7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a67e3644-13fa-4196-a2ef-ea1e26f5726c-20_442_967_283_486}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A light inextensible string of length \(a\) has one end attached to a fixed point \(O\) on a horizontal plane. A particle \(P\) is attached to the other end of the string. The particle is held at the point \(A\), where \(A\) is vertically above \(O\) and \(O A = a\). The particle is then projected horizontally with speed \(\sqrt { 10 a g }\), as shown in Figure 2. The particle strikes the plane at the point \(B\). After rebounding from the plane, \(P\) passes through \(A\). The coefficient of restitution between the plane and \(P\) is \(e\).
- Show that \(e \geqslant \frac { 1 } { 2 }\)
The point \(C\) is above the horizontal plane such that \(O C = a\) and angle \(C O B = 120 ^ { \circ }\)
As the particle reaches \(C\), the string breaks. The particle now moves freely under gravity and strikes the plane at the point \(D\).
Given that \(e = \frac { \sqrt { 3 } } { 2 }\) - find the size of the angle between the horizontal and the direction of motion of \(P\) at \(D\).