Let the speed of the planet at the perhelion Pin Fig. 8.1 (a) be `v_(p)` and the sun-planat distance SP be `r_(p)`. Relate `(r_(p')v_(p))` to the corresponding quantities at the aphelion `(r_(A).v_(A))` will the planat take equal times to traverse BAC and CPB?
Let the speed of the planet at the perhelion Pin Fig. 8.1 (a) be `v_(p)` and the sun-planat distance SP be `r_(p)`. Relate `(r_(p')v_(p))` to the corresponding quantities at the aphelion `(r_(A).v_(A))` will the planat take equal times to traverse BAC and CPB?
Text Solution
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Referring fig we note that `vecr_(P)` and `vecv_(P)` are perpendicular to each other. Similarly `vecr_(A)` and `vecv_(A)` are perpendicular to each. Using the law of conservation of angular momentum,
Angular momentum of the plane at `P=` Angular momentum of the planet at `A`
`impliesm_(P)v_(P)r_(P)=m_(A)v_(A)r_(A)`
or `(v_(P))/(v_(A))=(r_(A))/(r_(P))`
Since `r_(A)gtr_(P), v_(P)gtv_(A)`
Her area `SBAC` is greater than area `SCPB`. As the areal velocity of a planet is contant aroung the Sun i.e. equal areas are swept in equal times. Hence the planet will take longer time to reverse `BAC` than `CPB`.
Angular momentum of the plane at `P=` Angular momentum of the planet at `A`
`impliesm_(P)v_(P)r_(P)=m_(A)v_(A)r_(A)`
or `(v_(P))/(v_(A))=(r_(A))/(r_(P))`
Since `r_(A)gtr_(P), v_(P)gtv_(A)`
Her area `SBAC` is greater than area `SCPB`. As the areal velocity of a planet is contant aroung the Sun i.e. equal areas are swept in equal times. Hence the planet will take longer time to reverse `BAC` than `CPB`.
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