A particle \(P\) is projected from \(O\) with speed \(U\) at angle \(45^\circ\) above the horizontal and moves freely under gravity.

(a) State the vertical and horizontal components of velocity at time \(t\).

At time \(T\), \(P\) is moving at angle \(60^\circ\) below the horizontal.

(b) Show that \(T=\dfrac{U}{2g}(\sqrt2+\sqrt6)\).

At time \(T\), the particle strikes a smooth horizontal plane at a point a horizontal distance \(D\) from \(O\) and a vertical distance \(H\) below \(O\).

(c) Find \(H:D\).

After striking the plane, \(P\) rebounds with speed \(w\). The coefficient of restitution is \(\frac23\).

(d) Find \(w\) in terms of \(U\).