# Planet A has mass M and radius R.

Question:

Planet $\mathrm{A}$ has mass $\mathrm{M}$ and radius $\mathrm{R}$. Planet $\mathrm{B}$ has half the mass and half the radius of Planet A. If the escape velocities from the Planets $\mathrm{A}$ and $\mathrm{B}$ are $v_{\mathrm{A}}$ and $v_{\mathrm{B}}$, respectively, then

$\frac{v_{\mathrm{A}}}{v_{\mathrm{B}}}=\frac{n}{4}$. The value of $n$ is :

1. (1) 4

2. (2) 1

3. (3) 2

4. (4) 3

Correct Option: 1

Solution:

(1) Escape velocity of the planet $A$ is $V_{A}=\sqrt{\frac{2 G M_{A}}{R_{A}}}$

where $M_{A}$ and $R_{A}$ be the mass and radius of the planet $A$.

According to given problem

$M_{B}=\frac{M_{A}}{2}, R_{B}=\frac{R_{A}}{2}$

$\therefore \quad V_{B}=\sqrt{\frac{2 G \frac{M_{A}}{2}}{\frac{R_{A}}{2}}} \therefore \frac{V_{A}}{V_{B}}=\sqrt{\frac{\frac{2 G M_{A}}{R_{A}}}{\frac{2 G M_{A} / 2}{R_{A} / 2}}}=\frac{n}{4}=1$

$\Rightarrow \quad n=4$