A planet is revolving round the sun in an elliptical orbit, If v is the velocity of the planet when its position vector from the sun is r, then areal velocity of the planet is

Options

A. `|v xx r|`
B. `2|r xx v|`
C. `|(1)/(2)(r xx v)|`
D. None of these

Explanation

To find the areal velocity of a planet revolving around the sun in an elliptical orbit, we can use Kepler's second law, which states that the line joining a planet to the sun sweeps out equal areas in equal times.

### 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 the planet. Mathematically, it is given by:
\[ \text{Areal Velocity} = \frac{dA}{dt} \]
where \( dA \) is the area swept out in a small time interval \( dt \).

2. **Using the Formula for Area**:
The area \( dA \) swept out by the radius vector in a time interval \( dt \) can be expressed as:
\[ dA = \frac{1}{2} r \times v \sin(\theta) dt \]
where \( r \) is the position vector (distance from the sun), \( v \) is the velocity of the planet, and \( \theta \) is the angle between the position vector and the velocity vector.

3. **Finding Areal Velocity**:
Dividing the area \( dA \) by the time interval \( dt \), we have:
\[ \frac{dA}{dt} = \frac{1}{2} r v \sin(\theta) \]

4. **Using the Cross Product**:
The term \( r \times v \) can be interpreted as the cross product of the position vector and the velocity vector. The magnitude of this cross product gives us the area swept out per unit time:
\[ \frac{dA}{dt} = \frac{1}{2} (r \times v) \]

5. **Final Expression for Areal Velocity**:
Thus, the areal velocity of the planet is given by:
\[ \text{Areal Velocity} = \frac{1}{2} r \times v \]

### Conclusion:
The areal velocity of the planet when its position vector from the sun is \( r \) and its velocity is \( v \) is:
\[ \text{Areal Velocity} = \frac{1}{2} r \times v \]