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

Answer

A, B, C

Explanation

Let there are two stars `1` and `2` as shows below.

Let `P` is a point between `C_(1)` and `C_(2)`, where gravitational field strength is zero. Or at `P` field stregth due to star `1` is equal and opposite to the field stength 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) = 10a`
`:. r_(2) ((4)/(4 + 1)) (10a) = 8a`
and `r_(1) = 2a`
NOw, the body of mass `m` is projected from the surface of larger star to wards 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`. point `P` becouse beyond that the particle is attracted towards the smallar star itself.
From conservation of mechanical energy`(1)/(2) m nu_(min)^(2)`
`=` Potential energy of body at `P`
`- `Potential energy at the surface of the larger star.
`:. (1)/(2) m nu_(min)^(2) = [-(GMm)/(r_(1)) - (16 GMm)/(r_(2))]`
`- [-(GMm)/(10 a - 2a) - (16 GMm)/(2a)]`
`= [-(GMm)/( 2a) - (16 GMm)/(8a)] - [-(GMm)/( 8a) - (16 GMm)/(a)]`
or `(1)/(2) m nu_(min)^(2) = ((84)/(8)) (GMm)/(a)`
`:. nu_(min) = (3sqrt(5))/(2) (sqrt(GM)/(a))`