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 = 2xx10^(30)kg`
Initial distance between two stas,
`r = 10 xx10^(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` .
`:.` Final potential energy of two stars `=-(GMM)/(2R)`
Since gain in K.E. is at the cost of loss in P.E.
`:. Mv^(2)=-(GMM)/r-(GMM)/r-(-(GMM)/(2R))=(-GMM)/r+(GMM)/(2R)`
or `2xx10^(30)v^(2) = -(6.67 xx10^(-11) xx(2xx10^(30))^(2))/10^(12)`
`+ (6.67xx10^(-11) xx(2xx10^(30))^(2))/(2xx10^(7))`
`=-2.668 xx10^(38) + 1.334 xx10^(43) =1.334 xx10^(43)J`
`:. v = sqrt((1.334 xx10^(43))/(2xx10^(30)))=2.583xx10^(6)ms^(-1)`