First slide

Gravitaional potential energy

Question

Two planets A and B have the same material density. If the radius of A is twice that of B, then the ratio of escape velocity $\frac{V_{A}}{V_{B}}$ is

Easy

A

2
B

$\sqrt{2}$
C

$\frac{1}{\sqrt{2}}$
D

$\frac{1}{2}$

Solution

escape velocity = $\sqrt{\frac{2 G m}{R}} = \sqrt{\frac{2 G \frac{4}{3} π R^{3} ρ}{R}}$
$\begin{array}{l} V e \propto \sqrt{ρ} R \\ \frac{V_{1}}{V_{2}} = \frac{R_{1}}{R_{2}} = \frac{2}{1} \end{array}$

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