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

As mass of each star, `M = 2 xx 10^(30) kg`,
Initia distance between the stars `= r = 10^(9) km = 10^(12) m`
Final distance between the centres of two stars during collision
= R + R = 2R
`= 2 xx 10^(4) km`
`= 2 xx 10^(7) m`
Let speed with which the stars collide = v
Initial potential energy of the system `= -(GM M)/(r )`
Final potential energy of the stars `= -(GM m)/(2R)`
Total kinetic energy of the stars `= (1)/(2) Mv^(2) + (1)/(2) M v^(2)`
Since
Gain in K.E. = Loss in P.E.
`(1)/(2) Mv^(2) + (1)/(2) Mv^(2) = -(GM M)/(r ) - (-(GM M)/(2R))`
`Mv^(2) = -[(GM M)/(r ) - (GM M)/(2R)]`
`v^(2) = - GM[(2R - r)/(2R r)]`
`= -6.67 xx 10^(-11) xx 2 xx 10^(30) xx [(2 xx 10^(7) - 10^(12))/(2 xx 10^(19))]`
`v^(2) = 6.67 xx 10^(12)`
`v = 2.58 xx 10^(6) ms^(-1)`