# Collisions in Center-of-Mass coordinates

1. ### Xyius

496
1. The problem statement, all variables and given/known data
A particle of mass m approaches a stationary particle of mass 3m. They bounce off elastically. Assume 1D. Find the final velocities using the center of mass coordinate system.

2. Relevant equations
(All quantities with r or v are vectors r1 and r2 represent the vectors from the origin to each particle.)
Coordinates of the Center of mass vector.
$$R=\frac{m_1v_1+m_2v_2}{m_1+m_2}$$

Coordinates from vector 2 to vector 1
$$r=r_1-r_2$$

Inverse transformations
$$r_1=R+\frac{m_2}{m_1+m_2}$$
$$r_2=R+\frac{m_1}{m_1+m_2}$$

3. The attempt at a solution
Those equations are from my notes/book. I applied them so..
$$r^{CM}_1=r_1-R$$
and
$$r^{CM}_2=r_2-R$$

I really do not know how to solve for velocities this way. It is late and maybe I am having a brain fart but how am I supposed to use the conservation of momentum relation to find velocity if it always equals zero in this coordinate system? All I have to work with is the conservation of energy. Can anyone help me out? :(??

2. ### diazona

2,155
For starters, this equation is slightly wrong:
Also,
It's not true that velocity always equals zero in the CM coordinate system. That would be inconsistent with the fact that the objects are moving relative to each other.

As a first step to solving the problem, write out the equations for conservation of momentum and conservation of energy.

3. ### Xyius

496
Wow I must have been tired because the amount of errors in my original post is astounding! Firstly, I meant to write POSITION vectors in the R equation, not velocity vectors. Second, the inverse transformations are missing an "r" multiple after the term with the masses as well as a minus in the r2 expression. And THIRD I meant to say the MOMENTUM is equal to zero not velocity! Oh geez haha.
Anyway I know this following work is wrong because when I do it the normal way, v1f=-1/2v1i (If I remember correctly, I cant find my work from last night.)

The problem is, I can get the velocity of "v" which is the vector between r2 and r1, but I am confused on transforming it back to get v1.

4. ### diazona

2,155
I'm not sure if I'm reading your work properly - could you type it out rather than including an image? In any case, it seems like there are some variables $v_i$ and $v_f$, which I'm not clear on the meanings of, so what you're doing looks incorrect.