Question

The planet mercury is revolving in an elliptical orbit around the sun as shown in the figure. The kinetic energy of mercury will be greatest at

(A) A
(B) B
(C) C
(D) D

Answer 1

Hint: We will be solving this question using the conservation laws in planetary motion. We will also be using the relation between the velocity and the distance of mercury from the sun.
(1) Conservation of angular momentum $ mvr $ = constant
Where $ m $ is the mass, $ v $ is the velocity and $ r $ is the radius.
(2) The formula for the kinetic energy $ K=\dfrac{1}{2}m{{v}^{2}} $
Where $ m $ is the mass, $ v $ is the velocity.

Complete step by step answer:
As we are aware of Kepler's laws of planetary motion and the conservation laws of planetary motion, we can conclude that for all the planets that are revolving around the sun, the angular momentum is always conserved.
Therefore, we can write
 $ mvr $ = constant
Here, $ m $ is the mass of the mercury, $ v $ is the velocity of mercury and $ r $ is the radius, i.e., the distance between the sun and the planet mercury.
Now, as we know that the total mass of the mercury is conserved and is constant. Therefore, the only way the angular momentum of the planet can remain constant is that the velocity varies inversely in accordance with the distance. Therefore, here we can say that the velocity is inversely proportional to the distance between the sun and the planet mercury.
We will now discuss the kinetic energy of the planet mercury. As we can see from the formula for kinetic energy, $ K=\dfrac{1}{2}m{{v}^{2}} $ the kinetic energy is directly proportional to the velocity of the planet. And as we have concluded above, the velocity is inversely proportional to the distance. Therefore, for the kinetic energy to be maximum, the velocity should be maximum, and the only way it is possible is when the distance of mercury from the sun is minimum.
Hence, looking at the figure that is given in our question, the highest kinetic energy of the planet is at point A which is where the planet mercury is the closest to the sun.
Therefore option (A) is the correct answer.

Note:
Keep in mind that it is asked that at which point the kinetic energy of mercury will be greatest which is at point A. But if it were to find the point at which the kinetic energy is the minimum, then the correct answer would be at point B which is the farthest point from the sun. And also, the angular momentum of a planet revolving around the sun is always conserved.