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Two stars each of one solar mass $\left(=2 \times 10^{30}\; kg \right)$ 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 $G$).
Solution
Mass of each star, $M=2 \times 10^{30} kg$
Radius of each star, $R=10^{4} km =10^{7} m$
Distance between the stars, $r=10^{9} km =10^{12} m$
For negligible speeds, $v=0$ total energy of two stars separated at distance $r$ $=\frac{- GMM }{r}+\frac{1}{2} m v^{2}$
$=\frac{-G M M}{r}+0$
Now, consider the case when the stars are about to collide:
Velocity of the stars $=v$
Distance between the centers of the stars $=2 R$
Total kinetic energy of both stars $=\frac{1}{2} M v^{2}+\frac{1}{2} M v^{2}=M v^{2}$
Total potential energy of both stars $=\frac{- GMM }{2 R}$
Total energy of the two stars $= M v^{2}-\frac{G M M}{2 R}$
Using the law of conservation of energy, we can write:
$M v^{2}-\frac{ GMM }{2 R}=\frac{- GMM }{r}$
$v^{2}=\frac{- G M}{r}+\frac{ G M}{2 R}= G M\left(-\frac{1}{r}+\frac{1}{2 R}\right)$
$=6.67 \times 10^{-11} \times 2 \times 10^{30}\left[-\frac{1}{10^{12}}+\frac{1}{2 \times 10^{7}}\right]$
$=13.34 \times 10^{19}\left[-10^{-12}+5 \times 10^{-8}\right]$
$\approx 13.34 \times 10^{19} \times 5 \times 10^{-8}$
$\approx 6.67 \times 10^{12}$
$v=\sqrt{6.67 \times 10^{12}}=2.58 \times 10^{6} m / s$
