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

Explanation

Let `x` be the distance of point `A` from the centre of the smaller star, where the gravitational pull of two stars balance each other. If a body of mass `m` is placed at `A`, then

`(GM m)/(x^(2)) = (G(16 M)m)/((10a - x)^(2))`
or `(1)/(x^(2)) = (16)/((10a - x)^(2))` or `x = 2a`
So the body projected from bigger star with velocity `upsilon` will reach the smaller star due to star's gravitational field it has sufficient energy to cross the point `A`. `(x = 2 a)` i.e.,
`KE` of body at `B gt (PE` of body at `A - PE` of body at `B`)
i.e., `(1)/(2) m upsilon^(2) gt m (V_(A) - V_(B))` where
`V_(A) = - [(16 GM)/((10a - 2a)) + (GM)/(2a)] = - (20 GM)/(8a)`
`V_(B) = - [(16 GM)/( 2a) + (GM)/((10 a-2a)] = - (65 GM)/(8a)`
So `(1)/(2) m upsilon^(2) gt m [-(20 GM)/(8a) + (65GM)/(8a)]`
or `upsilon_(min) = (3)/(2) sqrt((5GM)/(a))`