Two stars each of one solar mass `(= 2 xx 10^(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

`implies` Mass of each star `M = 2 xx 10^(30)` kg
Initial distance between two stars,
`r = 10^(9)km = 10^(12)` m
Radius of each star `R = 10^(4) km = 10^(7) m`
Initial position :

From figure (a),
` v_1=0` so initial kinetic energy `K_1 = 0`
and potential energy `V_1 = - ("GMM")/r`
`:.` Total energy `E=K_(i)+V_(l)=-(GM^2)/r " "....(1)`
From figure (b),
Final kinetic energy `K_(2) =1/2 Mv^2 +1/2 Mv^2 `
`=Mv^2 `
and potential energy `V_(2) = - ("GMM")/(2R)`
`=- (GM^2)/(2R)`
`:.` Total energy `E_(2) = K_(2) + V_(2)`
`= Mv^(2) - (GM)^2/(2R) ....(2)`
From conservation of energy ,
`:. E_(1) =E_(2)`
`:. - (GM^2)/r =Mv^2 -(GM^2)/(2R)`
`:. Mv^2=(GM^2)/(2R)-(GM^2)/r`
`:. v^2 = GM [1/(2R) -1/r]`
`:. v^2 = 6.67 xx10^(-11) xx 2 xx 10^(30)[1/(2xx10^(7)) -1/10^(9)]`
`:. v^2 = 13.34 xx10^(19) xx 10^(-7) [0.5 - 0.1 ]`
` :. v^2 = 13.34 xx 10^(12) xx 0.49`
`:. v^(2) = 6.5366 xx10^(12)`
`:. v = 2.5566xx10^(6) ms^(-1)`
`:. v = 2.6 xx10^(6)ms^(-1)`