Two satellite of equal mass are revolving around earth in elliptical orbits of different semi - major axis . If their angular momenta about earth centre are in the ratio `3 :4` then ratio of their area l velocity is

Options

A. `3/4`
B. `2/3`
C. `1/3`
D. `4/3`

Answer

To solve the problem, we need to find the ratio of the areal velocities of two satellites revolving around the Earth, given that their angular momenta are in the ratio of 3:4.

### Step-by-Step Solution:

1. **Understanding Areal Velocity**:
Areal velocity (v) is defined as the area swept out by the radius vector of a satellite per unit time. For a satellite, this can be expressed as:
\[ v = \frac{dA}{dt} = \frac{L}{2m} \]
where \(L\) is the angular momentum and \(m\) is the mass of the satellite.

2. **Angular Momentum**:
The angular momentum \(L\) for a satellite in orbit is given by:
\[ L = m \cdot r^2 \cdot \omega \]
where \(r\) is the distance from the center of the Earth to the satellite and \(\omega\) is the angular velocity.

3. **Ratio of Angular Momenta**:
We are given that the angular momenta of the two satellites are in the ratio:
\[ \frac{L_1}{L_2} = \frac{3}{4} \]

4. **Finding Areal Velocities**:
Using the formula for areal velocity:
\[ v_1 = \frac{L_1}{2m} \quad \text{and} \quad v_2 = \frac{L_2}{2m} \]
The ratio of the areal velocities can be expressed as:
\[ \frac{v_1}{v_2} = \frac{L_1}{L_2} \]

5. **Substituting the Ratio**:
Substituting the given ratio of angular momenta into the equation for areal velocities:
\[ \frac{v_1}{v_2} = \frac{3}{4} \]

6. **Conclusion**:
Therefore, the ratio of the areal velocities of the two satellites is:
\[ \frac{v_1}{v_2} = \frac{3}{4} \]

### Final Answer:
The ratio of their areal velocities is \( \frac{3}{4} \).