Q38 / 50 · Manipal (MET) 2022 · Physics

If the radius of a planet is three times the radius of the earth. Both have same mass-densities. $v_P$ and $v_E$ are the escape velocities of the planet and the earth respectively, then

A
$v_P=1.5 v_E$
Correct answer
B
$v_P=3 v_E$
C
$v_E=2 v_P$
D
$v_P=2 v_E$

Solution

The escape velocity of a planet is given by, $v=\sqrt{\frac{2 G M}{R}}=\sqrt{\frac{2 G}{R} \times \frac{4}{3} \pi R^3 \cdot \rho}$ $\therefore \quad V \propto R \sqrt{\rho}$ $\therefore \quad \frac{v_P}{v_E}=\frac{R_P \sqrt{\rho_P}}{R_E \sqrt{\rho_E}}=\frac{R_P}{R_E} \quad\left(\because \rho_P=\rho_E\right)$ $\therefore \quad \frac{v_P}{v_E}=\frac{3 R_E}{R_E}=3 \quad\left(\because R_P=3 R_E\right)$ $\therefore \quad v_P=3 v_E$

Hence, the correct answer is option: 'B'.

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