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Find the maximum and minimum distances ...

Find the maximum and minimum distances of the planet `A` from the sun `S`, if at a certain moment of times it was at a distance `r_(0)` and travelling with the velocity `upsilon_(0)`. With the angle between the radius vector and velocity vector being equal to `phi`.

Text Solution

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At minimum and maximum distance velocity vector `(V)` makes an angle of `90^(@)` with radius vector. Hence, from conservation of angular momentum,
`mu upsilon_(0) r_(0) sin phi = mr upsilon` ..(i)
Hence, `m` is the mass of the plant.
From energy conservation law, it follows that
`(m upsilon_(0)^(2))/(2) - (GMm)/(r_(0)) = (m upsilon^(2))/(2) - (GMm)/(r)` ..(ii)
Hence, `M` is the mass of the sun.
Solving Eqs. (i) and (ii) for `r`, we fet two values of `r`, one is `r_(max)` and another is `r_(min)`. So,
`r_(max) = (r_(0))/(2 - K) (1 + sqrt(1 - K(2 - K) sin^(2) phi))`
and `r_(min) = (r_(0))/(2 - K) (1 - sqrt(1 - K(2 - K) sin^(2) phi))`
Here, `K = (r_(0)^(2) upsilon_(0)^(2))/(GM)`
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