Question

Assertion: Kepler's second law can be understood by the conservation of angular momentum principle.
Reason: Kepler's second law is related to areal velocity which can further be proved to be based on conservation of angular momentum as $\left( {\dfrac{{dA}}{{dt}}} \right) = \dfrac{{\left( {{r^2}\omega } \right)}}{2}$
(A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
(B) Both Assertion and Reason are correct and Reason is not the correct explanation for Assertion.
(C) Assertion is correct but Reason is incorrect.
(D) Assertion is incorrect but Reason is correct.

Answer 1

Hint : Kepler’s second law tells us that the areal velocity of a planet revolving around a certain point is constant it can also be said that the area covered by the planet in a given amount of time is constant irrespective of the position of the planet.

Complete step by step answer
Kepler’s second law is a consequence of the law of conservation of angular momentum which tells us that the angular momentum of a planet that is revolving around a certain point is conserved if the torque acting on it is zero. Thus, the assertion statement is correct.
When a planet is revolving around a certain point, the amount of area it covers in a time $dt$ such that it is a distance $r$ from one of the foci of its elliptical path and $d\theta$ is the angle it travels can be calculated as
$\Rightarrow dA = \dfrac{1}{2}{r^2}d\theta$
Then the areal velocity can be calculated as the time derivative of the above equation
$\Rightarrow \dfrac{{dA}}{{dt}} = \dfrac{1}{2}{r^2}\dfrac{{d\theta }}{{dt}}$
Since the angular momentum of the system is constant, assuming that the moment of inertia is constant, from the relation
$\Rightarrow L = I\omega$
We can say that the angular velocity of the planet will also be constant so $d\theta /dt$ is also constant and hence the areal velocity of the planet will be constant. So, the reason statement is also correct and it is also the correct explanation for the assertion statement.
So the correct option is A.

Note
Planets move in elliptical orbits around their centres so they can be closer to the centre at some points and further away at other points however when they are closer to their centres, they will move slower and vice versa so the areal velocity of the planet will remain constant. The main criteria that has to be satisfied for Kepler’s second law is that the law of conservation of angular momentum applies and there is no external torque acting on the planet.