To solve this question, we will use the fact that for an object in a circular orbit, the centripetal force required to keep the object in orbit is provided by the gravitational force between the object and the planet it is orbiting. This principle gives us the relationship between the speed of the satellite, its orbital radius, and the mass of the planet.

The formula for the gravitational force is given by:

$$ F = \frac{G M m}{r^2} $$

where:

• $F$ is the gravitational force between the two masses,
• $G$ is the gravitational constant,
• $M$ is the mass of the planet,
• $m$ is the mass of the satellite, and
• $r$ is the distance between the center of the planet and the satellite (radius of the orbit).

The centripetal force required to keep the satellite in orbit is given by:

$$ F_c = \frac{m v^2}{r} $$

where:

• $F_c$ is the centripetal force,
• $v$ is the orbital speed of the satellite, and
• $r$ is the radius of the orbit.

Since the gravitational force is providing the centripetal force, we can set the two forces equal to each other to find the relationship between speed and radius:

$$ \frac{G M m}{r^2} = \frac{m v^2}{r} $$

By simplifying this, we find the equation for the orbital speed of a satellite:

$$ v = \sqrt{\frac{G M}{r}} $$

Now, we can compare the speeds of satellites $A$ and $B$ based on their radii. Satellite $A$ orbits at a radius of $4R$ with a speed of $3v$, and we need to find the speed of satellite $B$ which orbits at a radius of $R$.

Substituting the respective radii into the speed equation:

• For satellite $A$, $v_A = 3v = \sqrt{\frac{G M}{4R}}$
• For satellite $B$, we want to find $v_B$ given that its radius is $R$: $v_B = \sqrt{\frac{G M}{R}}$

To find the ratio of $v_B$ to $3v$ (or the speed of $A$), we can write:

$$ v_B = \sqrt{\frac{G M}{R}} = \sqrt{4} \sqrt{\frac{G M}{4R}} = 2 \cdot \sqrt{\frac{G M}{4R}} $$

Given that $\sqrt{\frac{G M}{4R}}$ is the speed of satellite $A$ divided by 3 ($3v$), we find that:

$$ v_B = 2 \cdot 3v = 6v $$

Therefore, the correct answer is Option A: $6 v$.