Question

Two stars of one solar mass are approaching each other for a head-on collision. When they are a distance of ${10}^9$km, their speeds are negligible . What is the speed with which they collide ? The radius of each other is ${10}^4$ km . Assume the stars to remain undistorted until they colide .

Answer 1

Mass of each star $=2\times {10}^{30}kg$

Radius of each star $R={10}^4$ km=${10}^7$ m

Distance between stars $r={10}^9$ km=${10}^{12}$ m

For negligible speed,$v=0$

Total energy of two stars separated at a distance $r$

$=\frac{-GMM}{r}+\frac{1}{2}m{v}^2$

$=\frac{-GMM}{r}$-----------(1)

Now when the stars are about to collide let the velocity of stars be $v$.

here the distance between the center of the stars =2R

Kinetic energy of both stars $=\frac{1}{2}m{v}^2+\frac{1}{2}m{v}^2=m{v}^2$

and total potential energy of both the stars $=M{v}^2-\frac{GMM}{2R}=\frac{-GMM}{r}$

$v^2=\frac{-GM}{r}+\frac{GM}{2R}=GM[\frac{-1}{r}+\frac{1}{2R}]$

$v^2=6.67\times {10}^{-11}\times 2\times {10}^{30}\left[\right(\frac{-1}{{10}^{12}})+(\frac{1}{2}\times {10}^7\left)\right]$

$v=\sqrt{6.67\times {10}^{12}}=2.58\times {10}^6$