# Collisions in Center-of-Mass coordinates

by Xyius
Tags: centerofmass, collisions, coordinates
 P: 417 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? :(??
HW Helper
P: 2,156
For starters, this equation is slightly wrong:
 Quote by Xyius $$R=\frac{m_1v_1+m_2v_2}{m_1+m_2}$$
Also,
 Quote by Xyius 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?
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.
 P: 417 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.
HW Helper
P: 2,156

## Collisions in Center-of-Mass coordinates

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.

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