There is a crater of depth $\frac{R}{100}$ on the surface of moon of $R =$radius of moon. A projectile is fired vertically upward from the crater, with a velocity, which is equal to escape velocity on the moon. Find the maximum height obtained by the projectile.

If $M$ is the mass of the moon, then escape velocity from the moon surface

                                  $v = \sqrt{\frac{2GM}{R}}$

By conservation of mechanical energy, we can write

                        $K_i+U_i= K_f+U_{f ..............\left(i\right)}$

where      $$\begin{gathered} K_i = \frac{1}{2}m{v}^2 = \frac{1}{2}m\left(\frac{2GM}{R}\right) = \frac{GMm}{R} \\ K_{f } = 0 \\ U_i = m\times potential of point A \\ = m\left[-\frac{GM}{2R^3}\left(3R^2-r\right)\right] \\ = m\left[-\frac{GM}{2R^3}\left(3R^2-{\left(R-\frac{R}{100}\right)}^2\right)\right] \\ = \frac{-1.01GM}{R} \end{gathered}$$

If $H$ is the height attained, then

                           $U_f = \frac{-GMm}{R+H}$

On substituting these values in equation $\left(i\right)$ and simplifying, we get

               $H = 99.5 R$