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 :

• A. $\frac{1}{\sqrt{2}}$
• B. $2$
• C. $1$
• D. $\sqrt{2}$

Answer 1

The correct option is A $\frac{1}{\sqrt{2}}$

Orbital speed of the body is,

$V_0=\sqrt{\frac{GM}{R+h}}$

As the body is revolving very close to the surface of the planet, $R+h\approx R$

$\Rightarrow V_0=\sqrt{\frac{GM}{R}}-\left(i\right)$

Escape speed from the surface of planet,

$V_e=\sqrt{\frac{2GM}{R}}-\left(ii\right)$

Dividing (i) by (ii), we have

$\frac{V_0}{V_e}=\sqrt{\frac{\frac{GM}{R}}{\frac{2GM}{R}}}=\frac{1}{\sqrt{2}}$.

Hence, $\left(A\right)$ is the correct answer.