A projectile is fired upwards from the surface of the earth with a velocity kve where ve 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
Answer & explanation
Correct answer: option 4
$\left(\begin{array}{c}\text { Total } \\ \text { Mechanical } \\ \text { Energy }\end{array}\right)_{\text {surface }}=\left(\begin{array}{c}\text { Total } \\ \text { Mechanical } \\ \text { Energy }\end{array}\right)_{r}$
$\Rightarrow-\frac{GMm}{R}+\frac{1}{2} m\left(kv_{e}\right)^2=-\frac{GMm}{r}+0 $
where $v_{e}=\sqrt{\frac{2 GM}{R}}$
$\Rightarrow r=\frac{R}{1-k^2}$