91Ó°ÊÓ

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) \(8 \mathrm{Ns}\).

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 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 \(9.81 \mathrm{~ms}^{-2}\) ).

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 $$ \frac{U^{2}}{2 m}(M+m)\left(M \sec ^{2} \alpha+m \tan ^{2} \alpha\right) $$

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 \(\mathrm{BC}\).