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.

Explanation

The distance x (from the smaller planet) where the gravitational pull of two planet.s balance each other will be given by
`(-GMm)/x^2=(-G(16M)m)/((10x-x)^2)`
i.e., so the body will reach the smaller planet due to planet.s gravitational field if it has sufficient energy to cross the point B (x=2a) , i.e
`1/2"m"upsilon^2gtm(V_B-V_s)`
but `V_(s)=-[(16Gm)/(2a)+(GM)/((10-2a))]=-(65Gm)/(8a)`

and `V_(B)=-[(16GM)/(8a)+(GM)/(2a)]=-(20GM)/(8a)`
so `1/2"m"upsilon^(2) gt m [(65GM)/(8a)-(20GM)/(8a)]`
i.e `upsilon_("min")=3/2sqrt((5GM)/a)`