Collisions in Center-of-Mass coordinates

[Xyius]

Homework Statement

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.

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

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? :(??

 

[diazona]

For starters, this equation is slightly wrong:

$$R=\frac{m_1v_1+m_2v_2}{m_1+m_2}$$

Also,

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.

 

[Xyius]

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 can't find my work from last night.)
[PLAIN]http://img511.imageshack.us/img511/9606/pfq.gif
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.

 

[diazona]

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.

 

[paranormal]

I would first clarify what is meant by "center of mass coordinate system." This is not a commonly used term in physics. It could refer to either the center of mass frame, where the total momentum of the system is zero, or the center of mass coordinate system, where the origin is at the center of mass of the system.

Assuming that the problem is referring to the center of mass coordinate system, the equations provided seem to be correct for finding the coordinates of the center of mass (R) and the coordinates of each particle (r1 and r2) in this coordinate system. However, these equations do not directly relate to the velocities of the particles.

To solve for the final velocities in this coordinate system, we can use the conservation of momentum and conservation of energy equations. Since the collision is elastic, both momentum and energy are conserved.

Conservation of momentum:
m1v1i + m2v2i = m1v1f + m2v2f

Conservation of energy:
1/2m1v1i^2 + 1/2m2v2i^2 = 1/2m1v1f^2 + 1/2m2v2f^2

Using the equations provided, we can rewrite these equations in terms of the coordinates in the center of mass coordinate system:

Conservation of momentum:
m1(r1i - R) + m2(r2i - R) = m1(r1f - R) + m2(r2f - R)

Conservation of energy:
1/2m1(r1i - R)^2 + 1/2m2(r2i - R)^2 = 1/2m1(r1f - R)^2 + 1/2m2(r2f - R)^2

From here, we can solve for the final velocities (v1f and v2f) using algebraic manipulation. This approach may be more complicated than solving in the lab frame, but it can be a useful exercise in understanding how to use different coordinate systems in physics problems.

If the problem is referring to the center of mass frame, where the total momentum of the system is zero, then the equations provided are not relevant. In this frame, we can simply use conservation of energy to solve for the final velocities:

1/2m1v1i^2 + 1/2