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Let the speed of the planet at the pe...

Let the speed of the planet at the perihelion pin be vP and the Sun-planet distance SP be `r_(P)`. Relate `{r_(P), v_(P)}` to the corresponding quantities at the aphelion `{r_(A), v_(A)}`. Will the planet take equal times to traverse BAC and CPB ?

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The magnitude of the angular momentum at P is `L_(p)` = `m_(p) r_(p) v_(p)`, since inspection tells us that rp and vp are mutually perpendicular. Similarly,` L_(A) = m_(p) r_(A) v_(A)`. From angular momentum conservation
`m_(p) r_(p) v_(p) = m_(p) r_(A) v_(A)`
or `(v_(p))/(v_(A)) = (r_(A))/(r_(p))`
since `r_(A) gt r_(p), v_(p) gt v_(A).`
the area SBAC bounded by the ellipse and the radius vectors SB and SC is larger than SBPC in From Kepler’s second law, equal areas are swept in equal times. Hence the planet will take a longer time to traverse BAC than CPB.
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