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Escape Speed

Two planets o...

Question

Two planets of same density, have radii $R_{1}$ and $R_{2}$ such that $R_{1} = 2 R_{2}$ . If the escape velocity on the planets are $v_{1}$ and $v_{2}$ respectively. Then ratio $\frac{v_{2}}{v_{1}}$ is equal to

A

1

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B

4

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C

\frac{1}{2}

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D

2

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Solution

The correct option is C $\frac{1}{2}$
Gravitational Field $g = \frac{G M}{R^{2}} = \frac{G}{R^{2}} \times ρ \times \frac{4}{3} π R^{3} = \frac{4}{3} π ρ G R \Rightarrow g \propto R$
Escape Velocity $v = \sqrt{2 g R}$
$v \propto \sqrt{R \times R} \propto R$
Ratio of escape velocities
$\frac{v_{2}}{v_{1}} = \frac{R_{2}}{R_{1}} = \frac{1}{2}$

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