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Q.
The distances of two planets from the sun are $10^{13}\,m$ and $10^{12}\,m$ , respectively. The ratio of speeds of two planets around sun is
NTA AbhyasNTA Abhyas 2022
Solution:
The time period of satellite is
$T^{2}=kr^{3}$
$\Rightarrow T=kr^{\frac{3}{2}}$
or $\frac{T_{1}}{T_{2}}=\left(\frac{r_{1}}{r_{2}}\right)^{\frac{3}{2}}$
If $v_{1}$ and $v_{2}$ be the orbital speeds of the planets,
$v_{1}=\frac{2 \pi r_{1}}{T_{1}}$
and $v_{2}=\frac{2 \pi r_{2}}{T_{2}}$
$\therefore \frac{v_{1}}{v_{2}}=\frac{r_{1}}{r_{2}}\times \frac{T_{2}}{T_{1}}$
$=\frac{r_{1}}{r_{2}}\left(\frac{r_{2}}{r_{1}}\right)^{\frac{3}{2}}=\left(\frac{r_{2}}{r_{1}}\right)^{\frac{1}{2}}$
$=\frac{\left(10\right)^{12}}{\left(10\right)^{13}}^{\frac{1}{2}}=\frac{1}{\sqrt{10}}$