5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-09_485_442_221_758}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
A particle \(P\) of mass \(2 m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). Initially the particle is at the point \(A\) where \(O A = a\) and \(O A\) makes an angle \(60 ^ { \circ }\) with the downward vertical. The particle is projected downwards from \(A\) with speed \(u\) in a direction perpendicular to the string, as shown in Figure 3. The point \(B\) is vertically below \(O\) and \(O B = a\). As \(P\) passes through \(B\) it strikes and adheres to another particle \(Q\) of mass \(m\) which is at rest at \(B\).
- Show that the speed of the combined particle immediately after the impact is
$$\frac { 2 } { 3 } \sqrt { u ^ { 2 } + a g } .$$
- Find, in terms of \(a , g , m\) and \(u\), the tension in the string immediately after the impact.
The combined particle moves in a complete circle.
- Show that \(u ^ { 2 } \geqslant \frac { 41 a g } { 4 }\).