Escape velocity from the earth is $$11.2\,km/s.$$   Another planet of same mass has radius $$\frac{1}{4}$$ times that of the earth. What is the escape velocity from another planet?

Solution :
Compare the equation of escape velocity of earth and planet. Escape velocity is given by,
$${v_{es}} = \sqrt {\frac{{2G{M_e}}}{{{R_e}}}} $$
From a planet, $${{v'}_{es}} = \sqrt {\frac{{2G{M_p}}}{{{R_p}}}} $$
Therefore, $$\frac{{{{v'}_{es}}}}{{{v_{es}}}} = \sqrt {\frac{{2G{M_p}}}{{{R_p}}}} \times \sqrt {\frac{{{R_e}}}{{2G{M_e}}}} $$
It is given that,
mass of the planet = mass of the earth
i.e. $${M_p} = {M_e}$$
So, $$\frac{{{{v'}_{es}}}}{{{v_{es}}}} = \sqrt {\frac{{{R_e}}}{{{R_p}}}} \,......\left( {\text{i}} \right)$$
Given, $${R_p} = \frac{{{R_e}}}{4} \Rightarrow \frac{{{R_p}}}{{{R_e}}} = \frac{1}{4}\,\,{\text{and}}\,\,{v_{es{\text{ }}}} = 11.2\,km/s$$
Substituting in Eq. (i), we have
$$\frac{{{{v'}_{es}}}}{{11.2}} = \sqrt {\frac{4}{1}} = 2,{{v'}_{es}} = 11.2 \times 2 = 22.4\,km/s$$