5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t \mathrm {~s}\) are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Show that the equation of the trajectory is given by $$\mathrm { y } = \mathrm { x } \tan \theta - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \left( 1 + \tan ^ { 2 } \theta \right)$$ In the subsequent motion \(P\) passes through the point with coordinates \(( 30,20 )\).
2. Given that one possible value of \(\tan \theta\) is \(\frac { 4 } { 3 }\), find the other possible value of \(\tan \theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-10_451_1339_258_404} A light inextensible string is threaded through a fixed smooth ring \(R\) which is at a height \(h\) above a smooth horizontal surface. One end of the string is attached to a particle \(A\) of mass \(m\). The other end of the string is attached to a particle \(B\) of mass \(\frac { 6 } { 7 } m\). The particle \(A\) moves in a horizontal circle on the surface. The particle \(B\) hangs in equilibrium below the ring and above the surface (see diagram). When \(A\) has constant angular speed \(\omega\), the angle between \(A R\) and \(B R\) is \(\theta\) and the normal reaction between \(A\) and the surface is \(N\). When \(A\) has constant angular speed \(\frac { 3 } { 2 } \omega\), the angle between \(A R\) and \(B R\) is \(\alpha\) and the normal reaction between \(A\) and the surface is \(\frac { 1 } { 2 } N\).
3. Show that \(\cos \theta = \frac { 4 } { 9 } \cos \alpha\).
4. Find \(N\) in terms of \(m\) and \(g\) and find the value of \(\cos \alpha\).
   \includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-12_413_974_255_587} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac { 1 } { 2 } m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2 u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 5 } { 8 }\) and \(\alpha + \beta = 90 ^ { \circ }\).
5. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\).
   The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.
6. Find the value of \(\tan \alpha\).
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