A particle \(P\) of mass \(m\,\text{kg}\) is attached to one end of a light elastic string of natural length \(2\) m and modulus of elasticity \(2mg\) N. The other end is attached to a fixed point \(O\). The particle hangs in equilibrium vertically below \(O\).

The particle is pulled vertically downwards a distance \(d\) m below its equilibrium position and released from rest.

(a) Given that the particle just reaches \(O\) in the subsequent motion, find \(d\).

(b) Hence find the speed of \(P\) when it is \(2\) m below \(O\).