I know that g changes w.r.t. height, so I need to integrate the total g from a height R- radius of the earth to a height h.

If I project a body at a very large velocity such that it reaches a height h which is comparable to Earth's radius, then how to derive an equation for 'h' in terms of initial velocity 'u' and other constants.

This is where I ended up with this problem

S = u²/2g

But g is variable with height as h approaches R

So dg = GM/(R+dr)² --- this is where I think I might be wrong

Hence dS = u²(R+dr)²/GM

So how do I go ahead?