Two satellites of masses 3 m and m orbit the earth in circular orbits of radii r and 3 r respectively. The ratio of the their speeds is

Options

A. `1:1`
B. `sqrt3:1`
C. `3:1`
D. `9:1`

Explanation

To solve the problem of finding the ratio of the speeds of two satellites orbiting the Earth, we can follow these steps:

### Step 1: Understand the formula for orbital speed
The orbital speed \( v \) of a satellite in a circular orbit is given by the formula:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the radius of the orbit.

### Step 2: Identify the parameters for both satellites
- For the first satellite (mass \( 3m \)), the radius of the orbit is \( r \).
- For the second satellite (mass \( m \)), the radius of the orbit is \( 3r \).

### Step 3: Calculate the orbital speed for both satellites
1. **For the first satellite:**
\[ v_1 = \sqrt{\frac{GM}{r}} \]

2. **For the second satellite:**
\[ v_2 = \sqrt{\frac{GM}{3r}} = \sqrt{\frac{GM}{3} \cdot \frac{1}{r}} = \frac{1}{\sqrt{3}} \sqrt{\frac{GM}{r}} \]

### Step 4: Find the ratio of the speeds
To find the ratio of the speeds \( \frac{v_1}{v_2} \):
\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{GM}{r}}}{\frac{1}{\sqrt{3}} \sqrt{\frac{GM}{r}}} \]
This simplifies to:
\[ \frac{v_1}{v_2} = \frac{\sqrt{3}}{1} = \sqrt{3} \]

### Final Answer
The ratio of the speeds of the two satellites is:
\[ \frac{v_1}{v_2} = \sqrt{3} \]

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