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 \(O P = \frac { 3 } { 4 } a\).
- Use an energy method to show that \(k = \frac { 16 u ^ { 2 } } { a g }\).
- Show that \(P\) performs simple harmonic motion and find the period of this motion, giving your answer in terms of \(u\) and \(a\).
- Find, in terms of \(u\) and \(a\), the time that elapses before \(P\) first loses \(25 \%\) of its initial kinetic energy.