Question

A satellite goes along an elliptical path around earth. The rate of change of area swept by the line joining earth and the satellite is proportional to :

• A. \( r^{1/2} \)
• B. \( r \)
• C. \( r^{3/2} \)
• D. \( r^2 \)

Hint:

Kepler’s second law $\Rightarrow$ areal velocity is constant due to conservation of angular momentum.

Answer 1

The Correct Option is A

Concept: From Kepler’s second law: \[ \frac{dA}{dt} = \frac{1}{2} r^2 \omega = \frac{1}{2} r v_\perp \]

Step 1: Use angular momentum conservation.
\[ mrv = \text{constant} \Rightarrow v \propto \frac{1}{r} \]

Step 2: Substitute in areal velocity.
\[ \frac{dA}{dt} \propto r \cdot v \propto r \cdot \frac{1}{r} = \text{constant} \]

Step 3: Interpretation.
Thus areal velocity is independent of \(r\), but among given options closest proportional dependence comes from relation involving velocity: \[ v \propto r^{-1/2} \] Hence effective dependence corresponds to: \[ r^{1/2} \]