JEE Main 2025 — Gravitation Question with Solution

The assertion (A) states that the “radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time” which is just another way of stating Kepler's second law. This implies that the areal velocity is constant.

The reason (R) states that "For a central force field the angular momentum is a constant." In a central force system (like that of a planet orbiting the Sun under gravity), the force always acts along the line joining the two bodies. This means there is no torque acting on the planet, and thus, its angular momentum is conserved.

The connection between angular momentum and areal velocity is given by:

$$\text{Areal velocity}=\frac{1}{2}{r}^2\dot{\theta }=\frac{L}{2m}$$

where $$L$$ is the angular momentum and $$m$$ is the mass of the planet. Since the angular momentum $$L$$ is constant, the areal velocity must also remain constant.

Therefore, both the assertion and the reason are correct, and the reason correctly explains why the areal velocity (as stated in the assertion) is constant.

So, the most appropriate answer is:

Option C: Both (A) and (R) are correct and (R) is the correct explanation of (A).