A planet revolving in elliptical orbit has : A. a constant velocity of revolution. B. has the least velocity when it is nearest to the sun. C. its areal velocity is directly proportional to its velocity. D. areal velocity is inversely proportional to its velocity. E. to follow a trajectory such that the areal velocity is constant. Choose the correct answer from the options given below :

Gravitation PYQ — JEE Main 2021 · 26 Feb · Shift 1 · Q57 with Solution

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Correct answer: B

A planet revolving around the Sun in an elliptical orbit obeys:

Speed is not constant in an ellipse.
It moves fastest at perihelion (nearest to the Sun).
It moves slowest at aphelion (farthest from the Sun).
By Kepler’s second law, the line joining the planet and the Sun sweeps out equal areas in equal times.

So the areal velocity is constant:

\frac{dA}{dt} = \text{constant}

"a constant velocity of revolution"

This is false. In elliptical motion, the speed changes from point to point.

"has the least velocity when it is nearest to the sun"

This is false. It has the greatest velocity when nearest to the Sun.

"its areal velocity is directly proportional to its velocity"

This is false. Areal velocity is

\frac{dA}{dt} = \frac{1}{2} r v_\perp

Since for planetary motion this remains constant, it is not simply directly proportional to speed alone.

"areal velocity is inversely proportional to its velocity"

This is also false. Areal velocity is not generally inversely proportional to velocity alone.

"to follow a trajectory such that the areal velocity is constant"

This is true by Kepler’s second law.

Only statement E is correct.

Given options:

Hence the correct option is:

\boxed{\text{B: E only}}