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  4. The escape velocity of a body on a planet 'A' is 12 kms -1. The escape velocity of the body on another planet ' B ', whose density is four times and radius is half of the planet ' A ', is :

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Q. The escape velocity of a body on a planet '$A$' is $12\, kms ^{-1}$. The escape velocity of the body on another planet ' $B$ ', whose density is four times and radius is half of the planet ' $A$ ', is :

JEE MainJEE Main 2022Gravitation

Solution:

$ V _{\text {escape }}=\sqrt{\frac{2 Gm }{ R }} $
$\Rightarrow \sqrt{\frac{2 G \rho \times \frac{4}{3} \pi R ^{3}}{ R }}$
$ V _{\text {escape }} \propto \sqrt{\rho R ^{2}}$
$\therefore$ if $\rho$ is $4$ times and Radius is halved.
$\Rightarrow V _{\text {escape }} $ will remain same