Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : 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.
Reason (R) : For a central force field the angular momentum is a constant.
In the light of about statement, Choose the most appropriate answer  from the option  given below : 

According to Kepler’s second law, the line joining the Sun and any planet sweeps out equal areas in equal intervals of time. This statement is the same as saying that the areal velocity $$\frac{dA}{dt}$$ of the planet is constant.

In mechanics the areal velocity of a particle of mass $$m$$ moving with linear momentum $$\mathbf{p}=m\mathbf{v}$$ at an instantaneous position vector $$\mathbf{r}$$ from the origin is given by

$$\frac{dA}{dt}=\frac{1}{2m}\,|\mathbf{r}\times\mathbf{p}|$$

The quantity $$\mathbf{L}=\mathbf{r}\times\mathbf{p}$$ is the angular momentum of the particle about the origin. Hence

$$\frac{dA}{dt}=\frac{|\mathbf{L}|}{2m}$$

If a particle is subjected to a central force, the force vector $$\mathbf{F}$$ is always directed along the radius vector $$\mathbf{r}$$. Therefore the torque about the origin is

$$\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}=0$$

Zero torque implies $$\frac{d\mathbf{L}}{dt}=0$$, so the angular momentum $$\mathbf{L}$$ is conserved (constant in both magnitude and direction).

Because $$|\mathbf{L}|$$ is constant, the expression $$\frac{dA}{dt}=\frac{|\mathbf{L}|}{2m}$$ shows that the areal velocity $$\frac{dA}{dt}$$ must also be constant. This directly yields Kepler’s second law.

Thus:

  • Assertion (A) is correct: the radius vector sweeps out equal areas in equal times, so the areal velocity is constant.
  • Reason (R) is correct: in a central force field the angular momentum is conserved.
  • Reason (R) correctly explains why the areal velocity is constant, since constant angular momentum leads to constant $$\frac{dA}{dt}$$.

Therefore the appropriate choice is Option A: Both (A) and (R) are correct and (R) is the correct explanation of (A).