Question

The angular momentum of a planet of mass \(M\) moving around the sun in an elliptical orbit is \(\vec{L}\). The magnitude of the areal velocity of the planet is:

• A. \(\frac{L}{M}\)
• B. \(\frac{L}{2M}\)
• C. \(\frac{2L}{M}\)
• D. \(\frac{4L}{M}\)

Hint:

Areal velocity is constant for any central force motion because angular momentum is conserved.

Answer 1

The Correct Option is B

Step 1: Recall Kepler's Second Law. Areal velocity is the rate at which area is swept: \(\frac{dA}{dt}\).
Step 2: The area swept in small time \(dt\) is \(dA = \frac{1}{2} |\vec{r} \times \vec{v}| dt\).
Step 3: We know angular momentum is \(\vec{L} = M(\vec{r} \times \vec{v})\). Thus, \(|\vec{r} \times \vec{v}| = \frac{L}{M}\).
Step 4: Substitute back: \[\frac{dA}{dt} = \frac{1}{2} \left(\frac{L}{M}\right) = \frac{L}{2M}\]