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Escape Speed

Three planets...

Question

Three planets of same density and with radii $R_{1}, R_{2}$ and $R_{3}$ , such that $R_{1} = 2 R_{2} = 3 R_{3},$ have gravitation fields on the surfaces $E_{1}, E_{2}, E_{3}$ and escape velocities $v_{1}, v_{2}, v_{3}$ respectively, Then

A

\frac{E_{1}}{E_{2}} = \frac{1}{2}

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B

\frac{E_{1}}{E_{3}} = 3

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C

\frac{v_{1}}{v_{2}} = 2

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D

\frac{v_{1}}{v_{3}} = \frac{1}{3}

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Solution

The correct options are
B $\frac{v_{1}}{v_{2}} = 2$
C $\frac{E_{1}}{E_{3}} = 3$

$E = \frac{G M}{R^{2}}$ $= \frac{G (\frac{4}{3} π R^{3} ρ)}{R^{2}}$

$\Rightarrow E \propto R$

$\Rightarrow \frac{E_{1}}{E_{2}} = \frac{R_{1}}{R_{2}}$ $= 2$

$\Rightarrow (A)$ is not true

$\frac{E_{1}}{E_{3}} = \frac{R_{1}}{R_{3}}$ $= 3$

$\Rightarrow (B)$ is true

$v = \sqrt{g R} \Rightarrow v α R$

$\frac{v_{1}}{v_{2}} = \frac{R_{1}}{R_{2}} = 2$

$\Rightarrow (C)$ is true

$\frac{v_{1}}{v_{3}} = \frac{R_{1}}{R_{3}} = 3$

$\Rightarrow (D)$ is not true

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