Collisions in Center-of-Mass coordinates

  1. 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.
    [tex]R=\frac{m_1v_1+m_2v_2}{m_1+m_2}[/tex]

    Coordinates from vector 2 to vector 1
    [tex]r=r_1-r_2[/tex]

    Inverse transformations
    [tex]r_1=R+\frac{m_2}{m_1+m_2}[/tex]
    [tex]r_2=R+\frac{m_1}{m_1+m_2}[/tex]

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

    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. jcsd
  3. diazona

    diazona 2,156
    Homework Helper

    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.
     
  4. 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.)
    [​IMG]
    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.
     
  5. diazona

    diazona 2,156
    Homework Helper

    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 [itex]v_i[/itex] and [itex]v_f[/itex], which I'm not clear on the meanings of, so what you're doing looks incorrect.
     
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