A body is moving in a low circular orbit about a planet of mass M and radius R.

Question:

A body is moving in a low circular orbit about a planet of mass $M$ and radius $R$. The radius of the orbit can be taken to be $R$ itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is :

  1. (1) $\frac{1}{\sqrt{2}}$

  2. (2) 2

  3. (3) 1

  4. (4) $\sqrt{2}$


Correct Option: 1

Solution:

(1) Orbital speed of the body when it revolves very close to the surface of planet

$V_{0}=\sqrt{\frac{G M}{R}}$                   ....(1)

Here, $G=$ gravitational constant

Escape speed from the surface of planet

$V_{e}=\sqrt{\frac{2 G M}{R}}$                 ....(2)

Dividing (i) by (ii), we have

$\frac{V_{0}}{V_{e}}=\frac{\sqrt{\frac{G M}{R}}}{\sqrt{\frac{2 G M}{R}}}=\frac{1}{\sqrt{2}}$

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