Chapter 8

A force of \(10 \mathrm{~N}\) acts on a mass of \(2 \mathrm{~kg}\) for three seconds. If the initial velocity was \(50 \mathrm{~ms}^{-1}\) what is the final velocity?

A ball of mass \(0.4 \mathrm{~kg}\) hits a wall at right angles with a speed of \(12 \mathrm{~ms}^{-1}\) and bounces off, again at right angles to the wall, with a speed of \(8 \mathrm{~ms}^{-1}\). The impulse exerted by the wall on the ball is: (a) \(1.6 \mathrm{Ns}\) (b) \(20 \mathrm{Ns}\) (c) \(4 \mathrm{Ns}\) (d) \(8 \mathrm{Ns}\).

A stone weighing \(S \mathbb{N}\) is thrown vertically upwards, with velocity \(80 \mathrm{~ms}^{-} .\) What is its velocity after two seconds and after twenty seconds? \(\left(\right.\) take \(\left.g=10 \mathrm{~ms}^{-2}\right)\)

Water issues from a pipe. whose cross section is \(c \mathrm{~m}^{2}\), in a horizontal jet with velocity \(v \mathrm{~ms}^{-1}\). What force must be exerted by a shield placed perpendicular to the jet to bring the water to a horizontal stop? (The mass of \(1 \mathrm{~m}^{3}\) of water is \(10^{3} \mathrm{~kg}\) ).

A gun which is free to recoil horizontally fires a bullet when the barrel is inclined at \(30^{\circ}\) to the horizontal. When the bullet leaves the barrel it will be travelling at an angle to the horizontal of: (a) \(30^{\circ}\), (b) a little less than \(30^{\circ}\), (c) a little more than \(30^{\circ}\), (d) zero.

Two masses of 20 and 10 units. moving in the same direction at speeds of 16 and 12 units respectively collide and stick together. Find the velocity of the combined mass immediately afterwards.

A gun of mass \(1000 \mathrm{~kg}\) can launch a shell of mass \(1 \mathrm{~kg}\) with a horizontal velocity of \(1200 \mathrm{~ms}^{-1}\). What is the horizontal velocity of recoil of the gun?

A particle of mass \(2 \mathrm{~kg}\) moving with speed \(4 \mathrm{~ms}^{-1}\) is given a blow which changes the speed to \(1 \mathrm{~ms}^{-1}\) without deflecting the particle from a straight line. The impulse of the blow is: (a) \(10 \mathrm{Ns}\), (b) \(6 \mathrm{Ns}\), (c) we do not know whether it is \(10 \mathrm{Ns}\) or \(6 \mathrm{Ns}\).

A sphere of mass \(m\) falls from rest at a height \(h\) above a horizontal plane and rebounds to a height \(\frac{h}{2}\), Find the coefficient of restitution, the impulse exerted by the plane and the loss in \(K E\) due to impact.

A body of mass \(m\) is moving with speed \(v\) when a constant force \(F\) newtons is applied to it in the direction of motion for \(t\) seconds: (a) The impulse of the force is \(F t\) newton second. (b) \(F t=m v\), (c) The body loses an amount of kinetic energy equal to \(F t\). (d) The final speed of the body is \(v+\frac{F t}{m}\).

A sphere \(\mathrm{A}\), of mass \(2 m\) and velocity \(2 u\), overtakes and collides with sphere \(\mathrm{B}\), of mass \(m\) and velocity \(u\) travelling in the same line which is perpendicular to a vertical smooth wall. After being struck by A. sphere B goes on to strike the wall. If the coefficient of restitution between \(\mathrm{A}\) and \(\mathrm{B}\) is \(\frac{1}{2}\) and that between \(\mathrm{B}\) and the wall is \(\frac{3}{4}\) show that there is a second collision between \(\mathrm{A}\) and \(\mathrm{B}\) and describe what happens after the second impact.

A sphere \(A\), of mass \(m_{1}\), and velocity \(u\), collides with a stationary sphere B of mass \(m_{2}\). If sphere \(\mathrm{A}\) is brought to rest by the collision. find the velocity of \(\mathrm{B}\) after impact. and the coefficient of restitution. If sphere B now collides with a stationary sphere \(\mathrm{C}\) and is brought to rest find the mass of sphere \(\mathrm{C}\) assuming the same coefficient of restitution between \(\mathrm{A}\) and \(\mathrm{B}\), and \(\mathrm{B}\) and \(\mathrm{C}\).

A smooth sphere \(\mathrm{A}\) of mass \(2 m\), moving on a horizontal plane with speed \(u\), collides directly with another smooth sphere B of equal radius and of mass \(m\), which is at rest. If the coefficient of restitution between the spheres is \(e\), find their speeds after impact. The sphere B later rebounds from a perfectly elastic vertical wall, and then collides directly with A. Prove that after this collision the speed of B is \(9(1+e)^{2} u\) and find the speed of \(\mathrm{A}\).

State the law of conservation of linear momentum for two interacting particles. Show how the law of conservation of linear momentum applied to two particles which collide directly follows from Newton's laws of motion. Three smooth spheres A, B, C. equal in all respects, lie at rest and separated from one another on a smooth horizontal table in the order \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) with their centres in a straight line. Sphere A is projected with speed \(V\) directly towards sphere \(\mathrm{B}\). If the coefficient of restitution at each collision is \(e\), where \(0

A pump raises water from a depth of \(10 \mathrm{~m}\) and discharges it horizontally through a pipe of \(0.1 \mathrm{~m}\) diameter at a velocity of \(8 \mathrm{~ms}^{-1} .\) Calculate the work done by the pump in one second. If the water impinges directly with the same velocity on a vertical wall, find the force exerted by the water on the wall if it is assumed that none of the water bounces back (Take \(g\) as \(9.81 \mathrm{~ms}^{-2}, \pi\) as \(3.142\) and the mass of \(1 \mathrm{~m}^{3}\) of water as \(1000 \mathrm{~kg}\) ).

Two equal spheres \(\mathrm{B}\) and \(\mathrm{C}\), each of mass \(4 m\), lie at rest on a smooth horizontal table. A third sphere \(\mathrm{A}\), of the same radius as \(\mathrm{B}\) and \(\mathrm{C}\) but of mass \(m\), moves with velocity \(V\) along the line of centres of \(\mathrm{B}\) and \(\mathrm{C}\). The sphere A collides with B which then collides with \(C\). If \(\mathrm{A}\) is brought to rest by the first collision show that the coefficient of restitution between \(\mathrm{A}\) and \(\mathrm{B}\) is \(\frac{1}{4}\). If the coefficient of restitution between \(\mathrm{B}\) and \(\mathrm{C}\) is 1 find the velocities of \(\mathrm{B}\) and \(\mathrm{C}\) after the second collision. Show that the total loss of kinetic energy due to the two collisions is \(\frac{27 m V^{2}}{64}\).

(a) The coefficient of restitution between two colliding objects is less than } 1 \text {. }

A particle of mass \(m\) is projected vertically upward with speed \(u\) and when it reaches its greatest height a second particle, of mass \(2 m\), is projected vertically upward with speed \(2 u\) from the same point as the first. Prove that the time that elapses between the projection of the second particle and its collision with the first is \(\frac{u}{4 g}\), and find the height above the point of projection at which the collision occurs. If, on collision, the particles coalesce, prove that the combined particle will reach a greatest height of \(\frac{19 u^{2}}{18 g}\) above the point of projection.

Three particles A, B, C of masses \(m, 2 m, 3 m\) respectively lie at rest in that order in a straight line on a smooth horizontal table. The distance between consecutive particles is \(a .\) A slack light inelastic string of length \(2 a\) connects \(\mathrm{A}\) and \(\mathrm{B}\). An exactly similar slack string connects \(\mathrm{B}\) and \(\mathrm{C}\). If \(\mathrm{A}\) is projected in the direction CBA with speed \(V\), find the time which elapses before \(C\) begins to move. Find also the speed with which C begins to move. Show that the ratio of the impulsive tensions in \(\mathrm{BC}\) and \(\mathrm{AB}\) when \(\mathrm{C}\) is jerked into motion is \(3: 1 .\) Find the total loss of kinetic energy when C has started to move. (J.M.B.)

A smooth plane is fixed at an inclination \(30^{\circ}\) with its lower edge at a height \(a\) above a horizontal table. Two particles \(\mathrm{P}\) and \(\mathrm{Q}\), each of mass \(m\), are connected by a light inextensible string of length \(2 a\), and \(\mathrm{P}\) is held at the lower edge of the inclined plane while \(\mathbf{Q}\) rests on the table vertically below \(\mathrm{P}\). The particle \(\mathrm{P}\) is then projected with velocity \(u(u>\sqrt{g a}\) ) upwards along a line of greatest slope of the plane. Find the impulsive tension in the string when \(\mathrm{Q}\) is jerked into motion. Determine the magnitude of \(u\) if \(\mathrm{Q}\) just reaches the lower edge of the plane, and the tension in the string while \(Q\) is moving.

Two particles \(\mathrm{A}\) and \(\mathrm{B}\) collide directily head-on and bounce. Find their speeds immediately after impact. (a) The mass of \(\mathrm{A}\) is twice the mass of B. (b) Just before impact the speed of \(\mathrm{A}\) is \(4 \mathrm{~ms}^{-1}\) and that of \(\mathrm{B}\) is \(3 \mathrm{~ms}^{-1} .\) (c) No kinetic energy is lost by the impact.

A ball moving on a horizontal floor hits a smooth vertical wall normally. Calculate the speed with which it leaves the wall if: (a) the speed when approaching the wall is \(3 \mathrm{~ms}^{-1}\), (b) the coefficient of restitution is \(\frac{1}{2}\), (c) the mass of the ball is \(0.4 \mathrm{~kg}\).

18) Two particles of masses \(m\) and \(3 m\) are connected by a light inelastic string of length \(2 l\) which passes over a small smooth fixed peg. The particles are held in contact with the peg and then allowed, at the same instant, to fall from rest under gravity, one on either side of the peg. Prove that: (i) the speed of each particle just after the string tightens is \(\sqrt{(g l / 2)}\), (ii) the sudden tightening of the string causes a loss of energy equal to \(3 \mathrm{mgl}\), (iii) the lighter particle reaches the peg again after a total time \(\sqrt{61 / g}\).

A ball falls vertically on to a horizontal plane and bounces. Find the impulse the ball exerts on the plane if: (a) the ball is initially \(2 \mathrm{~m}\) above the plane, (b) it rises after bouncing to a height \(1.2 \mathrm{~m}\), (c) the coefficient of restitution is \(\sqrt{\frac{2}{3}}\) (d) the mass of the ball is \(0.5 \mathrm{~kg}\).

A golf ball, initially at rest, is dropped on to a horizontal surface and bounces directly up again with velocity \(v\). If the coefficient of restitution between the ball and the surface is \(e\), show that the ball will go on bouncing for a time \(\frac{2 v}{g(1-e)}\) after the first impact. \(\left\\{\right.\) You may assume \(\left.1+e+e^{2}+e^{3}+\ldots .=(1-e)^{-1}\right\\}\). If the golf ball is dropped from a height of \(19.62 \mathrm{~m}\) and comes to rest 12 seconds later, find the value of \(e\). (Take \(g\) as \(9.81 \mathrm{~ms}^{-2}\) ).

A particle of mass \(m\) is thrown vertically upwards with speed \(u\) from a point A on the ground. Simultaneously an identical particle is thrown vertically downwards also with speed \(u\) from a point B vertically above \(\mathrm{A}\) and at a height \(h\) above the ground \(\left(h<4 u^{2} / g\right)\). On impact the particles adhere and move subsequently as a single particle. Calculate the loss in kinetic energy caused by the impact and the speed of the combined particle on reaching the ground.

An inelastic string has a particle \(\mathrm{A}\) attached to one end and a particle \(\mathrm{B}\) attached to the other end. If \(\mathrm{A}\) is projected in the direction \(\overrightarrow{\mathrm{BA}}\) find the initial speed of \(\mathrm{B}\) if: (a) initially the string is slack. (b) the speed of projection of \(\mathrm{A}\) is \(4 \mathrm{~ms}^{-1}\), (c) the particles are of equal mass, (d) the string is \(2 \mathrm{~m}\) long.

The barrel of a gun of mass \(M\) resting on a smooth horizontal plane is elevated at an angle \(\alpha\) to the horizontal. The gun fires a shell of mass \(m\) and recoils with horizontal velocity \(U .\) If the velocity of the shell on leaving the gun has horizontal and vertical components \(v\) and \(w\) respectively, prove that \(w=(v+U) \tan \alpha\), and hence or otherwise prove that the initial inclination of the path of the shell to the horizontal is arctan \(\left[\left(1+\frac{m}{M}\right) \tan \alpha\right]\). Prove that the kinetic energy generated by the explosion is $$ \frac{U^{2}}{2 m}(M+m)\left(M \sec ^{2} \alpha+m \tan ^{2} \alpha\right) $$

A particle of mass \(4 \mathrm{~kg}\) is attached to one end \(\mathrm{X}\) of a light inextensible string which passes over a smooth light pulley and supports particies of masses \(2 \mathrm{~kg}\) and \(3 \mathrm{~kg}\) at the other end \(\mathrm{Y}\). The end \(\mathrm{X}\) is held in contact with a horizontal table at a depth \(6 \mathrm{~m}\) below the pulley, both portions of the string being vertical and the particles at \(\mathrm{Y}\) hanging freely. The system is released from rest. When \(\mathrm{Y}\) has descended a distance of \(2.5 \mathrm{~m}\), the particle of mass \(2 \mathrm{~kg}\) is disconnected and begins to fall freely. Calculate the greatest height reached by \(\mathrm{X}\) above the table and the momentum of the \(4 \mathrm{~kg}\) particle when it strikes the table. Take \(g\) to be \(9.81 \mathrm{~m} / \mathrm{s}^{2}\)

The momentum of a system remains constant in any direction in which no external force force acts.

Three particles \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\), each of mass \(m\), lie at rest on a smooth horizontal table. Light inextensible strings connect \(\mathrm{A}\) to \(\mathrm{B}\) and \(\mathrm{B}\) to \(\mathrm{C}\). The strings are just taut with the angle \(A B C=135^{\circ}\), when a blow of impulse \(J\) is applied to \(C\) in a direction parallel to \(\overrightarrow{\mathrm{AB}}\). Prove that A begins to move with speed \((J / 7) m\) and find the impulsive tension in the string \(\mathrm{BC}\).

A bullet of mass \(m\) is fired with speed \(u\) into a fixed block of wood and emerges with speed \(2 u / 3 .\) When the experiment is repeated with the block free to move the bullet emerges with speed \(u / 2\) relative to the block. Assuming the same constant resistance to penetration in both cases, find the mass and the final speed of the block in the second case. (Neglect the effect of gravity throughout)

28) The masses of three perfectly elastic spheres \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are \(M, M\) and \(m\) respectively \((M>m)\). The spheres are initially at rest with their centres in a straight line, \(\mathrm{C}\) lying between \(\mathrm{A}\) and \(\mathrm{B}\). If \(\mathrm{C}\) is given a velocity towards \(\mathrm{A}\) along the line of centres, show that after colliding first with \(\mathrm{A}\) and then with \(\mathrm{B}\) it will not collide a second time with \(\mathrm{A}\) if \(M<(\sqrt{5}+2) m\). Find the ratios of the kinetic energies of the three spheres after the second collision and verify that no energy has been lost.