Question

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

• A. $\frac{3}{2}v$
• B. $\sqrt{\frac{3}{2}}v$
• C. $\sqrt{\frac{2}{3}}v$
• D. $\frac{2}{3}v$

Answer 1

The correct option is C $\sqrt{\frac{2}{3}}v$

The orbital velocity of a satellite in circular orbit of radius $r$ is given by

$v=\sqrt{\frac{GM}{r}}$

Since, $G$ and $M$ are constant here, so

$v\propto \sqrt{\frac{1}{r}}$

$\Rightarrow \frac{v_2}{v_1}=\sqrt{\frac{r_1}{r_2}}\Rightarrow v_2=\sqrt{\frac{r_1}{r_2}}{v}_1$

since, $v_1=v;r_1=R;r_2=R+0.5R=1.5R$

$\therefore v_2=\sqrt{\frac{R}{1.5R}}v$

$\Rightarrow v_2=\sqrt{\frac{2}{3}}v$

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

Key Concept: The orbital velocity of a satellite is given by $v=\sqrt{\frac{GM}{r}}$