A flexible cable is wound in one row round a drum with a radius $\displaystyle R$. The weight of a unit of cable length is $\displaystyle \rho$. The entire cable weighs $\displaystyle G$. The drum moves by inertia without slipping along a horizontal surface, and the cable is wound off it. At the initial moment, when the cable was completely wound on the drum, the velocity of the drum centre was $\displaystyle v$.

Find the velocity of the drum centre at the moment of time when a part of the cable with a length $\displaystyle x$ lies on the surface, neglecting the radius of the cable cross section(in comparison with $\displaystyle R$) and the mass of the drum.

For diagram, refer: Drum - Mechanics problem on Flickr - Photo Sharing!

$\displaystyle \sqrt{\frac{Gv^2 + \rho gxR}{G - \rho x}}$

How to solve it?