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

Answer 1

The correct option is D

$\frac{1}{\sqrt{2}}$

Step 1. Given data

Mass of body, $M$

The radius of orbit, $R$

Step 2. Finding the ratio of the speed of the body to the escape velocity.

Escape velocity is the minimum velocity that a moving boy must have to escape from the Gravitational field of earth and move outward into space.

By using the formula,

Escape velocity is $V_e=\sqrt{\frac{2GM}{R}}$ [Where, $G$ is Gravitational constant.]

Orbital speed or speed of the body in the orbit is defined as the speed at which it orbits around the sun.

By using the formula,

The speed of the body in the orbit is $V_s=\sqrt{\frac{GM}{R}}$

The ratio of the speed of the body to the escape velocity is,

$\frac{V_s}{V_e}=\sqrt{\frac{GM}{R}}\times \sqrt{\frac{R}{2GM}}$

$\frac{V_s}{V_e}=\frac{1}{\sqrt{2}}$

Therefore, the ratio of the speed of the body in orbit to the escape velocity is $\frac{1}{\sqrt{2}}$.

Hence, option $\left(D\right)$ is correct.