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Is distance traveled proportional to relativistic momentum?

  1. Apr 24, 2016 #1
    1. The problem statement, all variables and given/known data
    Hi I've been modelling a particle travelling in a particle detector that has a momentum vector Px, Py, Pz which we've conveniently been using Pperpendicular (i.e. in the xy plane) and Pz.

    I can calculate the distance traveled in the xy plane and I need to calculate the distance traveled in the z direction.

    2. Relevant equations

    Knowing that

    [tex]\vec{p}=\frac{m\vec{v}}{\sqrt{1-\vec{v}^2/c^2}}[/tex]

    3. The attempt at a solution

    My guess was Momentum is proportional to the distance traveled which I can convince myself of in the case of classical momentum but knowing the momentum of this particle is of the order GeV/c I'm unsure whether I can actually do this? My assumption comes from being able to do this,

    [tex]\vec{d}=\frac{m \Delta t}{\sqrt{1-\vec{v}^2/c^2}}\vec{v}[/tex]

    where I assumed there was a proportionality constant k such that

    [tex]p_\perp = k d_\perp[/tex]
    [tex]p_z = k d_z[/tex]

    Therefore

    [tex]d_z=\frac{p_z}{p_\perp}d_\perp[/tex]

    But I'm unsure if I can just throw in the gamma factor and the time interval into that constant k such that these equations are valid. Would this be correct or would I have to do it another way?
     
  2. jcsd
  3. Apr 25, 2016 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    The direction of motion is the direction of the momentum vector. Everything else follows from basic geometry.
     
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