6 A ball of mass \(m \mathrm {~kg}\) is held at rest at a height \(h\) metres above a horizontal surface. The ball is released and bounces on the surface.
The coefficient of restitution between the ball and the surface is \(e\)
Prove that the kinetic energy lost during the first bounce is given by $$m g h \left( 1 - e ^ { 2 } \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{4975a2a9-1e45-44b4-b525-fc902627d03e-09_63_44_735_331}
\(7 \quad\) A light string has length 1.5 metres. A small sphere is attached to one end of the string.
The other end of the string is attached to a fixed point \(O\)
A thin horizontal bar is positioned 0.9 metres directly below \(O\)
The bar is perpendicular to the plane in which the sphere moves.
The sphere is released from rest with the string taut and at an angle \(\alpha\) to the downward vertical through \(O\) The string becomes slack when the angle between the two sections of the string is \(60 ^ { \circ }\) Ben draws the diagram below to show the initial position of the sphere, the bar and the path of the sphere.
\includegraphics[max width=\textwidth, alt={}, center]{4975a2a9-1e45-44b4-b525-fc902627d03e-10_623_748_1123_644}