One end of a light elastic string of natural length 0.8 m and modulus of elasticity 36 N is attached to a fixed point \(O\) on a smooth plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha=\frac{3}{5}\). A particle \(P\) of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of \(O\). The particle is projected with speed \(\sqrt{2} \mathrm{~ms}^{-1}\) directly down the plane from the position where \(O P\) is equal to the natural length of the string.

Find the maximum extension of the string during the subsequent motion.