Question

Let the speed of the planet at the perihelion `P` in figure be `v_(P)` and the Sun planet distance `SP` be `r_(P)`. Relater `r_(P), v_(P)` to the corresponding quantities at the aphelion `(r_(A),v_(A))`. Will the planet take equal times to transverse `BAC` and `CPB`?

Answer 1

Refer to Fig. we note that `vec(r_(p))` and `vec(upsilon_(p))` are perpendicular to each other. Similarly, `vec (r_(A))` and `vec(upsilon_(A))` are perpendicular to each other. Using the law of conservation of angular momentum.

Angular momentum of planet at `P =` angular momentum of planet at `A`
`m upsilon_(P) = m upsilon_(A) r_(A)` or `(upsilon_(P))/(upsilon_(A)) = (r_(A))/(r_(P))`
Since, `r_(A) gt r_(P)`, so `upsilon_(P) gt upsilon_(A)`.
Here, area `SBAC` is greater than area `SCPB`.
As the areal velocity of a planet is constant around the sun, i.e., equal areas are swept in equal intervals of time. Hence, planet will take longer time to traverse `BAC` than `CPB`.