Three stars of mass M and radius R are initially at rest and the distace between centres of any two stars is d and they from an equilateral triangle. They start moving towards the centroid due to mutual force of attraction. What are the velocities of the stars just before collision ?

Distance between the one star and centreoid is given by:x = 23d2-d241/2 =d3 Resultant force on one  star due to two is given by:F = GM2d22+GM2d22+2GM2d22cos60F = 3GM2d2 = Mawhere a is the acceleration of teh stara = 3GMd2ditance travel by star is d3initial velocity u = 0v2-u2 = 2asv2 -0 = 2×3GMd2×d3 = 2GMdv = 2GMdwhere v is the velocity before the collision.

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