Question 277: Gravitation

Question

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

Accepted Answer

Step 1: Expression for orbital velocity
The orbital velocity for a circular orbit at radius $r$:
$ v = \sqrt{\frac{GM}{r}} $
Here $GM$ is the planetÃ¢ÂÂs gravitational parameter.
Step 2: Relation between velocities
Let $v_P$ = orbital speed of satellite $P$ at radius $r_P = 3R$.
$ v_P = \sqrt{\frac{GM}{3R}} $
Given $v_P = 2V$.
For satellite $Q$, orbit radius $r_Q = R$:
$ v_Q = \sqrt{\frac{GM}{R}}. $
Step 3: Ratio of velocities
$ \frac{v_Q}{v_P} = \frac{\sqrt{	frac{GM}{R}}}{\sqrt{	frac{GM}{3R}}} = \sqrt{\frac{1}{1/3}} = \sqrt{3}. $
So
$ v_Q = \sqrt{3}\, v_P. $
Step 4: Substitute $v_P = 2V$
$ v_Q = \sqrt{3} \cdot 2V = 2\sqrt{3}\,V. $
Final Answer:
The speed of satellite $Q$ is:
$ \boxed{2\sqrt{3}\,V} $
Correct option: A.

Answer

$2 \sqrt{3} \mathrm{~V}$

Answer

$\frac{2 \mathrm{~V}}{\sqrt{3}}$

Answer

$\frac{\mathrm{V}}{2}$

Answer

$\frac{\mathrm{V}}{\sqrt{3}}$

Gravitation

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

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