The orbital velocity of a body near the surface of a planet ' $A$ ' is equal to escape velocity of a body…
The orbital velocity of a body near the surface of a planet ' $A$ ' is equal to escape velocity of a body from the planet ' $B$ '. If the masses of planets A and B are same, the ratio of their radii is
$1$
$\frac{1}{2}$
$\frac{1}{3}$
$2$
Solution
Orbital velocity of a body of planet A,
$\mathrm{V}_{\mathrm{A}}=\sqrt{\frac{\mathrm{GM}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{A}}}}$
Escape velocity of a body of planet B,
$\begin{aligned}
& \mathrm{V}_{\mathrm{B}}=\sqrt{\frac{2 \mathrm{GM}_{\mathrm{B}}}{\mathrm{r}_{\mathrm{B}}}} \\
& \mathrm{V}_{\mathrm{A}}=\mathrm{V}_{\mathrm{B}} \\
& \sqrt{\frac{\mathrm{GM}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{A}}}}=\sqrt{\frac{2 \mathrm{GM}_{\mathrm{B}}}{\mathrm{r}_{\mathrm{B}}}}\left(\because \mathrm{M}_{\mathrm{A}}=\mathrm{M}_{\mathrm{B}}\right) \\
& \frac{\mathrm{r}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{B}}}=\frac{1}{2}
\end{aligned}$