Statement:
The line that joins a planet and the Sun sweeps equal areas in equal intervals of time.
Explanation:
(i) Kepler observed that planets move faster when they are nearer to the Sun while they move slower when they are farther from the Sun.
(ii) Suppose the Sun is at the origin. The position of planet is denoted by \(\overrightarrow{r}\) and its momentum is denoted by \(\overrightarrow{p}\) (component ⊥ \(\overrightarrow{r}\)).
(iii) The area swept by the planet of mass m in given interval ∆t is \(\overrightarrow{\triangle A}\) which is given by
As for small ∆t, \(\overrightarrow{V}\) is perpendiculer to \(\overrightarrow{r}\) and this is the area of the triangle.
(iv) Linear momentum \((\overrightarrow{P})\) is the product of mass and velocity.
(vi) For central force the angular momentum is conserved. Hence, from equations (4) and (5),
This proves the law of areas.
