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 `rgt gtR`. If their speed at this separation are negligible, the speed `v` with which they collide would be

Explanation

Here, mass of each star, `M = 2 xx 10^(30) kg`.
Initial distance between two stars, `r = 10^(9) km = 10^(12) m`.
Initial potential energy of the systam `= (GM m)/(r )`
Total K.E. of the mass `= (1)/(2) M upsilon^(2) + (1)/(2) M upsilon^(2) = M upsilon^(2)`
where `upsilon` is the speed of stars with which they collide. when the stars are about to collide, the distance between their centres, `r = 2R`.
`:.` Final potential energy of two stars `= - (GM M)/(2R)`.
Since, gain in `K.E.` is at the cost of loss in `P.E.`
`:. M upsilon^(2) = - (GM M)/(r ) - (-(GM M)/(2R)) = (-GM M)/(r) + (GM M)/(2R)`
or `2 xx 10^(30) upsilon^(2) = - (6.67 xx 10^(-11) xx (2 xx 10^(30))^(2))/(10^(12)) + (6.67 xx 10^(-11) (2 xx 10^(30))^(2))/(2 xx 10^(7))`
`= - 2.668 xx 10^(38) + 1.334 xx 10^(43) = 1.334 xx 10^(43) J`
`:. upsilon = sqrt((1.334 xx 10^(43))/(2 xx 10^(30))) = 2.583 xx 10^(6) ms^(-1)`