Question

Two satellites P and Q go round a planet in circular orbits having radii ' $3R$ ' and ' $R$ ' respectively. If the speed of satellite $P$ is ' $2V$ ', the speed of the satellite $Q$ will be

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

Hint:

Satellite speed in circular orbit varies as: \[ v\propto \frac{1}{\sqrt{r}} \] Smaller radius means higher orbital speed.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Orbital velocity of a satellite depends on the mass of the planet and the radius of the orbit. A closer satellite must orbit faster to overcome gravity.
Step 2: Key Formula or Approach:
Orbital Speed: $v = \sqrt{\frac{GM}{r}} \implies v \propto \frac{1}{\sqrt{r}}$
Step 3: Detailed Explanation:
Let $v_P, r_P$ and $v_Q, r_Q$ be the speed and radius of satellites $P$ and $Q$.
$\frac{v_Q}{v_P} = \sqrt{\frac{r_P}{r_Q}}$
Given $r_P = 3R$ and $r_Q = R$.
\[ \frac{v_Q}{2V} = \sqrt{\frac{3R}{R}} = \sqrt{3} \]
\[ v_Q = 2\sqrt{3}V \]
Step 4: Final Answer:
The speed of satellite $Q$ is $2\sqrt{3}V$.