A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4 m g\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac { 1 } { 8 } l\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi \sqrt { } \left( \frac { l } { g } \right)\). At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac { 7 } { 16 } l\) under gravity, it strikes a fixed smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac { 1 } { 3 }\). Show that the speed of \(P\) immediately after the impact is \(\frac { 1 } { 4 } \sqrt { } ( 5 g l )\).