Examine Assertion (A)

Assertion says: The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.

This is exactly Kepler’s second law.

Areal velocity is $\frac{dA}{dt}$ and Kepler’s second law states that $\frac{dA}{dt}=\text{constant}.$

Hence, Assertion (A) is correct.

Examine Reason (R)

Reason says: For a central force field the angular momentum is a constant.

In a central force field, force is always along the radius vector, so torque about the center is zero: $\vec{\tau}=\vec{r}\times \vec{F}=0.$

Therefore, $\frac{d\vec{L}}{dt}=\vec{\tau}=0 \implies \vec{L}=\text{constant}.$

So, Reason (R) is also correct.

Check whether (R) explains (A)

Areal velocity of a particle is related to angular momentum by $\frac{dA}{dt}=\frac{1}{2}|\vec{r}\times \vec{v}|.$

Since linear momentum is $\vec{p}=m\vec{v}$, angular momentum is $\vec{L}=\vec{r}\times \vec{p}=m(\vec{r}\times \vec{v}).$

Therefore, $\frac{dA}{dt}=\frac{L}{2m}.$

If angular momentum $L$ is constant, then $\frac{dA}{dt}=\text{constant}.$

Thus, the constancy of angular momentum in a central force field directly explains why equal areas are swept in equal times.

Conclusion

• (A) is correct
• (R) is correct
• (R) is the correct explanation of (A)

Therefore, the correct option is: $\boxed{\text{C}}$