The aeral velocity and the angular moementum of the planet are related by which of the following relationos? (where `m_(p)` is the same mass of the planet)

Options

A. `(Delta vec(t))/(Deltat)=(vec(L))/(2m_(p))`
B. `(Delta vec(t))/(Deltat)=(vec(L))/(m_(p))`
C. `(Delta vec(t))/(Deltat)=(2vec(L))/(m_(p))`
D. `(Delta vec(t))/(Deltat)=(vec(L))/(sqrt(2)m_(p))`

Explanation

To solve the problem of relating the aerial velocity and the angular momentum of a planet, we can follow these steps:

### Step-by-Step Solution:

1. **Define Aerial Velocity**:
Aerial velocity is defined as the area swept out by a planet in unit time. For a planet moving in a circular orbit around the Sun, the aerial velocity \( A \) can be expressed as:
\[ A = \frac{\text{Area}}{\text{Time}} = \frac{\pi r^2}{T} \]
where \( r \) is the radius of the orbit and \( T \) is the time period of revolution.

2. **Calculate the Angular Momentum**:
The angular momentum \( L \) of the planet can be expressed as:
\[ L = m_p v r \]
where \( m_p \) is the mass of the planet, \( v \) is the tangential velocity, and \( r \) is the radius of the orbit.

3. **Relate Velocity and Time Period**:
The velocity \( v \) of the planet can be related to the time period \( T \) by the formula:
\[ v = \frac{2 \pi r}{T} \]

4. **Substitute Velocity in Angular Momentum**:
Substituting the expression for \( v \) into the angular momentum equation:
\[ L = m_p \left(\frac{2 \pi r}{T}\right) r = \frac{2 \pi m_p r^2}{T} \]

5. **Express Time Period in Terms of Angular Momentum**:
Rearranging the equation gives:
\[ T = \frac{2 \pi m_p r^2}{L} \]

6. **Substitute Time Period in Aerial Velocity**:
Now substituting \( T \) back into the aerial velocity equation:
\[ A = \frac{\pi r^2}{\frac{2 \pi m_p r^2}{L}} = \frac{L}{2 m_p} \]

7. **Final Relation**:
Therefore, we find the relation between aerial velocity \( A \) and angular momentum \( L \):
\[ A = \frac{L}{2 m_p} \]

### Conclusion:
The aerial velocity \( A \) and angular momentum \( L \) of the planet are related by the equation:
\[ A = \frac{L}{2 m_p} \]