A satellite of mass m moves along an elliptical path arouned the earth. The areal velocity of the satellite is proportional to

Options

A. m
B. `m^(-1)`
C. `m^(0)`
D. `m^(1//2)`

Answer

To solve the problem of determining what the areal velocity of a satellite moving in an elliptical path around the Earth is proportional to, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Areal Velocity**: Areal velocity is defined as the rate at which area is swept out by the radius vector of a satellite as it moves in its orbit. Mathematically, it is expressed as \( \frac{dA}{dt} \), where \( A \) is the area swept out in time \( t \).

2. **Visualizing the Orbit**: Imagine the satellite moving in an elliptical orbit around the Earth (or Sun). The focus of the ellipse is at one of the foci (where the Earth is located).

3. **Using Geometry**: Consider a triangle formed by the radius vector from the focus to the satellite and the line segment connecting two positions of the satellite at different times. The area \( dA \) of this triangle can be calculated using the formula:
\[ dA = \frac{1}{2} \times \text{base} \times \text{height} \]

4. **Identifying the Base and Height**:
- The base of the triangle is the distance \( R \) from the focus to the satellite.
- The height can be determined using the velocity \( v \) of the satellite and the angle \( \theta \) that the radius vector makes with the line of motion. The height is given by \( v \cdot t \cdot \sin(\theta) \).

5. **Calculating the Area**: Thus, the area swept out in a small time interval \( dt \) can be expressed as:
\[ dA = \frac{1}{2} \cdot R \cdot (v \cdot dt \cdot \sin(\theta)) \]

6. **Finding Areal Velocity**: Dividing both sides by \( dt \) gives us the areal velocity:
\[ \frac{dA}{dt} = \frac{1}{2} R v \sin(\theta) \]

7. **Analyzing the Proportionality**: In the expression \( \frac{1}{2} R v \sin(\theta) \), we observe that:
- The areal velocity depends on \( R \), \( v \), and \( \sin(\theta) \).
- Importantly, there is **no dependence on the mass \( m \)** of the satellite.

8. **Conclusion**: Since the areal velocity does not depend on the mass of the satellite, we conclude that the areal velocity is proportional to \( m^0 \) (which means it is independent of mass).

### Final Answer:
The areal velocity of the satellite is proportional to \( m^0 \).