One end of a light elastic string of natural length l passing through a small smooth ring of mass m is attached to a point O of a ceiling. A particle P of mass M attached to the other end of the string hangs in equilibrium with the ring being held at rest at the point O. if 2Mg is the modulus of elasticity of the string show that the extension of the string in the equilibrium position is L/2.

The Ring now released from rest at O slide vertically downward along the string ,under gravity, collides and coalesces with P. Find the velocity of the composite body consisting of the ring and the particle when it begins to move vertically downwards.

I attempted the question as shown in the image but I got the velocity wrong.What is the mistake I have made here?

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