Grandad was correct in his method.

An alternative approach that would not require integrating the acceleration term would be to use the physical grounds that as the particle starts at rest it's final KE must be equal to the PE it gains in the 'fall'. Hence you would arrive at the same equation as grandad with

i.e.

where is the velocity at and is simply a dummy variable in place of so we can use the limits to take care of the constant of integration.

So rearranging the above equation we have that

thus integrating wrt the dummy variable in place of using the limits at and at we have that

.

Now to deal with the integral wrt let so

hence

.

Finally this gives us that

.

I'll leave it to you to check in case I made any algebraic errors!