7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-11_671_1077_276_429}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
A particle is projected from the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g a )\). The particle leaves the surface of the sphere at the point \(C\), where \(\angle A O C = \theta\), and strikes the plane at the point \(P\), as shown in Figure 5.
- Show that \(\cos \theta = \frac { 3 } { 4 }\).
- Find the angle that the velocity of the particle makes with the horizontal as it reaches \(P\).