# The distance between the centres of two stars is 10alpha. The masses of these stars are M and 16M and their radii alpha  and 2alpha. A body of m

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

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The force of attraction is zero at say x from the bigger star.
Then force on mass m due to bigger star = Force on mass m
due to small star
(GM_Bm)/(x^2) = (GM_Sm)/((10a-x)^2) rArr (16M)/(x^2 = (M)/((10a -x))^2 rArr x = 8a
If we throw a mass m from bigger star giving it such a velocity
that is sufficient to bring it to P, then later on due to greater
force by the star M_s it will pull it towards itself [without any
external energy thereafter].
The energy of the system (of these masses) initially
= Final energy when m is at P
-(GM_BM_s)/(10a) - (GMB_m)/(2a) - (GM_Sm)/(8a) +(1)/(2) mv^2
=-(GM_BM_S)/(10a) - (GM_B m)/(8a) - (GM_Sm)/(2a)
[:. M_B = 16M, M_S = M)
:.v = (3)/(2)sqrt((5GM)/(a))