Question 11 EITHER alternative.

A light spring has natural length \(a\) and modulus of elasticity \(k m g\). The spring lies on a smooth horizontal surface with one end attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The system is in equilibrium with \(O P=a\). The particle is projected towards \(O\) with speed \(u\) and comes to instantaneous rest when \(O P=\frac{3}{4} a\).
(i) Use an energy method to show that \(k=\frac{16 u^{2}}{a g}\).
(ii) Show that \(P\) performs simple harmonic motion and find the period of this motion, giving your answer in terms of \(u\) and \(a\).
(iii) Find, in terms of \(u\) and \(a\), the time that elapses before \(P\) first loses 25% of its initial kinetic energy.