Question

Distance between the centres of two stars is `10a`. The masses of these stars are `M` and `16 M` ans their radit a nd `2a` respectively. A body of mass `m` is fired straight from the surface of the larger star tpwards the surface of the smaller star. What should be its minimum intial speed to reach the surface of the smaller star? Obtain the expression in terms of `G`, `M` and a.

Answer 1

Correct Answer - A::B::C::D
Let there are two stars `1` and `2` as shows below.

Let `P` is a points between `C_(1)` and `C_(2)` where gravitational field strength is zero or at `p` field strength due to star `1` is equal and opposite to the field strength due to star `2`. Hence,
`(GM)/(r_(1)^(2)) = (G(16 M))/(r_(2)^(2))` or `(r_(2))/(r_(1))= 4` also `r_(1) + r_(2) = 10 a`
`:. r_(2) = ((4)/(4 + 1)) (10 a) = 8 a` and `r_(1) = 2a` ltbegt Now, the body of mass `m` is projected from the surface of larger star towards the smaller one. Between `C_(2)` and `P` it is attracted towards `2` and between `C_(1)` and `p` it will be attracted towards `1`. Therefore , the body should be projected to just cross `p` because beyond that the partical is attracted towards the smaller star itself.

From conservation of machanical energy `(1)/(2) m upsilon_(min)^(2)` ltbr. `= Potential energy of the body at `P` - Potential energy at the surface of larger star.
`:. (1)/(2) m upsilon_(min)^(2) = [-(GMm)/(r_(1)) - (16 GMm)/(r_(2))] - [-(GMm)/(10 a - 2a) - (16 GMm)/(2a)]`
`=[-(GMm)/(2a) - (16 GMm)/(8a)] - [-(GMm)/(2a) - (16 GMm)/(a)]`
or `(1)/(2) m upsilon_(min)^(2) = ((45)/(8)) rArr :. upsilon_(min) = (3sqrt(5))/(2) (sqrt((GM)/(a)))`