Question

If a planet were suddenly stopped in its orbit supposed to be circular, show that it would fall into the sun in a time $T\times \left(\frac{\sqrt{2}}{8}\right)$, where T is the time period of revolution.

Answer 1

Let, radius of planetary orbit :- $r$

Time taken to reach sun :- $t$

Time period of revolution = $T$

We considered the planet to reach the sun is an elliptical path with time period $2t^{\prime }=T^{\prime }$

$\frac{T^{\prime }}{T}={\left(\frac{r^{\prime }}{r}\right)}^{3/2}[r^{\prime }=\frac{r}{2}]T^{\prime }=T{\left(\frac{1}{2}\right)}^{3/2}{t}^{\prime }=T{\left(\frac{1}{2}\right)}^{3/2}{t}^{\prime }=T{\left(\frac{1}{2}\right)}^{3/2}\times \frac{1}{2}\therefore t^{\prime }=\frac{\sqrt{2}}{8}T$