A light elastic string has natural length \(a\) and modulus of elasticity \(4 m g\). One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac{5}{4} a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac{3}{2} a\), the speed of \(P\) is \(\frac{1}{2} v\).
(a) Find an expression for \(v\) in terms of \(a\) and \(g\).
(b) Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac{3}{2} a\).