Explanation

This is a standard gravitation problem frequently asked in JEE Main, JEE Advanced and IIT JEE.

Escape velocity of a planet is given by:

$$ v = \sqrt{\frac{2GM}{R}} $$

Mass of a spherical planet can be written in terms of density:

$$ M = \frac{4}{3}\pi R^3 \rho $$

Substituting this in escape velocity formula:

$$ v = \sqrt{\frac{2G}{R} \cdot \frac{4}{3}\pi R^3 \rho} $$

$$ v \propto R\sqrt{\rho} $$

Given data:

Escape velocity of planet A:

$$ v_A = 10\ \text{km/s} = 10000\ \text{m/s} $$

For planet B:

$$ R_B = 0.1R_A $$

$$ \rho_B = 0.1\rho_A $$

Using proportionality:

$$ \frac{v_B}{v_A} = \frac{R_B}{R_A}\sqrt{\frac{\rho_B}{\rho_A}} $$

$$ \frac{v_B}{10000} = 0.1 \times \sqrt{0.1} $$

$$ \frac{v_B}{10000} = \frac{0.1}{\sqrt{10}} $$

$$ v_B = \frac{10000}{\sqrt{10}} = 100\sqrt{10}\ \text{m/s} $$

Therefore, the escape velocity of planet B is $100\sqrt{10}$ m/s.

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