Question

A planet was suddenly stooped in its orbit supposed to be circular. The time it will fall on to the sun is, if time period of planet's revolution is $T$.

• A. $\frac{T}{2}$
• B. $\frac{\sqrt{2}T}{4}$
• C. $\frac{\sqrt{2}T}{8}$
• D. $\sqrt{2}T$

Answer 1

The correct option is C $\frac{\sqrt{2}T}{8}$

Let $r$ be the radius of the orbit of the planet around the sun.

Let time taken to reach the sun be ${}^{\prime }{t}^{\prime }$.

Time taken for the planet to revolve around the sun of radius $r=T$

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

By Kepler's law,

$\begin{array}{c}\frac{T^{\prime }}{T}={\left(\frac{r^{\prime }}{r}\right)}^{\frac{3}{2}}\\ r^{\prime }=\frac{1}{2}r\\ T^{\prime }=T{\left(\frac{1}{2}\right)}^{\frac{3}{2}}\\ t^{\prime }=T{\left(\frac{1}{2}\right)}^{\frac{3}{2}}\times \frac{1}{2}\\ t^{\prime }=\frac{\sqrt{2}}{8}T\end{array}$