A projectile is fired upwards from the surface of the earth with a velocity kv_e where v_e is the escape velocity and k < 1. If r is the maximum distance from the centre of the earth to which it rises and R is the radius of the earth, then r is

$\frac{R}{1- k^2}$

$\left(\begin{array}{c}\text { Total } \\ \text { Mechanical } \\ \text { Energy }\end{array}\right)_{\text {surface }}=\left(\begin{array}{c}\text { Total } \\ \text { Mechanical } \\ \text { Energy }\end{array}\right)_{r}$

$\Rightarrow-\frac{GMm}{R}+\frac{1}{2} m\left(kv_{e}\right)^2=-\frac{GMm}{r}+0 $

where $v_{e}=\sqrt{\frac{2 GM}{R}}$

$\Rightarrow r=\frac{R}{1-k^2}$