Question

The orbital velocity of an artificial satellite in a circular orbit just above the earth's surface is $\nu$. For a satellite orbiting at an altitude of half of the earth's radius, the orbital velocity is

• A. $\left(\frac{3}{2}\right)\nu$
• B. $\sqrt{\left(\frac{3}{2}\right)}\nu$
• C. $\sqrt{\left(\frac{2}{3}\right)}\nu$
• D. $\left(\frac{2}{3}\right)\nu$

Answer 1

The correct option is C $\sqrt{\left(\frac{2}{3}\right)}\nu$

Orbital velocity, $v=\sqrt{g^{\prime }r}$ where $r$ is the radius of the orbit and $g^{\prime }$ is the acceleration due to gravity in that orbit.

Thus, $\nu =\sqrt{gR}$ Where R is the radius of Earth

As $g^{\prime }=g[1+\frac{h}{R}{]}^{-2}$

Now, at $h=R/2⟹g^{\prime }=g[\frac{2}{3}{]}^2$

Thus ${\nu }^{\prime }=\sqrt{g[\frac{2}{3}{]}^2\frac{3R}{2}}⟹{\nu }^{\prime }=\nu \sqrt{\frac{2}{3}}$