Question

The escape velocity on a planet is $v$. If the radius of the planet contracts to ${\frac{1}{4}}^{th}$ of present value without any change in its mass, the escape velocity will be

• A. halved
• B. doubled
• C. quadrupoled
• D. becomes one fourth

Answer 1

The correct option is B doubled

Escape speed, $v_{min}=\sqrt{2gR}$
Since we talking about another planet with different radius, gravitational force also changes.
$v_{min}=\sqrt{\frac{2GM}{R}}\Rightarrow v_{min}\propto \sqrt{\frac{1}{R}}\Rightarrow \frac{v_2}{v_1}=\sqrt{\frac{R_1}{R_2}}$
Given: $R_2=\frac{R_1}{4}$
$\therefore \frac{v_2}{v_1}=\sqrt{\frac{R_1}{\left(\frac{R_1}{4}\right)}}\Rightarrow \frac{v_2}{v_1}=\sqrt{4}\Rightarrow v_2=2v_1$