A chain of length l and mass m falls through a small hole

A 30-g bullet is fired with a horizontal velocity of 450 m/s and becomes embedded in block B which has a mass of 3 kg. After the impact, block B slides on 30-kg carrier C until it impacts the end of the carrier. Knowing the impact between B and C is perfectly plastic and the coefficient of kinetic friction between B and C is 0.2, determine (a) the velocity of the bullet and B after the first impact, (b) the final velocity of the carrier.

A 30-g bullet is fired with a horizontal velocity of 450 m/s through 3-kg block B and becomes embedded in carrier C which has a mass of 30 kg. After the impact, block B slides 0.3 m on C before coming to rest relative to the carrier. Knowing the coefficient of kinetic friction between B and C is 0.2, determine (a) the velocity of the bullet immediately after passing through B, (b) the final velocity of the carrier.

Car A weighing 4000 lb and car B weighing 3700 lb are at rest on a 22-ton flatcar which is also at rest. Cars A and B then accelerate and quickly reach constant speeds relative to the flatcar of 7 ft/s and 3.5 ft/s, respectively, before decelerating to a stop at the opposite end of the flatcar. Neglecting friction and rolling resistance, determine the velocity of the flatcar when the cars are moving at constant speeds.

A bullet is fired with a horizontal velocity of 1500 ft/s through a 6-lb block A and becomes embedded in a 4.95-lb block B. Knowing that blocks A and B start moving with velocities of 5 ft/s and 9 ft/s, respectively, determine (a) the weight of the bullet, (b) its velocity as it travels from block A to block B.

Two swimmers A and B, of weight 190 lb and 125 lb, respectively, are at diagonally opposite corners of a floating raft when they realize that the raft has broken away from its anchor. Swimmer A immediately starts walking toward B at a speed of 2 ft/s relative to the raft. Knowing that the raft weighs 300 lb, determine (a) the speed of the raft if B does not move, (b) the speed with which B must walk toward A if the raft is not to move.

A 180-lb man and a 120-lb woman stand side by side at the same end of a 300-lb boat, ready to dive, each with a 16-ft/s velocity relative to the boat. Determine the velocity of the boat after they have both dived, if (a) the woman dives first, (b) the man dives first.

A 40-Mg boxcar A is moving in a railroad switchyard with a velocity of 9 km/h toward cars B and C, which are both at rest with their brakes off at a short distance from each other. Car B is a 25-Mg flatcar supporting a 30-Mg container, and car C is a 35-Mg boxcar. As the cars hit each other they get automatically and tightly coupled. Determine the velocity of car A immediately after each of the two couplings, assuming that the container (a) does not slide on the flatcar, (b) slides after the first coupling but hits a stop before the second coupling occurs, (c) slides and hits the stop only after the second coupling has occurred.

Packages in an automobile parts supply house are transported to the loading dock by pushing them along on a roller track with very little friction. At the instant shown packages B and C are at rest and package A has a velocity of 2 m/s. Knowing that the coefficient of restitution between the packages is 0.3, determine (a) the velocity of package C after A hits B and B hits C, (b) the velocity of A after it hits B for the second time.

A system consists of three particles A, B, and C. We know that \(m_{A} = 3 \ kg\), \(m_{B} = 2 \ kg\), and \(m_{C} = 4 \ kg\) and that the velocities of the particles expressed in m/s are, respectively, \(v_{A} = 4i + 2j + 2k\), \(v_{B} = 4i + 3j\), and \(v_{C} = 22i + 4j + 2k\). Determine the angular momentum \(H_{O}\) of the system about O.

For the system of particles of Prob. 14.9, determine (a) the position vector \(\overline{\mathbf{r}}\) of the mass center G of the system, (b) the linear momentum \(m \overline{\mathbf{v}}\) of the system, (c) the angular momentum \(H_{G}\) of the system about G. Also verify that the answers to this problem and to Prob. 14.9 satisfy the equation given in Prob. 14.27.

A system consists of three particles A, B, and C. We know that \(W_{A}=5 \ \mathrm{lb}, \ W_{B}=4 \ \mathrm{lb} \text {, and } W_{C}=3 \ \mathrm{lb}\) and that the velocities of the particles expressed in ft/s are, respectively, \(\mathbf{v}_{A}=2 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}, \ \mathbf{v}_{B}=v_{x} \mathbf{i}+v_{y} \mathbf{j}+v_{z} \mathbf{k}, \text { and } \mathbf{v}_{C}=-3 \mathbf{i}-2 \mathbf{j}+\mathbf{k}\). Determine (a) the components \(v_{x}\) and \(v_{y}\) of the velocity of particle B for which the angular momentum \(\mathbf{H}_{O}\) of the system about O is parallel to the x axis, (b) the value of \(\mathbf{H}_{O}\).

For the system of particles of Prob. 14.11, determine (a) the components \(v_{x}\) and \(v_{z}\) of the velocity of particle B for which the angular momentum \(\mathbf{H}_{O}\) of the system about O is parallel to the z axis, (b) the value of \(\mathbf{H}_{O}\).

A system consists of three particles A, B, and C. We know that \(m_A=3\mathrm{\ kg}\), \(m_B=4\mathrm{\ kg}\), and \(m_c=5\mathrm{\ kg}\) and that the velocities of the particles expressed in m/s are, respectively, \(\mathbf{v}_{A}=-4 \mathbf{i}+4 \mathbf{j}+6 \mathbf{k}\), \(\mathbf{v}_{B}=-6 \mathbf{i}+8 \mathbf{j}+4 \mathbf{k}\), and \(\mathbf{v}_{C}=2 \mathbf{i}-6 \mathbf{j}-4 \mathbf{k}\). Determine the angular momentum \(\mathbf{H}_{O}\) of the system about O.

For the system of particles of Prob. 14.13, determine (a) the position vector \(\overline{\mathbf{r}}\) of the mass center G of the system, (b) the linear momentum \(m \overline{\mathbf{v}}\) of the system, (c) the angular momentum \(\mathbf{H}_{G}\) of the system about G. Also verify that the answers to this problem and to Prob. 14.13 satisfy the equation given in Prob. 14.27.

A 13-kg projectile is passing through the origin O with a velocity \(\mathbf{v}_0=(35\mathrm{\ m}/\mathrm{s})\mathbf{i}\) when it explodes into two fragments A and B, of mass 5 kg and 8 kg, respectively. Knowing that 3 s later the position of fragment A is (90 m, 7 m, -14 m), determine the position of fragment B at the same instant. Assume \(a_y=-g=-9.81\mathrm{\ m}/\mathrm{s}^2\) and neglect air resistance.

A 300-kg space vehicle traveling with a velocity \(\mathbf{v}_0=(360\mathrm{\ m}/\mathrm{s})\mathbf{i}\) passes through the origin O at t = 0. Explosive charges then separate the vehicle into three parts A, B, and C, with mass, respectively, 150 kg, 100 kg, and 50 kg. Knowing that at t = 4 s, the positions of parts A and B are observed to be A (1170 m, -290 m, -585 m) and B (1975 m, 365 m, 800 m), determine the corresponding position of part C. Neglect the effect of gravity.

A 2-kg model rocket is launched vertically and reaches an altitude of 70 m with a speed of 30 m/s at the end of powered flight, time t = 0. As the rocket approaches its maximum altitude it explodes into two parts of masses \(m_A=0.7\mathrm{\ kg}\) and \(m_B=1.3\mathrm{\ kg}\). Part A is observed to strike the ground 80 m west of the launch point at t = 6 s. Determine the position of part B at that time.

An 18-kg cannonball and a 12-kg cannonball are chained together and fired horizontally with a velocity of 165 m/s from the top of a 15-m wall. The chain breaks during the flight of the cannonballs and the 12-kg cannonball strikes the ground at t = 1.5 s, at a distance of 240 m from the foot of the wall, and 7 m to the right of the line of fire. Determine the position of the other cannonball at that instant. Neglect the resistance of the air.

Car A was traveling east at high speed when it collided at point O with car B, which was traveling north at 45 mi/h. Car C, which was traveling west at 60 mi/h, was 32 ft east and 10 ft north of point O at the time of the collision. Because the pavement was wet, the driver of car C could not prevent his car from sliding into the other two cars, and the three cars, stuck together, kept sliding until they hit the utility pole P. Knowing that the weights of cars A, B, and C are, respectively, 3000 lb, 2600 lb, and 2400 lb, and neglecting the forces exerted on the cars by the wet pavement, solve the problems indicated. Knowing that the speed of car A was 75 mi/h and that the time elapsed from the first collision to the stop at P was 2.4 s, determine the coordinates of the utility pole P.

Car A was traveling east at high speed when it collided at point O with car B, which was traveling north at 45 mi/h. Car C, which was traveling west at 60 mi/h, was 32 ft east and 10 ft north of point O at the time of the collision. Because the pavement was wet, the driver of car C could not prevent his car from sliding into the other two cars, and the three cars, stuck together, kept sliding until they hit the utility pole P. Knowing that the weights of cars A, B, and C are, respectively, 3000 lb, 2600 lb, and 2400 lb, and neglecting the forces exerted on the cars by the wet pavement, solve the problems indicated. Knowing that the coordinates of the utility pole are \(x_p=46\ \mathrm{ft}\) and \(y_p=59\ \mathrm{ft}\), determine (a) the time elapsed from the first collision to the stop at P, (b) the speed of car A.

An expert archer demonstrates his ability by hitting tennis balls thrown by an assistant. A 2-oz tennis ball has a velocity of (32 ft/s)i - (7 ft/s)j and is 33 ft above the ground when it is hit by a 1.2-oz arrow traveling with a velocity of (165 ft/s)j + (230 ft/s)k where j is directed upwards. Determine the position P where the ball and arrow will hit the ground, relative to point O located directly under the point of impact.

Two spheres, each of mass m, can slide freely on a frictionless, horizontal surface. Sphere A is moving at a speed \(v_0=16\mathrm{\ ft}/\mathrm{s}\) when it strikes sphere B which is at rest and the impact causes sphere B to break into two pieces, each of mass m/2. Knowing that 0.7 s after the collision one piece reaches point C and 0.9 s after the collision the other piece reaches point D, determine (a) the velocity of sphere A after the collision, (b) the angle u and the speeds of the two pieces after the collision.

In a game of pool, ball A is moving with a velocity \(\mathbf{v}_{0}\) when it strikes balls B and C which are at rest and aligned as shown. Knowing that after the collision the three balls move in the directions indicated and that \(v_0=12\mathrm{\ ft}/\mathrm{s}\) and \(v_C=6.29\mathrm{\ ft}/\mathrm{s}\), determine the magnitude of the velocity of (a) ball A, (b) ball B.

A 6-kg shell moving with a velocity \(\mathbf{v}_0=(12\mathrm{\ m}/\mathrm{s})\mathbf{i}-(9\mathrm{\ m}/\mathrm{s})\mathbf{j}-(360\mathrm{\ m}/\mathrm{s})\mathbf{k}\) explodes at point D into three fragments A, B, and C of mass, respectively, 3 kg, 2 kg, and 1 kg. Knowing that the fragments hit the vertical wall at the points indicated, determine the speed of each fragment immediately after the explosion. Assume that elevation changes due to gravity may be neglected.

A 6-kg shell moving with a velocity \(\mathbf{v}_0=(12\mathrm{\ m}/\mathrm{s})\mathbf{i}-(9\mathrm{\ m}/\mathrm{s})\mathbf{j}-(360\mathrm{\ m}/\mathrm{s})\mathbf{k}\) explodes at point D into three fragments A, B, and C of mass, respectively, 2 kg, 1 kg, and 3 kg. Knowing that the fragments hit the vertical wall at the points indicated, determine the speed of each fragment immediately after the explosion. Assume that elevation changes due to gravity may be neglected.

In a scattering experiment, an alpha particle A is projected with the velocity \(\mathbf{u}_0=-(600\mathrm{\ m}/\mathrm{s})\mathbf{i}+(750\mathrm{\ m}/\mathrm{s})\mathbf{j}-(800\mathrm{\ m}/\mathrm{s})\mathbf{k}\) into a stream of oxygen nuclei moving with a common velocity \(\mathbf{v}_0=(600\mathrm{\ m}/\mathrm{s})\mathbf{j}\). After colliding successively with the nuclei B and C, particle A is observed to move along the path defined by the points \(A_1\ (280,\ 240,\ 120)\) and \(A_2\ (360,\ 320,\ 160)\), while nuclei B and C are observed to move along paths defined, respectively, by \(B_1\ (147,\ 220,\ 130)\) and \(B_2\ (114,\ 290,\ 120)\), and by \(C_1\ (240,\ 232,\ 90)\) and \(C_2\ (240,\ 280,\ 75)\). All paths are along straight lines and all coordinates are expressed in millimeters. Knowing that the mass of an oxygen nucleus is four times that of an alpha particle, determine the speed of each of the three particles after the collisions.

Derive the relation \(\mathbf{H}_{O}=\bar{\mathbf{r}} \times m \bar{\mathbf{v}}+H_{G}\) between the angular momenta \(\(\mathbf{H}_{O}\) and \(\(\mathbf{H}_{G}\) defined in Eqs. (14.7) and (14.24), respectively. The vectors \(\bar{\mathbf{r}}\) and \(\bar{\mathbf{v}}\) define, respectively, the position and velocity of the mass center G of the system of particles relative to the newtonian frame of reference Oxyz, and m represents the total mass of the system.

Show that Eq. (14.23) may be derived directly from Eq. (14.11) by substituting for \(\mathbf{H}_{O}\) the expression given in Prob. 14.27.

Consider the frame of reference \(A x^{\prime} y^{\prime} z^{\prime}\) in translation with respect to the newtonian frame of reference Oxyz. We define the angular momentum \(\mathbf{H}_{A}^{\prime}\) of a system of n particles about A as the sum \(\mathbf{H}_{A}^{\prime}=\sum_{i=1}^{n} \mathbf{r}_{i}^{\prime} \times m_{i} \mathbf{v}_{i}^{\prime}\) (1) of the moments about A of the momenta \(m_{i} \mathbf{v}_{i}^{\prime}\) of the particles in their motion relative to the frame \(A x^{\prime} y^{\prime} z^{\prime}\). Denoting by \(\mathbf{H}_{A}\) the sum \(\mathbf{H}_{A}=\sum_{i=1}^{n} \mathbf{r}_{i}^{\prime} \times m_{i} \mathbf{v}_{i}\) of the moments about A of the momenta \(m_{i} \mathbf{v}_{i}\) of the particles in their motion relative to the newtonian frame Oxyz, show that \(\mathbf{H}_{A}=\mathbf{H}_{A}^{\prime}\) at a given instant if, and only if, one of the following conditions is satisfied at that instant: (a) A has zero velocity with respect to the frame Oxyz, (b) A coincides with the mass center G of the system, (c) the velocity \(\mathbf{v}_{A}\) relative to Oxyz is directed along the line AG.

Show that the relation \(\Sigma \mathbf{M}_{A}=\dot{\mathbf{H}}_{A}^{\prime}\), where \(\mathbf{H}_{A}^{\prime}\) is defined by Eq. (1) of Prob. 14.29 and where \(\Sigma \mathbf{M}_{A}\) represents the sum of the moments about A of the external forces acting on the system of particles, is valid if, and only if, one of the following conditions is satisfied: (a) the frame \(A x^{\prime} y^{\prime} z^{\prime}\) is itself a newtonian frame of reference, (b) A coincides with the mass center G, (c) the acceleration \(\mathbf{a}_{A}\) of A relative to Oxyz is directed along the line AG.

Determine the energy lost due to friction and the impacts for Prob. 14.1.

In Prob. 14.4, determine the energy lost as the bullet (a) passes through block A, (b) becomes embedded in block B.

In Prob. 14.6, determine the work done by the woman and by the man as each dives from the boat, assuming that the woman dives first.

Determine the energy lost as a result of the series of collisions described in Prob. 14.8.

Two automobiles A and B, of mass \(m_{A}\) and \(m_{B}\), respectively, are traveling in opposite directions when they collide head on. The impact is assumed perfectly plastic, and it is further assumed that the energy absorbed by each automobile is equal to its loss of kinetic energy with respect to a moving frame of reference attached to the mass center of the two-vehicle system. Denoting by \(E_{A}\) and \(E_{B}\), respectively, the energy absorbed by automobile A and by automobile B, (a) show that \(E_{A} / E_{B}=m_{B} / m_{A}\), that is, the amount of energy absorbed by each vehicle is inversely proportional to its mass, (b) compute \(E_{A}\) and \(E_{B}\), knowing that \(m_A=1600\mathrm{\ kg}\) and \(m_B=900\mathrm{\ kg}\) and that the speeds of A and B are, respectively, 90 km/h and 60 km/h.

It is assumed that each of the two automobiles involved in the collision described in Prob. 14.35 had been designed to safely withstand a test in which it crashed into a solid, immovable wall at the speed \(v_{0}\).. The severity of the collision of Prob. 14.35 may then be measured for each vehicle by the ratio of the energy it absorbed in the collision to the energy it absorbed in the test. On that basis, show that the collision described in Prob. 14.35 is \(\left(m_{A} / m_{B}\right)^{2}\) times more severe for automobile B than for automobile A.

Solve Sample Prob. 14.4, assuming that cart A is given an initial horizontal velocity \(\mathbf{v}_{0}\) while ball B is at rest.

Two hemispheres are held together by a cord which maintains a spring under compression (the spring is not attached to the hemispheres). The potential energy of the compressed spring is 120 J and the assembly has an initial velocity \(\mathbf{v}_{0}\) of magnitude \(v_0=8\mathrm{\ m}/\mathrm{s}\). Knowing that the cord is severed when \(\mathrm{u}=30^{\circ}\), causing the hemispheres to fly apart, determine the resulting velocity of each hemisphere.

A 15-lb block B starts from rest and slides on the 25-lb wedge A, which is supported by a horizontal surface. Neglecting friction, determine (a) the velocity of B relative to A after it has slid 3 ft down the inclined surface of the wedge, (b) the corresponding velocity of A.

A 40-lb block B is suspended from a 6-ft cord attached to a 60-lb cart A, which may roll freely on a frictionless, horizontal track. If the system is released from rest in the position shown, determine the velocities of A and B as B passes directly under A.

In a game of pool, ball A is moving with a velocity \(\mathbf{v}_{0}\) of magnitude \(v_0=15\mathrm{\ ft}/\mathrm{s}\) when it strikes balls B and C, which are at rest and aligned as shown. Knowing that after the collision the three balls move in the directions indicated and assuming frictionless surfaces and perfectly elastic impact (i.e., conservation of energy), determine the magnitudes of the velocities \(\mathbf{v}_{A}\), \(\mathbf{v}_{B}\), and \(\mathbf{v}_{C}\).

In a game of pool, ball A is moving with a velocity \(\mathbf{v}_{0}\) of magnitude \(v_0=15\mathrm{\ ft}/\mathrm{s}\) when it strikes balls B and C, which are at rest and aligned as shown. Knowing that after the collision the three balls move in the directions indicated and assuming frictionless surfaces and perfectly elastic impact (i.e., conservation of energy), determine the magnitudes of the velocities \(\mathbf{v}_{A}\), \(\mathbf{v}_{B}\), and \(\mathbf{v}_{C}\).

Three spheres, each of mass m, can slide freely on a frictionless, horizontal surface. Spheres A and B are attached to an inextensible, inelastic cord of length l and are at rest in the position shown when sphere B is struck squarely by sphere C which is moving to the right with a velocity \(\mathbf{v}_{0}\). Knowing that the cord is slack when sphere B is struck by sphere C and assuming perfectly elastic impact between B and C, determine (a) the velocity of each sphere immediately after the cord becomes taut, (b) the fraction of the initial kinetic energy of the system which is dissipated when the cord becomes taut.

In a game of pool, ball A is moving with the velocity \(\mathbf{v}_{0}=v_{0} \mathbf{i}\) when it strikes balls B and C, which are at rest side by side. Assuming frictionless surfaces and perfectly elastic impact (i.e., conservation of energy), determine the final velocity of each ball, assuming that the path of A is (a) perfectly centered and that A strikes B and C simultaneously, (b) not perfectly centered and that A strikes B slightly before it strikes C.

Two small spheres A and B, of mass 2.5 kg and 1 kg, respectively, are connected by a rigid rod of negligible mass. The two spheres are resting on a horizontal, frictionless surface when A is suddenly given the velocity \(\mathbf{v}_0=(3.5\mathrm{\ m}/\mathrm{s})\mathbf{i}\). Determine (a) the linear momentum of the system and its angular momentum about its mass center G, (b) the velocities of A and B after the rod AB has rotated through \(180^{\circ}\).

A 900-lb space vehicle traveling with a velocity \(\mathbf{v}_0=(1500\ \mathrm{ft}/\mathrm{s})\mathbf{k}\) passes through the origin O. Explosive charges then separate the vehicle into three parts A, B, and C, with masses of 150 lb, 300 lb, and 450 lb, respectively. Knowing that shortly thereafter the positions of the three parts are, respectively, A(250, 250, 2250), B(600, 1300, 3200), and C(-475, -950, 1900), where the coordinates are expressed in feet, that the velocity of B is \(\mathbf{v}_B=(500\ \mathrm{ft}/\mathrm{s})\mathbf{i}+(1100\mathrm{\ ft}/\mathrm{s})\mathbf{j}+(2100\mathrm{\ ft}/\mathrm{s})\mathbf{k}\), and that the x component of the velocity of C is -400 ft/s, determine the velocity of part A.

Four small disks A, B, C, and D can slide freely on a frictionless horizontal surface. Disks B, C, and D are connected by light rods and are at rest in the position shown when disk B is struck squarely by disk A which is moving to the right with a velocity \(\mathbf{v}_0=(38.5\mathrm{\ ft}/\mathrm{s})\mathbf{i}\). The weights of the disks are \(W_A=W_B=W_C=15\ \mathrm{lb}\), and \(W_D=30\mathrm{\ lb}\). Knowing that the velocities of the disks immediately after the impact are \(\mathbf{v}_A=\mathbf{v}_B=(8.25\ \mathrm{ft}/\mathrm{s})\mathbf{i}\), \(\mathbf{v}_{C}=v_{C} \mathbf{i}\), and \(\mathbf{v}_{D}=v_{D} \mathbf{i}\), determine (a) the speeds \(v_{C}\) and \(v_{D}\), (b) the fraction of the initial kinetic energy of the system which is dissipated during the collision.

In the scattering experiment of Prob. 14.26, it is known that the alpha particle is projected from \(A_0(300,\ 0,\ 300)\) and that it collides with the oxygen nucleus C at Q(240, 200, 100), where all coordinates are expressed in millimeters. Determine the coordinates of point \(B_{0}\) where the original path of nucleus B intersects the zx plane. (Hint. Express that the angular momentum of the three particles about Q is conserved.)

Three identical small spheres, each of weight 2 lb, can slide freely on a horizontal frictionless surface. Spheres B and C are connected by a light rod and are at rest in the position shown when sphere B is struck squarely by sphere A which is moving to the right with a velocity \(\mathbf{v}_0=(8\ \mathrm{ft}/\mathrm{s})\mathbf{i}\). Knowing that \(\mathrm{u}=45^{\circ}\) and that the velocities of spheres A and B immediately after the impact are \(\mathbf{v}_{A}=0\) and \(\mathbf{v}_B=(6\ \mathrm{ft}/\mathrm{s})\mathbf{i}+\left(\mathbf{v}_B\right)_y\mathbf{j}\), determine \(\left(v_{B}\right)_{y}\) and the velocity of C immediately after impact.

Three small spheres A, B, and C, each of mass m, are connected to a small ring D of negligible mass by means of three inextensible, inelastic cords of length l. The spheres can slide freely on a frictionless horizontal surface and are rotating initially at a speed \(v_{0}\) about ring D which is at rest. Suddenly the cord CD breaks. After the other two cords have again become taut, determine (a) the speed of ring D, (b) the relative speed at which spheres A and B rotate about D, (c) the fraction of the original energy of spheres A and B which is dissipated when cords AD and BD again became taut.

In a game of billiards, ball A is given an initial velocity \(\mathbf{v}_{0}\) along the longitudinal axis of the table. It hits ball B and then ball C, which are both at rest. Balls A and C are observed to hit the sides of the table squarely at \(A^{\prime}\) and \(C^{\prime}\), respectively, and ball B is observed to hit the side obliquely at \(B^{\prime}\). Knowing that \(v_0=4\mathrm{\ m}/\mathrm{s}\), \(v_A=1.92\mathrm{\ m}/\mathrm{s}\), and a = 1.65 m, determine (a) the velocities \(\mathbf{v}_{B}\) and \(\mathbf{v}_{C}\) of balls B and C, (b) the point \(C^{\prime}\) where ball C hits the side of the table. Assume frictionless surfaces and perfectly elastic impacts (i.e., conservation of energy).

For the game of billiards of Prob. 14.51, it is now assumed that \(v_0=5\mathrm{\ m}/\mathrm{s}\), \(v_C=3.2\mathrm{\ m}/\mathrm{s}\), and c = 1.22 m. Determine (a) the velocities \(\mathbf{v}_{A}\) and \(\mathbf{v}_{B}\) of balls A and B, (b) the point \(A^{\prime}\) where ball A hits the side of the table.

Two small disks A and B, of mass 3 kg and 1.5 kg, respectively, may slide on a horizontal, frictionless surface. They are connected by a cord, 600 mm long, and spin counterclockwise about their mass center G at the rate of 10 rad/s. At t = 0, the coordinates of G are \(\bar{x}_{0}=0\), \(\bar{y}_0=2\mathrm{\ m}\), and its velocity \(\bar{\mathbf{v}}_0=(1.2\mathrm{\ m}/\mathrm{s})\mathbf{i}+(0.96\mathrm{\ m}/\mathrm{s})\mathbf{j}\). Shortly thereafter the cord breaks; disk A is then observed to move along a path parallel to the y axis and disk B along a path which intersects the x axis at a distance b = 7.5 m from O. Determine (a) the velocities of A and B after the cord breaks, (b) the distance a from the y axis to the path of A.

Two small disks A and B, of mass 2 kg and 1 kg, respectively, may slide on a horizontal and frictionless surface. They are connected by a cord of negligible mass and spin about their mass center G. At t = 0, G is moving with the velocity \(\bar{\mathbf{v}}_{0}\) and its coordinates are \(\bar{x}_{0}=0\), \(\bar{y}_0=1.89\mathrm{\ m}\). Shortly thereafter, the cord breaks and disk A is observed to move with a velocity \(\mathbf{v}_A=(5\mathrm{\ m}/\mathrm{s})\mathbf{j}\) in a straight line and at a distance a = 2.56 m from the y axis, while B moves with a velocity \(\mathbf{v}_B=(7.2\mathrm{\ m}/\mathrm{s})\mathbf{i}-(4.6\mathrm{\ m}/\mathrm{s})\mathbf{j}\) along a path intersecting the x axis at a distance b = 7.48 m from the origin O. Determine (a) the initial velocity \(\bar{\mathbf{v}}_{0}\) of the mass center G of the two disks, (b) the length of the cord initially connecting the two disks, (c) the rate in rad/s at which the disks were spinning about G.

Three small identical spheres A, B, and C, which can slide on a horizontal, frictionless surface, are attached to three 9-in.-long strings, which are tied to a ring G. Initially the spheres rotate clockwise about the ring with a relative velocity of 2.6 ft/s and the ring moves along the x axis with a velocity \(\mathbf{v}_0=(1.3\ \mathrm{ft}/\mathrm{s})\mathbf{i}\). Suddenly the ring breaks and the three spheres move freely in the xy plane with A and B following paths parallel to the y axis at a distance a = 1.0 ft from each other and C following a path parallel to the x axis. Determine (a) the velocity of each sphere, (b) the distance d.

Three small identical spheres A, B, and C, which can slide on a horizontal, frictionless surface, are attached to three strings of length l which are tied to a ring G. Initially the spheres rotate clockwise about the ring which moves along the x axis with a velocity \(\mathbf{v}_{0}\). Suddenly the ring breaks and the three spheres move freely in the xy plane. Knowing that \(\mathbf{v}_A=(3.5\ \mathrm{ft}/\mathrm{s})\mathbf{j}\), \(\mathbf{v}_C=(6.0\ \mathrm{ft}/\mathrm{s})\mathbf{i}\), a = 16 in., and d = 9 in., determine (a) the initial velocity of the ring, (b) the length l of the strings, (c) the rate in rad/s at which the spheres were rotating about G.

A stream of water of cross-section area \(A_{1}\) and velocity \(\mathbf{v}_{1}\) strikes a circular plate which is held motionless by a force P. A hole in the circular plate of area \(A_{2}\) results in a discharge jet having a velocity \(\mathbf{v}_{1}\). Determine the magnitude of P.

A jet ski is placed in a channel and is tethered so that it is stationary. Water enters the jet ski with velocity \(\mathbf{v}_{1}\) and exits with velocity \(\mathbf{v}_{2}\). Knowing the inlet area is \(A_{1}\) and the exit area is \(A_{2}\), determine the tension in the tether.

A stream of water of cross-section area A and velocity \(\mathbf{v}_{1}\) strikes a plate which is held motionless by a force P. Determine the magnitude of P, knowing that \(A=0.75\ \mathrm{in}^2\), \(v_1=80\mathrm{\ ft}/\mathrm{s}\), and V = 0.

A stream of water of cross-section area A and velocity \(\mathbf{v}_{1}\) strikes a plate which moves to the right with a velocity V. Determine the magnitude of V, knowing that \(A=1\ \mathrm{in}^2\), \(v_1=100\mathrm{\ ft}/\mathrm{s}\), and P = 90 lb.

A rotary power plow is used to remove snow from a level section of railroad track. The plow car is placed ahead of an engine which propels it at a constant speed of 20 km/h. The plow car clears 160 Mg of snow per minute, projecting it in the direction shown with a velocity of 12 m/s relative to the plow car. Neglecting friction, determine (a) the force exerted by the engine on the plow car, (b) the lateral force exerted by the track on the plow.

Tree limbs and branches are being fed at A at the rate of 5 kg/s into a shredder which spews the resulting wood chips at C with a velocity of 20 m/s. Determine the horizontal component of the force exerted by the shredder on the truck hitch at D.

Sand falls from three hoppers onto a conveyor belt at a rate of 90 lb/s for each hopper. The sand hits the belt with a vertical velocity \(v_1=10\ \mathrm{ft}/\mathrm{s}\) and is discharged at A with a horizontal velocity \(v_2=13\ \mathrm{ft}/\mathrm{s}\). Knowing that the combined mass of the beam, belt system, and the sand it supports is 1300 lb with a mass center at G, determine the reaction at E.

The stream of water shown flows at a rate of 550 L/min and moves Problems with a velocity of magnitude 18 m/s at both A and B. The vane is supported by a pin and bracket at C and by a load cell at D which can exert only a horizontal force. Neglecting the weight of the vane, determine the components of the reactions at C and D.

The nozzle discharges water at the rate of 340 gal/min. Knowing the velocity of the water at both A and B has a magnitude of 65 ft/s and neglecting the weight of the vane, determine the components of the reactions at C and D \(\left(1\mathrm{\ ft}^3=7.48\mathrm{\ gal}\right)\).

A high-speed jet of air issues from nozzle A with a velocity of \(\mathbf{v}_{A}\) and mass flow rate of 0.36 kg/s. The air impinges on a vane causing it to rotate to the position shown. The vane has a mass of 6 kg. Knowing that the magnitude of the air velocity is equal at A and B, determine (a) the magnitude of the velocity at A, (b) the components of the reactions at O.

Coal is being discharged from a first conveyor belt at the rate of 120 kg/s. It is received at A by a second belt which discharges it again at B. Knowing that \(v_1=3\mathrm{\ m}/\mathrm{s}\) and \(v_2=4.25\mathrm{\ m}/\mathrm{s}\) and that the second belt assembly and the coal it supports have a total mass of 472 kg, determine the components of the reactions at C and D.

A mass q of sand is discharged per unit time from a conveyor belt moving with a velocity \(\mathbf{v}_{0}\). The sand is deflected by a plate at A so that it falls in a vertical stream. After falling a distance h the sand is again deflected by a curved plate at B. Neglecting the friction between the sand and the plates, determine the force required to hold in the position shown (a) plate A, (b) plate B.

The total drag due to air friction on a jet airplane traveling at 900 km/h is 35 kN. Knowing that the exhaust velocity is 600 m/s relative to the airplane, determine the mass of air which must pass through the engine per second to maintain the speed of 900 km/h in level flight.

While cruising in level flight at a speed of 600 mi/h, a jet plane scoops in air at the rate of 200 lb/s and discharges it with a velocity of 2100 ft/s relative to the airplane. Determine the total drag due to air friction on the airplane.

In order to shorten the distance required for landing, a jet airplane is equipped with movable vanes which partially reverse the direction of the air discharged by each of its engines. Each engine scoops in the air at a rate of 120 kg/s and discharges it with a velocity of 600 m/s relative to the engine. At an instant when the speed of the airplane is 270 km/h, determine the reverse thrust provided by each of the engines.

The helicopter shown can produce a maximum downward air speed of 80 ft/s in a 30-ft-diameter slipstream. Knowing that the weight of the helicopter and its crew is 3500 lb and assuming \(\mathrm{g}=0.076\ \mathrm{lb}/\mathrm{ft}^3\) for air, determine the maximum load that the helicopter can lift while hovering in midair.

A floor fan designed to deliver air at a maximum velocity of 6 m/s in a 400-mm-diameter slipstream is supported by a 200-mm-diameter circular base plate. Knowing that the total weight of the assembly is 60 N and that its center of gravity is located directly above the center of the base plate, determine the maximum height h at which the fan may be operated if it is not to tip over. Assume \(\mathrm{r}=1.21\mathrm{\ kg}/\mathrm{m}^3\) for air and neglect the approach velocity of the air.

The jet engine shown scoops in air at A at a rate of 200 lb/s and discharges it at B with a velocity of 2000 ft/s relative to the airplane. Determine the magnitude and line of action of the propulsive thrust developed by the engine when the speed of the airplane is (a) 300 mi/h, (b) 600 mi/h.

A jet airliner is cruising at a speed of 900 km/h with each of its three engines discharging air with a velocity of 800 m/s relative to the plane. Determine the speed of the airliner after it has lost the use of (a) one of its engines, (b) two of its engines. Assume that the drag due to air friction is proportional to the square of the speed and that the remaining engines keep operating at the same rate.

A 16-Mg jet airplane maintains a constant speed of 774 km/h while climbing at an angle \(a=18^{\circ}\). The airplane scoops in air at a rate of 300 kg/s and discharges it with a velocity of 665 m/s relative to the airplane. If the pilot changes to a horizontal flight while maintaining the same engine setting, determine (a) the initial acceleration of the plane, (b) the maximum horizontal speed that will be attained. Assume that the drag due to air friction is proportional to the square of the speed.

The propeller of a small airplane has a 2-m-diameter slipstream and produces a thrust of 3600 N when the airplane is at rest on the ground. Assuming \(r=1.225\mathrm{\ kg}/\mathrm{m}^3\) for air, determine (a) the speed of the air in the slipstream, (b) the volume of air passing through the propeller per second, (c) the kinetic energy imparted per second to the air in the slipstream.

The wind turbine–generator shown has an output-power rating of 1.5 MW for a wind speed of 36 km/h. For the given wind speed, determine (a) the kinetic energy of the air particles entering the 82.5-m-diameter circle per second, (b) the efficiency of this energy conversion system. Assume \(r=1.21\mathrm{\ kg}/\mathrm{m}^3\) for air.

A wind turbine-generator system having a diameter of 82.5 m produces 1.5 MW at a wind speed of 12 m/s. Determine the diameter of blade necessary to produce 10 MW of power assuming the efficiency is the same for both designs and \(\mathrm{r}=1.21\mathrm{\ kg}/\mathrm{m}^3\) for air.

While cruising in level flight at a speed of 570 mi/h, a jet airplane scoops in air at a rate of 240 lb/s and discharges it with a velocity of 2200 ft/s relative to the airplane. Determine (a) the power actually used to propel the airplane, (b) the total power developed by the engine, (c) the mechanical efficiency of the airplane.

In a Pelton-wheel turbine, a stream of water is deflected by a series of blades so that the rate at which water is deflected by the blades is equal to the rate at which water issues from the nozzle \((\Delta m / \Delta t=\left.A r v_{A}\right)\). Using the same notation as in Sample Prob. 14.7, (a) determine the velocity V of the blades for which maximum power is developed, (b) derive an expression for the maximum power, (c) derive an expression for the mechanical efficiency.

A circular reentrant orifice (also called Borda’s mouthpiece) of diameter D is placed at a depth h below the surface of a tank. Knowing that the speed of the issuing stream is \(v=1\ \overline{2gh}\) and assuming that the speed of approach \(v_{1}\) is zero, show that the diameter of the stream is \(d=D/1\overline{\ 2}\). (Hint: Consider the section of water indicated, and note that P is equal to the pressure at a depth h multiplied by the area of the orifice.)

Gravel falls with practically zero velocity onto a conveyor belt at the constant rate q = dm/dt. (a) Determine the magnitude of the force P required to maintain a constant belt speed v. (b) Show that the kinetic energy acquired by the gravel in a given time interval is equal to half the work done in that interval by the force P. Explain what happens to the other half of the work done by P.

The depth of water flowing in a rectangular channel of width b at a speed \(v_{1}\) and a depth \(d_{1}\) increases to a depth \(d_{2}\) at a hydraulic jump. Express the rate of flow Q in terms of b, \(d_{1}\), and \(d_{2}\).

Determine the rate of flow in the channel of Prob. 14.84, knowing that b = 12 ft, \(d_1=4\ \mathrm{ft}\), and \(d_2=5\ \mathrm{ft}\).

A chain of length l and mass m lies in a pile on the floor. If its end A is raised vertically at a constant speed v, express in terms of the length y of chain which is off the floor at any given instant (a) the magnitude of the force P applied to A, (b) the reaction of the floor.

Solve Prob. 14.86, assuming that the chain is being lowered to the floor at a constant speed v.

The ends of a chain lie in piles at A and C. When released from rest at time t = 0, the chain moves over the pulley at B, which has a negligible mass. Denoting by L the length of chain connecting the two piles and neglecting friction, determine the speed v of the chain at time t.

A toy car is propelled by water that squirts from an internal tank at a constant 6 ft/s relative to the car. The weight of the empty car is 0.4 lb and it holds 2 lb of water. Neglecting other tangential forces, determine the top speed of the car.

A toy car is propelled by water that squirts from an internal tank. The weight of the empty car is 0.4 lb and it holds 2 lb of water. Knowing the top speed of the car is 8 ft/s determine the relative velocity of the water that is being ejected.

The main propulsion system of a space shuttle consists of three identical rocket engines which provide a total thrust of 6 MN. Determine the rate at which the hydrogen-oxygen propellant is burned by each of the three engines, knowing that it is ejected with a relative velocity of 3750 m/s.

The main propulsion system of a space shuttle consists of three identical rocket engines, each of which burns the hydrogen-oxygen propellant at the rate of 750 lb/s and ejects it with a relative velocity of 12,000 ft/s. Determine the total thrust provided by the three engines.

A rocket weighs 2600 lb, including 2200 lb of fuel, which is consumed at a rate of 25 lb/s and ejected with a relative velocity of 13,000 ft/s. Knowing that the rocket is fired vertically from the ground, determine its acceleration (a) as it is fired, (b) as the last particle of fuel is being consumed.

A space vehicle describing a circular orbit about the earth at a speed of \(24\times10^3\mathrm{\ km}/\mathrm{h}\) releases at its front end a capsule which has a gross mass of 600 kg, including 400 kg of fuel. If the fuel is consumed at the rate of 18 kg/s and ejected with a relative velocity of 3000 m/s, determine (a) the tangential acceleration of the capsule as its engine is fired, (b) the maximum speed attained by the capsule.

A 540-kg spacecraft is mounted on top of a rocket with a mass of 19 Mg, including 17.8 Mg of fuel. Knowing that the fuel is consumed at a rate of 225 kg/s and ejected with a relative velocity of 3600 m/s, determine the maximum speed imparted to the spacecraft if the rocket is fired vertically from the ground.

The rocket used to launch the 540-kg spacecraft of Prob. 14.95 is redesigned to include two stages A and B, each of mass 9.5 Mg, including 8.9 Mg of fuel. The fuel is again consumed at a rate of 225 kg/s and ejected with a relative velocity of 3600 m/s. Knowing that when stage A expels its last particle of fuel, its casing is released and jettisoned, determine (a) the speed of the rocket at that instant, (b) the maximum speed imparted to the spacecraft.

A communications satellite weighing 10,000 lb, including fuel, has been ejected from a space shuttle describing a low circular orbit around the earth. After the satellite has slowly drifted to a safe distance from the shuttle, its engine is fired to increase its velocity by 8000 ft/s as a first step to its transfer to a geosynchronous orbit. Knowing that the fuel is ejected with a relative velocity of 13,750 ft/s, determine the weight of fuel consumed in this maneuver.

Determine the increase in velocity of the communications satellite of Prob. 14.97 after 2500 lb of fuel has been consumed.

Determine the distance separating the communications satellite of Prob. 14.97 from the space shuttle 60 s after its engine has been fired, knowing that the fuel is consumed at a rate of 37.5 lb/s.

For the rocket of Prob. 14.93, determine (a) the altitude at which all of the fuel has been consumed, (b) the velocity of the rocket at this time.

Determine the altitude reached by the spacecraft of Prob. 14.95 when all the fuel of its launching rocket has been consumed.

For the spacecraft and the two-stage launching rocket of Prob. 14.96, determine the altitude at which (a) stage A of the rocket is released, (b) the fuel of both stages has been consumed.

In a jet airplane, the kinetic energy imparted to the exhaust gases is wasted as far as propelling the airplane is concerned. The useful power is equal to the product of the force available to propel the airplane and the speed of the airplane. If v is the speed of the airplane and u is the relative speed of the expelled gases, show that the mechanical efficiency of the airplane is h = 2v/(u + v). Explain why h = 1 when u = v.

In a rocket, the kinetic energy imparted to the consumed and ejected fuel is wasted as far as propelling the rocket is concerned. The useful power is equal to the product of the force available to propel the rocket and the speed of the rocket. If v is the speed of the rocket and u is the relative speed of the expelled fuel, show that the mechanical efficiency of the rocket is \(\mathrm{h}=2 u v /\left(u^{2}+v^{2}\right)\). Explain why h = 1 when u = v.

Three identical cars are being unloaded from an automobile carrier. Cars B and C have just been unloaded and are at rest with their brakes off when car A leaves the unloading ramp with a velocity of 5.76 ft/s and hits car B, which hits car C. Car A then again hits car B. Knowing that the velocity of car B is 5.04 ft/s after the first collision, 0.630 ft/s after the second collision, and 0.709 ft/s after the third collision, determine (a) the final velocities of cars A and C, (b) the coefficient of restitution for each of the collisions.

A 30-g bullet is fired with a velocity of 480 m/s into block A, which has a mass of 5 kg. The coefficient of kinetic friction between block A and cart BC is 0.50. Knowing that the cart has a mass of 4 kg and can roll freely, determine (a) the final velocity of the cart and block, (b) the final position of the block on the cart.

An 80-Mg railroad engine A coasting at 6.5 km/h strikes a 20-Mg flatcar C carrying a 30-Mg load B which can slide along the floor of the car \(\left(\mathrm{m}_{k}=0.25\right)\). Knowing that the car was at rest with its brakes released and that it automatically coupled with the engine upon impact, determine the velocity of the car (a) immediately after impact, (b) after the load has slid to a stop relative to the car.

In a game of pool, ball A is moving with a velocity \(\mathbf{v}_{0}\) when it strikes balls B and C which are at rest and aligned as shown. Knowing that after the collision the three balls move in the directions indicated and that \(v_0=12\mathrm{\ ft}/\mathrm{s}\) and \(v_C=6.29\mathrm{\ ft}/\mathrm{s}\), determine the magnitude of the velocity of (a) ball A, (b) ball B.

Mass C, which has a mass of 4 kg, is suspended from a cord attached to cart A, which has a mass of 5 kg and can roll freely on a frictionless horizontal track. A 60-g bullet is fired with a speed \(v_0=500\mathrm{\ m}/\mathrm{s}\) and gets lodged in block C. Determine (a) the velocity of C as it reaches its maximum elevation, (b) the maximum vertical distance h through which C will rise.

A 15-lb block B is at rest and a spring of constant k = 72 lb/in is held compressed 3 in. by a cord. After 5-lb block A is placed against the end of the spring the cord is cut causing A and B to move. Neglecting friction, determine the velocities of blocks A and B immediately after A leaves B.

Car A was at rest 9.28 m south of point O when it was struck in the rear by car B, which was traveling north at a speed \(v_{B}\). Car C, which was traveling west at a speed \(v_{C}\), was 40 m east of point O at the time of the collision. Cars A and B stuck together and, because the pavement was covered with ice, they slid into the intersection and were struck by car C which had not changed its speed. Measurements based on a photograph taken from a traffic helicopter shortly after the second collision indicated that the positions of the cars, expressed in meters, were \(\mathbf{r}_{A}=-10.1 \mathbf{i}+16.9 \mathbf{j}\), \(\mathbf{r}_{B}=-10.1 \mathbf{i}+20.4 \mathbf{j}\), and \(\mathbf{r}_{C}=-19.8 \mathbf{i}-15.2 \mathbf{j}\). Knowing that the masses of cars A, B, and C are, respectively, 1400 kg, 1800 kg, and 1600 kg, and that the time elapsed between the first collision and the time the photograph was taken was 3.4 s, determine the initial speeds of cars B and C.

The nozzle shown discharges water at the rate of 200 gal/min. Knowing that at both B and C the stream of water moves with a velocity of magnitude 100 ft/s, and neglecting the weight of the vane, determine the force-couple system which must be applied at A to hold the vane in place \(\left(1\mathrm{\ ft}^3=7.48\mathrm{\ gal}\right)\).

Prior to takeoff, the pilot of a 6000-lb twin-engine airplane tests the reversible-pitch propellers with the brakes at point B locked. Knowing that the velocity of the air in the two 6.6-ft-diameter slipstreams is 60 ft/s and that point G is the center of gravity of the airplane, determine the reactions at points A and B. Assume \(\mathrm{g}=0.075\mathrm{\ lb}/\mathrm{ft}^3\) and neglect the approach velocity of the air.

A railroad car of length L and mass \(m_{0}\) when empty is moving freely on a horizontal track while being loaded with sand from a stationary chute at a rate dm/dt = q. Knowing that the car was approaching the chute at a speed \(v_{0}\), determine (a) the mass of the car and its load after the car has cleared the chute, (b) the speed of the car at that time.

A garden sprinkler has four rotating arms, each of which consists of two horizontal straight sections of pipe forming an angle of \(120^{\circ}\) with each other. Each arm discharges water at a rate of 20 L/min with a velocity of 18 m/s relative to the arm. Knowing that the friction between the moving and stationary parts of the sprinkler is equivalent to a couple of magnitude \(M=0.375\mathrm{\ N}\cdot\mathrm{m}\), determine the constant rate at which the sprinkler rotates.

A chain of length l and mass m falls through a small hole in a plate. Initially, when y is very small, the chain is at rest. In each case shown, determine (a) the acceleration of the first link A as a function of y, (b) the velocity of the chain as the last link passes through the hole. In case 1 assume that the individual links are at rest until they fall through the hole; in case 2 assume that at any instant all links have the same speed. Ignore the effect of friction.