Two satellites P and Q have their radii ratio as 4 : 1. The speed of satellite P is 2V. The speed of satellite Q will be

Options

A. 4V
B. 3V
C. 2V
D. 5V

Explanation

To solve the problem of finding the speed of satellite Q given the speed of satellite P and the ratio of their radii, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Data**:
- The ratio of the radii of satellites P and Q is given as \( r_P : r_Q = 4 : 1 \).
- The speed of satellite P is given as \( V_P = 2V \).

2. **Express the Radii**:
- Let the radius of satellite Q be \( r_Q = r \).
- Then, the radius of satellite P will be \( r_P = 4r \).

3. **Use the Formula for Orbital Speed**:
- The formula for the orbital speed \( V \) of a satellite is given by:
\[ V = \sqrt{\frac{GM}{r}} \]
- Here, \( G \) is the gravitational constant and \( M \) is the mass of the celestial body around which the satellites are orbiting.

4. **Write the Speeds for Satellites P and Q**:
- For satellite P:
\[ V_P = \sqrt{\frac{GM}{r_P}} = \sqrt{\frac{GM}{4r}} \]
- For satellite Q:
\[ V_Q = \sqrt{\frac{GM}{r_Q}} = \sqrt{\frac{GM}{r}} \]

5. **Relate the Speeds of the Satellites**:
- We can express the ratio of the speeds of satellite Q to satellite P:
\[ \frac{V_Q}{V_P} = \frac{\sqrt{\frac{GM}{r}}}{\sqrt{\frac{GM}{4r}}} \]
- Simplifying this gives:
\[ \frac{V_Q}{V_P} = \frac{\sqrt{4}}{1} = 2 \]

6. **Substitute the Known Speed of Satellite P**:
- We know \( V_P = 2V \), so we can substitute this into our equation:
\[ V_Q = 2 \cdot V_P = 2 \cdot (2V) = 4V \]

7. **Conclusion**:
- The speed of satellite Q is \( V_Q = 4V \).

### Final Answer:
The speed of satellite Q will be \( 4V \).

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