Kepler's Second Law

Kepler's second law, the law of areas, states that the line joining a planet or comet to the Sun sweeps out equal areas in equal times, which is a direct consequence of the conservation of angular momentum because the Sun's gravity is a central force exerting no torque. Angular momentum at perihelion and aphelion gives $m{v}_p{r}_p=m{v}_a{r}_a$, since the velocity is perpendicular to the radius at both turning points. Rearranging, $\frac{v_p}{v_a}=\frac{r_a}{r_p}$. With $r_a=4r_p$, the ratio becomes $\frac{v_p}{v_a}=4$, that is 4 : 1. The option 1 : 4 inverts the relationship, implying the comet moves slower when closest, which is physically wrong. The option 2 : 1 wrongly takes a square root of the distance ratio. The option 16 : 1 incorrectly squares the distance ratio. This is the NCERT angular-momentum interpretation of the law of areas. The deeper reason the law holds is that gravity is a central force, always directed along the line joining the comet to the Sun, so it produces no torque about the Sun and angular momentum $L=mvr$ remains strictly constant throughout the orbit. The same conservation principle also explains why the comet sweeps equal areas in equal times, since the areal velocity $\frac{dA}{dt}=\frac{L}{2m}$ is itself fixed. A plausibility check confirms the comet whips fastest near the Sun and crawls slowest far away, consistent with everyday observations of cometary motion.