Question

Find the speeds of a planet of msdd `m` in its perihelion and aphelion postitions. The semimajor axis of its orbit is a, ecentricity is `e` and te mass of the sun is `M`. Also find the total energy of the planet in terms of the given parameters.

Answer 1

Correct Answer - A::B
Let `upsilon_(1)` and `upsilon_(2)` be the speeds of the planet at perihelion and aphelion positions.
`r_(1) = a (1 - e)`
and `r_(2) = a(1 + e)` ..(i)

Applying conservation of momentum of the planet at `p` (perihelion) and `A` (aphelion)
`m upsilon_(1)r_(1) sin 90^(@) = m upsilon_(2)r_(2) sin 90^(@)`
`upsilon_(1) r_(1) = upsilon_(2) r_(2)`
Applying conservation of mechanical energy in these two positions, we have
`(1)/(2) m upsilon_(1)^(2) - (GMm)/(r_(1) = (1)/(2) m upsilon_(2)^(2) - (GMm)/(r_(2)`
Solving Eqs. (i), (ii) and (iii), we get
`upsilon_(1) = sqrt((GM)/(a)((1 + e)/(1 - e))) ` `upsilon_(2) = sqrt((GM)/(a)((1 - e)/(1 + e))) `
Further, total energy of the planet
`E = (1)/(2) m upsilon_(1)^(2) - (GMm)/(r_(1) = (1)/(2) m [(GM)/(a)((1 - e)/(1 + e))] - (GMm)/(a(1 - e))`
`= (GMm)/(a(1 - e)) [((1 + e)/(2)) - 1]`
`= (GMm)/(a(1 - e)) ((e -1)/(2))` or `E = - (GMm)/(2a)`