6 A particle \(P\), of mass \(m\), is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) moves in complete vertical circles about \(O\) with the string taut. The points \(A\) and \(B\) are on the path of \(P\) with \(A B\) a diameter of the circle. \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(\sqrt { 5 a g }\). The ratio of the tension in the string when \(P\) is at \(A\) to the tension in the string when \(P\) is at \(B\) is \(9 : 5\).

1. Find the value of \(\cos \theta\).
2. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) during its motion.
   \includegraphics[max width=\textwidth, alt={}, center]{e4926d36-7246-4cde-b466-44ecc4c30a61-12_613_718_251_676} The smooth vertical walls \(A B\) and \(C B\) are at right angles to each other. A particle \(P\) is moving with speed \(u\) on a smooth horizontal floor and strikes the wall \(C B\) at an angle \(\alpha\). It rebounds at an angle \(\beta\) to the wall \(C B\). The particle then strikes the wall \(A B\) and rebounds at an angle \(\gamma\) to that wall (see diagram). The coefficient of restitution between each wall and \(P\) is \(e\).
3. Show that \(\tan \beta = e \tan \alpha\).
4. Express \(\gamma\) in terms of \(\alpha\) and explain what this result means about the final direction of motion of \(P\).
   As a result of the two impacts the particle loses \(\frac { 8 } { 9 }\) of its initial kinetic energy.
5. Given that \(\alpha + \beta = 90 ^ { \circ }\), find the value of \(e\) and the value of \(\tan \alpha\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.