A particle \(P\) is projected with speed \(u\) at an angle \(\tan^{-1}\left(\dfrac43\right)\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity.

When \(P\) is moving horizontally, it strikes a smooth inclined plane at point \(A\). The plane is inclined to the horizontal at an angle \(\alpha\), and the line of greatest slope through \(A\) lies in the vertical plane through \(O\) and \(A\).

As a result of the impact, \(P\) moves vertically upwards. The coefficient of restitution between \(P\) and the inclined plane is \(e\).

(a) Show that \(e\tan^2\alpha=1\).

In its subsequent motion, the greatest height reached by \(P\) above \(A\) is \(\dfrac3{16}\) of the vertical height of \(A\) above the horizontal plane.

(b) Find \(e\).