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 aerial velocities of two satellites revolving around the Earth in elliptical orbits. Given that their angular momenta are in the ratio of 3:4, we can use the relationship between angular momentum and aerial velocity.

### Step-by-Step Solution:

1. **Understanding Aerial Velocity**:
The aerial velocity (denoted as \( \frac{dA}{dt} \)) is defined as the area swept out by the radius vector in unit time. According to Kepler's second law, this is constant for a satellite in orbit.

2. **Angular Momentum Relationship**:
The angular momentum \( L \) of a satellite can be expressed as:
\[ L = m \cdot v \cdot r \]
where \( m \) is the mass of the satellite, \( v \) is its orbital velocity, and \( r \) is the distance from the center of the Earth.

3. **Using Kepler's Second Law**:
According to Kepler's second law, the aerial velocity is related to angular momentum:
\[ \frac{dA}{dt} = \frac{L}{2m} \]
This means the aerial velocity is proportional to the angular momentum divided by twice the mass of the satellite.

4. **Setting Up the Ratios**:
Let \( L_1 \) and \( L_2 \) be the angular momenta of the two satellites. We are given:
\[ \frac{L_1}{L_2} = \frac{3}{4} \]
Since both satellites have equal mass (\( m_1 = m_2 \)), we can write the ratio of their aerial velocities as:
\[ \frac{\frac{dA_1}{dt}}{\frac{dA_2}{dt}} = \frac{L_1 / 2m}{L_2 / 2m} = \frac{L_1}{L_2} \]

5. **Calculating the Ratio**:
Substituting the known ratio of angular momenta:
\[ \frac{dA_1/dt}{dA_2/dt} = \frac{3/4} \]

6. **Conclusion**:
Therefore, the ratio of the aerial velocities of the two satellites is:
\[ \frac{dA_1/dt}{dA_2/dt} = \frac{3}{4} \]

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

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