. The chain has a weight of 5 lb/ft and a total length of 20 ft.

The second stage of a 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 rocket’s acceleration just before all the fuel is consumed? Neglect the effect of gravitation.

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 B. The 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.

The jet is traveling at a speed of 720 km/h. If the fuel is being spent at 0.8 kg/s, and the engine takes in air at 200 kg/s, whereas the exhaust gas (air and fuel) has a relative speed of 12 000 m/s, determine the acceleration of the plane at this instant. The drag resistance of the air is \(F_{D} = (55 \ v^{2})\), where the speed is measured in m/s. The jet has a mass of 7 Mg.

The rope has a mass \(m^{\prime}\) per unit length. If the end length y = h is draped off the edge of the table, and released, determine the velocity of its end A for any position y, as the rope uncoils and begins to fall.

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 \(50 \ m^{3} /s\). 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 \(1.22 \ kg/m^{3}\). Hint: Since mass both enters and exits the plane, Eqs. 15–28 and 15–29 must be combined to yield \(\Sigma F_{s} = m \frac{dv}{dt} - v_{D/e} \frac{dm_{e}}{dt} + v_{D/i} \frac{dm_i}{dt}\).

The jet is traveling at a speed of 500 mi/h, \(30^{\circ}\) with the horizontal. If the fuel is being spent at 3 lb/s, and the engine takes in air at 400 lb/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 \(F_{D} = (0.7v^{2}) \ lb\), 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, \(F_{D} = cv^{2}\), where c is a constant to be determined. Neglect the loss of mass due to fuel consumption.

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.

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 lb/s and ejected with a velocity of 4400 ft/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 m/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.

The truck has a mass of 50 Mg when empty. When it is unloading \(5 \ m^{3}\) of sand at a constant rate of \(0.8 \ m^{3} /s\), 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 \(\rho_{s} = 1520 \ kg/m^{3}\).

The car has a mass \(m_{0}\) and is used to tow the smooth chain having a total length \(l\) and a mass per unit of length \(m^{\prime}\). If the chain is originally piled up, determine the tractive force F that must be supplied by the rear wheels of the car, necessary to maintain a constant speed v while the chain is being drawn out.

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.

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.

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.

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 \(30^{\circ}\) and BCA is \(60^{\circ}\). 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.

Packages having a mass of 6 kg slide down a smooth chute and land horizontally with a speed of 3 m/s on the surface of a conveyor belt. If the coefficient of kinetic friction between the belt and a package is \(\mu_{k} = 0.2\), determine the time needed to bring the package to rest on the belt if the belt is moving in the same direction as the package with a speed v = 1 m/s.

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 \(\mu_{k} = 0.4\). If the block is initially moving up the plane at \(v_{0} = 2 \ m/s\), and at this instant (t = 0) the motor develops a tension in the cord of \(T = (300 + 120 \sqrt{t}) \ N\), where t is in seconds, determine the velocity of the block when t = 2 s.

A 20-kg block is originally at rest on a horizontal surface for which the coefficient of static friction is \(\mu_{s} = 0.6\) and the coefficient of kinetic friction is \(\mu_{k} = 0.5\). If a horizontal force F is applied such that it varies with time as shown, determine the speed of the block in 10 s. Hint: First determine the time needed to overcome friction and start the block moving.

The three freight cars A, B, and C have masses of 10 Mg, 5 Mg, and 20 Mg, respectively. They are 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 after the two collisions have taken place.

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 \(\mu_{k} = 0.2\).

Block A has a mass of 3 kg and is sliding on a rough horizontal surface with a velocity \((v_{A})_{1} = 2 \ m/s\) when it makes a direct collision with block B, 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 \(\mu_{k} = 0.3\).

Two smooth billiard balls A and B have an equal mass of m = 200 g. If A strikes B with a velocity of \((v_{A})_{1} = 2 \ m/s\) as shown, determine their final velocities just after collision. Ball B is originally at rest and the coefficient of restitution is e = 0.75.

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 \(M = (8t^{2} + 5) \ N \cdot m\), where t 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 t = 2 s. The cylinder has a speed \(v_{0} = 2 \ m/s\) when t = 0.