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Q.
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 $V_{A} / V_{B}$ is
Let the density be $d$ for both the planets.
Given that $R _{ A }=2 R _{ B }$
Now, mass of $A, M_{A}=\frac{4 d \pi R_{A}{ }^{3}}{3}=\frac{32 d \pi R_{B}{ }^{3}}{3}$
similarly, $M _{ B }=\frac{4 d \pi R _{ B }{ }^{3}}{3}$
Escape velocity for a planet is given by
$V =\sqrt{\frac{2 GM }{ R }}$
So, $V_{A}=\sqrt{\frac{2 GM _{A}}{3 R_{A}}}=\sqrt{\frac{64 Gd \pi R_{B}^{3}}{6 R_{B}}}$
$=\sqrt{\frac{32 Gd \pi R_{B}^{2}}{3}}$
Similarly, $V _{ B }=\sqrt{\frac{8 Gd \pi R _{ B }^{2}}{3}}$
Taking the ratio, $\frac{ V _{ A }}{ V _{ B }}=\sqrt{\frac{32 Gd \pi R _{ B }^{2}}{3}} \times \sqrt{\frac{3}{8 Gd \pi R _{ B }{ }^{2}}}$
$=2$