What is escape velocity ? Obtain an expression for it.

Definition: The minimum velocity required for an object to escape from gravitational field of
planet is known as escape velocity. Consider an object mass “m” on the surface of a planet of mass “M” and radius “R”. Gravitational potential energy of the system

$=\frac{\mathrm{GMm}}{\mathrm{R}}$...(1)

Kinetic energy of object $=\frac{1}{2}{\mathrm{mv}}_{\mathrm{e}}^2$.......(2)

(1) & (2) KE = PE ;
$\begin{array}{l}\frac{1}{2}{\mathrm{mV}}_{\mathrm{e}}^2=\frac{\mathrm{GMm}}{\mathrm{R}}\\ {\mathrm{V}}_{\mathrm{e}}^2=\frac{2\mathrm{GM}}{\mathrm{R}};{\mathrm{V}}_{\mathrm{e}}=\sqrt{\frac{2\mathrm{GM}}{\mathrm{R}}}\end{array}$.......(3)

We know $\mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{R}}^2}\Rightarrow {\mathrm{gR}}^2=\mathrm{GM}$......(4)

Substitute (4) in (3)

we get ${\mathrm{V}}_{\mathrm{e}}=\sqrt{2\mathrm{gR}}$