Question

A satellite of mass m revolves around the earth of radius R at a height x from its surface. If g is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is

• A. gx
• B. $\frac{gR}{R-x}$
• C. $\frac{g{R}^2}{R+x}$
• D. ${\left(\frac{g{R}^2}{R+x}\right)}^{\frac{1}{2}}$

Answer 1

The correct option is D

${\left(\frac{g{R}^2}{R+x}\right)}^{\frac{1}{2}}$

Step 1: Given data

Acceleration due to gravity on the surface of the earth is g.

The radius of the earth is R and x is the height of the satellite above the earth's surface.

The mass of the satellite is m.

Step 2: Calculate the orbital speed of the satellite.

We know that,

$\frac{m{v}^2}{R+x}=\frac{GMm}{{\left(R+x\right)}^2}$

$v=\sqrt{\frac{GM}{R+x}}$

$v=\sqrt{\frac{GM}{R^2}\times \frac{R^2}{R+x}}$

Step 3: Final answer

$v=\sqrt{\frac{g{R}^2}{R+x}}$

We can also write it as

$v={\left(\frac{g{R}^2}{R+x}\right)}^{\frac{1}{2}}$

So, the correct answer is (d).