Two starts each of one solar mass `(=2xx10^(30)kg)` are approaching each other for a head on collision. When they are a distance `10^(9)`km. their speeds are negligible. What is the speed with which they collide? The radius of each star is `10^(4)`km. Assume the stars to remain undistorted until they collide. (Use the known value of G).

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 system `=-(GMm)/(r)`
Total K.E. of the stars `= (1)/(2)Mv^(2) + (1)/(2) Mv^(2) = Mv^(2)`
Where v is the speed of stars with which they collide. When the stars are about to collide, the distance between their centres, `r^(1) = 2R`.
`therefore` Final potential energy of two stars `= -(GM M)/(2R)`.
Since gain in K.E. is at the cost of loss in P.E.
`therefore Mv^(2) = -(GM M)/(r) - (-(GM M)/(2R)) = (-GM M)/(r) + (GM M)/(2R)`
or `2 xx 10^(30)v^(2) = -(6.67 xx 10^(-11) xx (2 xx 10^(30))^(2))/(10^(12)) + (6.67 xx 10^(-11) xx (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`
`therefore v = sqrt((1.334 xx 10^(43))/(2 xx 10^(30))) = 2.583 xx 10^(6) ms^(-1)`