A particle $P$ of mass $m$ 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 $O$. The particle is held so that the string is taut, with $O P$ horizontal. The particle is projected downwards with speed $\sqrt { } \left( \frac { 2 } { 5 } a g \right)$ and begins to move in a vertical circle. The string breaks when its tension is equal to $\frac { 11 } { 5 } m g$.\\
(i) Show that the string breaks when $O P$ makes an angle $\theta$ with the downward vertical through $O$, where $\cos \theta = \frac { 3 } { 5 }$. Find the speed of $P$ at this instant.\\

(ii) For the subsequent motion after the string breaks, find the distance $O P$ when the particle $P$ is vertically below $O$.\\

\hfill \mbox{\textit{CAIE FP2 2018 Q11 EITHER}}