Does choosing the system affect gravitation related results? So I was working out my homework exercises ( this question is not about exercise but about a concept) and I came across this question.
Two stars each of one solar mass $(= 2×10^{30})$ 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.
Now my attempt.
I tried to use energy conservation. I took one star as my system. 
Now initial energy of system is 
 $= -GM²/r$ (I used lower case $r$ for initial distance)
Now final energy of system is
$= (1/2) Mv^2 + -GM^2/2R$ (i.e.the distance between two stars is twice the $r$ radius of each star) 
Now applying energy conservation and solving we have
$$v^{2}=2GM ( 1/2R - 1/r)$$
This is my attempt. 
Now solution States that initial energy of the star is same as I wrote but final energy of system includes kinetic energy of both the stars. So final energy of the system is
\begin{align*}
& = (1/2) Mv^2 + (1/2)Mv^2 -GM^2/2R\\
& =Mv^2- GM^2/2R
\end{align*}
Solving using these we get,
$$v^2= GM( 1/2R -1/r).$$
Now I assume he has used both the stars as his system. But considering only one star as system the results should be same as former. Is there any problem in the argument. Or else what can be wrong.
 A: The concept which you have got wrong is that of gravitational potential energy.
A system of one mass on its own cannot have gravitational potential energy.
Your second misconception is the distance moved by each mass relative to the centre of mass of the two mass system.  
The use of gravitational potential with energy for the two mass system is perfectly valid and that method gives the correct answer.  
To use a system consisting of only one you must evaluate the work done  by the gravitational force $(\int_{\rm start} ^{\rm finish}\vec F_{\rm gravitational} \cdot d\vec r)$ which depends on the separation of the two masses when moving from a distance $\frac r2$ from the centre of mass of the two mass system to $\frac R2$.
This work done by the gravitational force will produce the increase in kinetic energy of the mass under consideration.  
If you do the (same) computation for the other mass you will note that adding the results of the two systems together will give you the same result as using the concept gravitational potential energy for the two mass sytem.
A: You assumed that one of the bodies is at rest (since you only have one term for kinetic energy) but both of the bodies are moving toward each other and they gain equal amounts of energy (by symmetry of the system).So you must take 2 terms of kinetic energy, one for each star.
