Two stars, each of mass $2 imes 10^{30} ext{ kg}$ and radius $10^4 ext{ km}$, are approaching ea

Question

Two stars, each of mass $2 	imes 10^{30}	ext{ kg}$ and radius $10^4	ext{ km}$, are approaching each other for a head-on collision. When they are at a distance of $10^9	ext{ km}$, their speeds are negligible. What is the approximate speed of each star just before they collide?

Professor PC Thomas Classes · Answer

Applying conservation of mechanical energy: initially, $E_i = -\frac{GM^2}{r}$. At collision, the distance between their centres is $2R$. Thus, $E_f = 2(\frac{1}{2}Mv^2) - \frac{GM^2}{2R}$. Equating $E_i$ and $E_f$ and neglecting the $\frac{1}{r}$ term since $r \gg 2R$, we get $v = \sqrt{\frac{GM}{2R}}$. Plugging in $G=6.67 	imes 10^{-11}$, $M=2 	imes 10^{30}$, and $R=10^7$, $v \approx 2.58 	imes 10^6	ext{ m/s}$.