Question

The time period of earth is taken as $T$ and its distance from sun as $R$ What will be the distance of a certain planet from sun whose time period is $64$ times of earth?

Answer 1

Time period of earth $=T$

distance of earth from sun $=R.$

Time period of planet $=64T$

let the distance from sun be $R_0$

So$,$ from Kepler's law's

The square of the time for a planet to complete a revolution about the sun is proportional to the cube root of its distance from sun$.$

So$,$ $T^2\alpha R^3$------------$\left(1\right)$

${\left(64T\right)}^2\alpha R_0^3$

dividing equation $\left(1\right)$ and $\left(2\right)$

$\frac{T^2}{{\left(64\right)}^2{T}^2}=\frac{R^3}{R_0^3}=R_0^3={\left(64\right)}^2{R}^3\$$
$R_0^3={\left(4^3\right)}^2{R}^3$

$R_0=16R$

Hence,

We get, $R_0=16R$