Two stars each of mass M and radius R are approaching each other for a head-on collision. They start approaching each other when their separation is r >> R. If their speeds at this separation are negligible, the speed v with which they collide would be

Since the speeds of the stars are negligible when they are at a distance r, hence the initial kinetic energy of the system is zero. Therefore, the initial total energy of the system isEi = KE + PE = 0+(-GMMr) = -GM2rwhere M represents the mass of each star and r is initial separation between them; twice the radius of star i.e. 2R.Let v be the speed with which two stars collide. Then total energy of the system at the instant of their collision is given byEf = 2 ×(12Mv2)+(-GMM2R)= Mv2-GM22RAccording to law of conservation of mechanical energyEf = EiMv2-GM22R = -GM2r or v2 = GM(12R-1r)or v = GM(12R-1r)