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Working with pulleys - rope sliding down a pulley

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A thick rope with a length of 5m is draped over a pulley, such that the right side y=3m is dangling free and on the left side L-y=2m is dangling free. The rope has a mass of 10kg. Since the longer side is more massive than the shorter side, the rope begins falling downward, with the shorter end going up towards the pulley. Assuming the initial velocity of the rope is zero, derive an expression for the velocity of the rope when the end of the rope reaches the pulley, and calculate a numerical answer.

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We will use the conservation of energy along with some calculus to do this problem
The mass per unit length = m/L. The mass of length dy = (the mass per unit length)*dy = (m/L)*dy

In the figure above, we take the gravitational potential energy equal to zero at the point where ...

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