Planet A has mass M and radius R. Planet B has half the mass and half the radius of planet A. If the escape velocities from the planets A and B are $u_{A} a n d u_{B}$ respectively , then $\frac{u_{A}}{u_{B}} = \frac{n}{4}$ . The value of $n$ is

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answer is D.

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Detailed Solution

$\begin{array}{l} E s c a p e V e l o c i t y = V_{e} = \sqrt{\frac{2 G M}{R}} \\ \Rightarrow \frac{V_{A}}{V_{B}} = \sqrt{\frac{M_{A}}{M_{B}} \times \frac{R_{B}}{R_{A}}} \\ \Rightarrow \frac{V_{A}}{V_{B}} = \sqrt{\frac{M}{\frac{M}{2}} \times \frac{\frac{R}{2}}{R}} \\ \Rightarrow \frac{V_{A}}{V_{B}} = 1 - - - (1) \\ \Rightarrow \frac{V_{A}}{V_{B}} = \frac{n}{4} (G i v e n) - - - - (2) \\ \Rightarrow f r o m (1) a n d (2) w e g e t \\ \Rightarrow n = 4 \end{array}$