Question

Two stars each of one solar mass (= $2\times 10^{30}$ kg) are approaching each other for a head on collision. When they are a distance 109 km, their speeds are negligible. What is the speed with which they collide ? The radius of each star is 104 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).

Answer 1

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

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

= $\frac{-GMM}{r}+0$ ...(i)

Now, consider the case when the stars are about to collide:
Velocity of the stars =$v$
Distance between the centers of the stars = $2R$

Total kinetic energy of both stars = $\frac{1}{2}Mv^2+\frac{1}{2}Mv^2$ = $Mv^2$

Total energy of both stars =$Mv^2-\frac{GMM}{2R}$ ...(ii)

Total energy of the two stars = Using the law of conservation of energy, we can write:

$MV^2-\frac{GMM}{2R}$ = $\frac{-GMM}{r}$

$v^2$= $\frac{-GM}{r}+\frac{GM}{2R}= GM$ $\bigg(-\frac{1}{r}+\frac{1}{2R}\bigg)$

= $6.67 \times 10^ {-11} \times2 \times10^{30} \bigg[-\frac{1}{ 10^{12}} + \frac{1}{2\times 10^7}\bigg ]$

= $13.34 \times 10^{19} [-10^{-12} + 5 \times 10^{-8}] ∼ 13.34 \times 10^{19} \times 5 \times10^{-8} ∼ 6.67 \times 10^{12}$

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