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Special Relativity -- Elastic Particle Collision Algebra

  1. Dec 9, 2015 #1
    1. The problem statement, all variables and given/known data

    Consider the following head-on elastic collision. Particle 1 has rest mass 2mo, and particle 2 has rest mass mo. Before the collision, particle 1 movies toward particle 2, which is initially at rest, with speed u (= 0.600c ). After the collision each particle moves in the forward direction with speeds of u1 and u2, respectively.

    a) Apply the laws of conservation of total energy (or, equivalently, of relativistic mass) and of relativistic momentum to this collision and solve the resulting equation to find u1 and u2 (the resulting speeds of the two particles).

    2. Relevant equations

    Well, I know the mass is conserved in elastic collisions.

    ##m_1\gamma_1 + m_2\gamma_2 = m_1\gamma_{1}^{'} + m_2\gamma_{2}^{'} ##

    where gamma is the Lorrentz factor. I also know energy is conserved, (Ei = Ef, where E = Moc2 + (M - Mo)c2 where Mois the rest mass and M is the relativistic mass), but that basically reduces to the same thing as the mass equation.

    I also know the equation for the conservation of momentum,

    ##p_1 + p_2 = p_{1}^{'} + p_{2}^{'} ##

    where

    ##p = M_o\gamma u ##

    Mo is the rest mass and u is the speed.

    3. The attempt at a solution

    So first I plugged in the values of the masses and speeds into the mass-conservation equation and got

    ##2m_o\gamma_1 + m_o\gamma_2 = 2m_o\gamma_1^{'} + m_o\gamma_2^{'} ##

    ##3.5 - 2\gamma_1^{'} = \gamma_2^{'} ##

    but when I got around to plugging the numbers into the momentum equation, I got

    ## 2m_o\gamma_1u_1 = 2m_o\gamma_{1}^{'}u_{1}^{'} + m_o\gamma_{2}^{'}u_{2}^{'} ##

    ## 2\gamma_1u_1 = 2\gamma_{1}^{'}u_{1}^{'} + \gamma_{2}^{'}u_{2}^{'} ##

    The problem arises when, no matter what I substitute in for any of the Lorrentz factors (gammas), I always get two unkonws in the equation, u1 prime and u2 prime. I can't think of a way to isolate just for one. Any help?
     
  2. jcsd
  3. Dec 10, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    The concept of relativistic mass is not used any more in physics.

    You have two equations, it is fine to have two unknowns.
    ##\gamma_1'## and ##u_1'## have a known relation, you can use one of them to express the other (same for 2 of course).
     
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