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One end of a light inextensible string of length $\frac { 3 } { 2 } a$ is attached to a fixed point $O$ on a horizontal surface. The other end of the string is attached to a particle $P$ of mass $m$. The string passes over a small fixed smooth peg $A$ which is at a distance $a$ vertically above $O$. The system is in equilibrium with $P$ hanging vertically below $A$ and the string taut. The particle is projected horizontally with speed $u$ (see diagram). When $P$ is at the same horizontal level as $A$, the tension in the string is $T$. Show that $T = \frac { 2 m } { a } \left( u ^ { 2 } - a g \right)$.

The ratio of the tensions in the string immediately before, and immediately after, the string loses contact with the peg is $5 : 1$.\\
(i) Show that $u ^ { 2 } = 5 a g$.\\
(ii) Find, in terms of $m$ and $g$, the tension in the string when $P$ is next at the same horizontal level as $A$.

\hfill \mbox{\textit{CAIE FP2 2015 Q10 EITHER}}