Question

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

• A. $10$
• B. $\sqrt{10}$
• C. $\frac{1}{\sqrt{10}}$
• D. $10^{25}$

Answer 1

Answer: (C)

The time period of satellite is
$T^2=k{r}^3$
$\Rightarrow T=k{r}^{\frac{3}{2}}$
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 $v_2=\frac{2\pi r_2}{T_2}$
$\therefore \frac{v_1}{v_2}=\frac{r_1}{r_2}\times \frac{T_2}{T_1}$
$=\frac{r_1}{r_2}{\left(\frac{r_2}{r_1}\right)}^{\frac{3}{2}}={\left(\frac{r_2}{r_1}\right)}^{\frac{1}{2}}$
$={\frac{{\left(10\right)}^{12}}{{\left(10\right)}^{13}}}^{\frac{1}{2}}=\frac{1}{\sqrt{10}}$