Question

A particle is projected vertically upwards from the surface of teh earth (radius $R_e$) with a kinetic energy equal to half of the minimum value needed for it to escape. The height to which it rises above the surface ot the earth is $\frac{R}{n}$ where $n$ is:

Answer 1

Escape velocity is given by $v_e=\sqrt{\frac{2GM}{R}}$

Thus, from energy conservation,

$\frac{1}{2}m\frac{v_e^2}{4}-\frac{GMm}{R}=-\frac{GMm}{h+R}$

$\frac{3}{4R}=\frac{1}{h+R}$

$h=\frac{R}{3}=\frac{R}{n}$

So, the answer is $n=3$