1. Use the orbital speed formula

For a satellite moving in a circular orbit of radius $r$ around a planet,

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

where $G$ is the gravitational constant and $M$ is the mass of the planet.

So, orbital speed is inversely proportional to the square root of radius:

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

2. Write the given radii

• Satellite $A$ has orbit radius $4R$
• Satellite $B$ has orbit radius $R$

Thus,

$\frac{v_B}{v_A} = \sqrt{\frac{r_A}{r_B}} = \sqrt{\frac{4R}{R}} = \sqrt{4} = 2$

3. Use the given speed of satellite A

Given:

$v_A = 3v$

Therefore,

$v_B = 2v_A = 2(3v) = 6v$

4. Match with the options

The correct option is:

$\boxed{6v}$

So, Option A is correct.