Two satellites P and Q go round a planet in circular orbits of radii 9R and R respectively. If the speed of the satellite P is 4V, then speed of satellite Q will be

Options

A. 6V
B. 12V
C. 2V
D. 36V

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the relationship between speed and radius
The speed of a satellite in a circular orbit is given by the formula:
\[ V \propto \frac{1}{\sqrt{R}} \]
This means that the speed of a satellite is inversely proportional to the square root of the radius of its orbit.

### Step 2: Set up the ratio of speeds
Let \( V_P \) be the speed of satellite P and \( V_Q \) be the speed of satellite Q. According to the relationship:
\[ \frac{V_Q}{V_P} = \sqrt{\frac{R_P}{R_Q}} \]
Where \( R_P \) is the radius of the orbit of satellite P and \( R_Q \) is the radius of the orbit of satellite Q.

### Step 3: Substitute the known values
From the problem, we know:
- \( R_P = 9R \)
- \( R_Q = R \)
- \( V_P = 4V \)

Now substituting these values into the equation:
\[ \frac{V_Q}{4V} = \sqrt{\frac{9R}{R}} \]

### Step 4: Simplify the equation
Simplifying the right side:
\[ \sqrt{\frac{9R}{R}} = \sqrt{9} = 3 \]

So we have:
\[ \frac{V_Q}{4V} = 3 \]

### Step 5: Solve for \( V_Q \)
Now, we can solve for \( V_Q \):
\[ V_Q = 4V \times 3 \]
\[ V_Q = 12V \]

### Conclusion
Thus, the speed of satellite Q is:
\[ V_Q = 12V \]

### Final Answer
The speed of satellite Q is \( 12V \).

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