Question

The distances of two planets from the sun are 10¹³ and 10¹² metre respectively. Find the ratio of time periods and speed of two planets.​

Answer 1

$\large \underline{\underline{\bf{ \pink{Given : }}}}$

✧ Distance of one planet from the Sun, $\sf r_1= 10^{13} m$

✧ Distance of second planet from the Sun, $\sf r_2= 10^{12} m$

$\large \underline{\underline{\bf{ \pink{To \: Find :}} }}$

✧ Ratio of time periods of two planets, $\sf \dfrac{T_1}{T_2}=?$

✧ Ratio of speed of two planets, $\sf \dfrac{v_1}{v_2}=?$

$\large \underline{\underline{\bf{ \pink{Solution : }}}}$

Now, We know that :

$\dag \underline{\boxed{\pink{\bf \dfrac{T_1 ^2}{T_2 ^2} = \dfrac{r_1 ^3}{r_2 ^3}}}}$

$\sf : \implies \dfrac{T_1}{T_2} =\Bigg( \dfrac{r_1}{r_2}\Bigg)^{\frac{3}{2}}$

By substituting values :

$\sf : \implies \dfrac{T_1}{T_2} = \Bigg( \dfrac{10^{13}}{10^{12}}\Bigg)^{\frac{3}{2}}$

$\sf : \implies \dfrac{T_1}{T_2} = (10)^{\frac{3}{2}}$

$\sf : \implies \dfrac{T_1}{T_2} =\sqrt{10^3}$

$\sf : \implies \dfrac{T_1}{T_2} =10\sqrt{10}$

$\underline{\boxed{\pink{\bf \dfrac{T_1}{T_2} = 10\sqrt{10}}}}$

Now,

$\dag \underline{\boxed{\pink{\bf v = \dfrac{2\pi r}{T}}}}$

$\sf : \implies \dfrac{v_1}{v_2}= \dfrac{\dfrac{2\pi r_1}{T_1}}{\dfrac{2\pi r_2}{T_2}}$

$\sf : \implies \dfrac{v_1}{v_2}= \dfrac{\cancel{2\pi} r_1}{T_1} \times \dfrac{T_2}{\cancel{2\pi} r_2}$

$\sf : \implies \dfrac{v_1}{v_2}= \dfrac{r_1}{T_1} \times \dfrac{T_2}{r_2}$

By substituting values :

$\sf : \implies \dfrac{v_1}{v_2}= \dfrac{10^{13}}{T_1} \times \dfrac{T_2}{10^{12}}$

$\sf : \implies \dfrac{v_1}{v_2}= \dfrac{10 \times T_2}{T_1}$

$\sf : \implies \dfrac{v_1}{v_2}=10\times \dfrac{ T_2}{T_1}$

$\sf : \implies \dfrac{v_1}{v_2}=10\times \dfrac{1}{10\sqrt{10}}$

$\sf : \implies \dfrac{v_1}{v_2}=\dfrac{1}{\sqrt{10}}$

$\underline{\boxed{\pink{\bf \dfrac{v_1}{v_2}=\dfrac{1}{\sqrt{10}}}}}$