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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt{ }(2 a g)\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
(i) Show that \(\cos \theta=\frac{2}{3}\).
(ii) Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion.