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?
As I cannot fully understand your question, I will somewhat provide you with the tools so that you can answer your own question. $$ g = \frac{GM}{R^2} $$ The above mentioned equation gives you the acceleration due to the gravitational at some point point that is R units of distance away from the center of any body of mass M. This can also be called as the gravitational field.
For earth, $M = 6 * 10^{24}$(approximately), and for any point on the surface of the earth $R$ will be approximately $6400$ $km$.
Now, only using the variables..
Let $g$ be the acceleration due to gravity on the earth's surface. Therefore, $$g = \frac{GM}{R^2}$$, where R = radius of earth.
Let $g*$ be the acceleration due to gravity at a height h from the earth's surface. Now, $$g* = \frac{GM}{(R + h)^2}$$
With the ratio $\frac{g*}{g}$, you can compare the two fields. For h << R, use binomial expansion for negative index and approximate. (Higher powers of $h/R$ are negligible).
If you want to find the potential energy, integrate with limits from infinity to the desired distance ($R + h$) the following expression. $$ W = \int Fdx = \int_\infty^{(R+h)} \frac{Gm_1m_2}{x^2}dx$$
Note: Gravitational potential energy at a point is defined as the work done to bring a mass from infinity to that point.
Hope I am right and hope this helps you.
