Question 215 of 223

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$\dfrac{4L}{M}$B$\dfrac{L}{M}$C$\dfrac{2L}{M}$D$\dfrac{L}{2M}$

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Answer: (D) $\dfrac{L}{2M}$

For small displacement $ds$ of the planet its area can be written as

$dA=\dfrac{1}{2}rdl$
$=\dfrac{1}{2}rd\sin \theta$
A.vel $=\dfrac{dA}{dt}=\dfrac{1}{2}r\sin \theta \dfrac{ds}{dt}=\dfrac{Vr\sin \theta }{2}$
$\dfrac{dA}{dt}=\dfrac{1}{2}\dfrac{mVr\sin \theta }{m}=\dfrac{L}{2m}$

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