9231 P33 - Jun 2025 - Q7 - 11 marks
6870
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 \(U\).
