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 follow these steps:

### Step 1: Understand Areal Velocity
Areal velocity is defined as the rate at which area is swept out by the position vector of the planet with respect to the sun. It can be expressed mathematically as:
\[ \text{Areal Velocity} = \frac{dA}{dt} \]
where \( dA \) is the area swept out in a small time interval \( dt \).

### Step 2: Relate Areal Velocity to Angular Momentum
The areal velocity can also be related to the angular momentum \( L \) of the planet. The formula for areal velocity \( A \) is given by:
\[ A = \frac{L}{m} \]
where \( L \) is the angular momentum and \( m \) is the mass of the planet.

### Step 3: Calculate Angular Momentum
The angular momentum \( L \) of the planet can be calculated using the formula:
\[ L = m \cdot r \cdot v_{\perp} \]
where \( v_{\perp} \) is the component of the velocity perpendicular to the position vector \( r \).

### Step 4: Determine the Perpendicular Velocity Component
In an elliptical orbit, the velocity \( v \) can be decomposed into two components: one along the radius vector \( r \) and one perpendicular to it. The perpendicular component \( v_{\perp} \) can be found using the relationship:
\[ v_{\perp} = v \cdot \sin(\theta) \]
where \( \theta \) is the angle between the position vector \( r \) and the velocity vector \( v \).

### Step 5: Substitute into Areal Velocity Formula
Substituting the expression for angular momentum into the areal velocity formula gives:
\[ \text{Areal Velocity} = \frac{m \cdot r \cdot v_{\perp}}{m} = r \cdot v_{\perp} \]

### Step 6: Final Expression for Areal Velocity
Thus, the final expression for the areal velocity of the planet is:
\[ \text{Areal Velocity} = \frac{1}{2} r \cdot v \cdot \sin(\theta) \]

### Summary
The areal velocity of the planet revolving around the sun in an elliptical orbit is given by:
\[ \text{Areal Velocity} = \frac{1}{2} r \cdot v \cdot \sin(\theta) \]