Question

How gravitational potential energy depends upon the difference in heights of initial and final position?

Answer 1

Gravitational potential energy is zero at infinity and decreases as the distance decreases.

This is because gravitational force exerted on the body by the earth is attractive.

Hence gravitational potential energy(U) is negative.

U = -$\frac{GMm}{r}$

Answer 2

Let M = Mass of planet 
m = mass of body 
r = distance of body from the center of the planet

The word Potential energy defined only for conservative force field.

So the change in Potential energy (dU) of a system corresponding to a conservative internal force is given by

                              dU =  F.dr x cos(180)       {since the displacement is  anti-                                                                            parallel to the gravitational pull}

⇒    $\int\limits^f_i {dU}$ = $-\int\limits^{r_f}_{r_i} {F} .\, dr$

⇒  $U_f-U_i$ = $-\int\limits^{r_f}_{r_i} {F} .\, dr$

⇒ ΔU = $-\int\limits^{r_f}_{r_i} {\frac{GMm}{r^2}} .\, dr$

⇒ ΔU = $-\frac{Gmm}{[r]^f_i}$

⇒   ΔU = $-\frac{GMm}{r_f-r_i}$

  which proves that gravitational potential depends on the difference of the initial and final heights of the body.