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 :
Correct Option: 1
(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
$M_{B}=\frac{M_{A}}{2}, R_{B}=\frac{R_{A}}{2}$
$\therefore \quad V_{B}=\sqrt{\frac{2 G \frac{M_{A}}{2}}{\frac{R_{A}}{2}}} \therefore \frac{V_{A}}{V_{B}}=\sqrt{\frac{\frac{2 G M_{A}}{R_{A}}}{\frac{2 G M_{A} / 2}{R_{A} / 2}}}=\frac{n}{4}=1$
$\Rightarrow \quad n=4$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.