The distance of the two planets A and B from the sun are $r_A$ and $r_B$ respectively. Also $r_B$ is equal…
The distance of the two planets A and B from the sun are $r_A$ and $r_B$ respectively. Also $r_B$ is equal to $100 r_A$. If the orbital speed of the planet $A$ is ' $v$ ' then the orbital speed of the planet $B$ is
$\frac{v}{10}$
$\frac{v}{2}$.
$\sqrt{2} \mathrm{v}$
10 v
Solution
Orbital or Critical velocity $v_{\text {orb }}=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}$
From the data given,
$\mathrm{v}_{\text {orb }_{\mathrm{A}}}=\sqrt{\frac{\mathrm{GM}}{\mathrm{r}_{\mathrm{A}}}}$
...(i) and
$\mathrm{v}_{\text {orb }_{\mathrm{B}}}=\sqrt{\frac{\mathrm{GM}}{\mathrm{r}_{\mathrm{B}}}}=\sqrt{\frac{\mathrm{GM}}{100 \mathrm{r}_{\mathrm{A}}}}=\frac{1}{10} \sqrt{\frac{\mathrm{GM}}{\mathrm{r}_{\mathrm{A}}}}$
Dividing equation (i) by (ii)
$\begin{aligned}
& \frac{v_{\text {orb }_A}}{v_{\text {orb }_B}}=\sqrt{\frac{G M}{r_A}} \times 10 \sqrt{\frac{r_A}{G M}} \\
& v_{\text {orb }_B}=\frac{v_{\text {orb }_A}}{10}=\frac{v}{10} \ldots\left(\text { given }_{\mathrm{orb}_A}=\mathrm{v}\right)
\end{aligned}$
.