6 A particle \(P\) is projected with speed \(u\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The direction of motion of \(P\) makes an angle \(\alpha\) above the horizontal when \(P\) first reaches three-quarters of its greatest height.

1. Show that \(\tan \alpha = \frac { 1 } { 2 } \tan \theta\).
2. Given that \(\tan \theta = \frac { 4 } { 3 }\), find the horizontal distance travelled by \(P\) when it first reaches three-quarters of its greatest height. Give your answer in terms of \(u\) and \(g\).
   \includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-12_241_1009_269_529} One end of a light spring of natural length \(a\) and modulus of elasticity \(4 m g\) is attached to a fixed point \(O\). The other end of the spring is attached to a particle \(A\) of mass \(k m\), where \(k\) is a constant. Initially the spring lies at rest on a smooth horizontal surface and has length \(a\). A second particle \(B\), of mass \(m\), is moving towards \(A\) with speed \(\sqrt { \frac { 4 } { 3 } \mathrm { ga } }\) along the line of the spring from the opposite direction to \(O\) (see diagram). The particles \(A\) and \(B\) collide and coalesce. At a point \(C\) in the subsequent motion, the length of the spring is \(\frac { 3 } { 4 } a\) and the speed of the combined particle is half of its initial speed.
3. Find the value of \(k\).
   At the point \(C\) the horizontal surface becomes rough, with coefficient of friction \(\mu\) between the combined particle and the surface. The deceleration of the combined particle at \(C\) is \(\frac { 9 } { 20 } \mathrm {~g}\).
4. Find the value of \(\mu\).
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