Question

A satellite is revolving in a circular orbit at a height $h$ from the earth's surface, such that$h<<R$ where $R$ is the radius of the earth. Assuming that the effect of earth's atmosphere can be neglected, the minimum increase in the speed required so that the satellite could escape from the gravitational field of the earth is:

• A. $\sqrt{gR}$
• B. $\sqrt{2gR}$
• C. $\sqrt{gR}(\sqrt{2}-1)$
• D. $\sqrt{\frac{gR}{2}}$

Answer 1

The correct option is C

$\sqrt{gR}(\sqrt{2}-1)$

Step 1. Given data :

A satellite is revolving in a circular orbit at a height = $h$

Radius of the earth =$R$

Orbital velocity = $V_{\circ }$

Escape velocity = $V_e$

gravity = $g$

Step 2. Find the minimum increase in the speed to escape from gravitational field :

$V_{\circ }=\sqrt{g(R+h)}\approx \sqrt{gR}$

$V_e=\sqrt{2g(R+h)}\approx \sqrt{2gR}$

$$\begin{gathered} ∆V=V_e-V_{\circ } \\ =(\sqrt{2}-1)\sqrt{gR} \end{gathered}$$

Hence, (C) option is correct.