One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\lambda m g\), is attached to a fixed point \(O\). The string lies 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 in the direction \(O P\). When the length of the string is \(\frac{4}{3} a\), the speed of \(P\) is \(\sqrt{2 a g}\). When the length of the string is \(\frac{5}{3} a\), the speed of \(P\) is \(\frac{1}{2} \sqrt{2 a g}\).

Find the value of \(\lambda\).