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

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Solution

As we know that gravity is given by
g = $\frac{G M}{R^{2}}$ and density $ρ = \frac{mass}{volume}$
So, for planets A and B, the gravity can be written as
$g_{A} = \frac{G \frac{4}{3} π R_{A}^{3} ρ}{R_{A}^{2}}; g_{B} = \frac{G \frac{4}{3} π R_{B}^{3} ρ}{R_{B}^{2}}$
As $R_{A} = 2 R_{B}$ $\Rightarrow g_{A} = 2 g_{B}$
Escape velocity
$V_{e s} = \sqrt{2 g R}$
Escape velocity of planet B
$(V_{e s})_{B} = \sqrt{2 g_{B} R_{B}}$
Escape velocity of planet A
$(V_{e s})_{A} = \sqrt{2 g_{A} R_{A}} = 2 \sqrt{2 g_{B} R_{B}}$

$\frac{(V_{e s})_{A}}{(V_{e s})_{B}} = 2$

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