Two satellites A and B go around a planet in circular orbits of radii 4 R and R respectively. If the speed of the satellite A is 3 V, then the speed of the satellite B will be

Options

A. 12 v
B. 6 v
C. `(4v)/3`
D. `(3v)/2 `

Answer

To solve the problem of finding the speed of satellite B given the speed of satellite A, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between gravitational force and centripetal force**:
The gravitational force acting on a satellite provides the necessary centripetal force for it to maintain its circular orbit. The gravitational force \( F_g \) is given by:
\[ F_g = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.

2. **Set up the equation for centripetal acceleration**:
The centripetal force required for circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the orbital speed of the satellite.

3. **Equate gravitational force and centripetal force**:
Since the gravitational force provides the centripetal force, we can set them equal:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]

4. **Cancel the mass of the satellite \( m \)**:
Since \( m \) appears on both sides of the equation, we can cancel it out:
\[ \frac{GM}{r^2} = \frac{v^2}{r} \]

5. **Rearrange to find the orbital speed**:
Multiplying both sides by \( r \) gives:
\[ v^2 = \frac{GM}{r} \]
Taking the square root, we find:
\[ v = \sqrt{\frac{GM}{r}} \]

6. **Relate the speeds of the two satellites**:
Since both satellites A and B orbit the same planet, the gravitational constant \( G \) and the mass \( M \) are the same for both. Therefore, the orbital speeds are related to the square root of the inverse of their respective radii:
\[ \frac{v_B}{v_A} = \sqrt{\frac{r_A}{r_B}} \]

7. **Substitute the given values**:
For satellite A, the radius \( r_A = 4R \) and for satellite B, the radius \( r_B = R \). The speed of satellite A is given as \( v_A = 3V \):
\[ \frac{v_B}{3V} = \sqrt{\frac{4R}{R}} = \sqrt{4} = 2 \]

8. **Calculate the speed of satellite B**:
Rearranging gives:
\[ v_B = 2 \times 3V = 6V \]

### Final Answer:
The speed of satellite B is \( 6V \).