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Relativistic Force and Momentum

  1. Oct 5, 2005 #1
    I've been having a lot of trouble with this problem. There's definitely something I'm missing and it most likely has to do with the force.

    "A mass [tex]m[/tex] is thrown from the origin at [tex]t = 0[/tex] with initial three-momentum [tex]p_0[/tex] in the y direction. If it is subject to a constant force [tex]F_0[/tex] in the x direction, find its velocity [tex]v[/tex] as a function of [tex]t[/tex] and by integrating [tex]v[/tex], find its trajectory. You will need to integrate functions such as [tex]t/\sqrt{a + bt^2}[/tex] and [tex]1/\sqrt{a + bt^2}[/tex].

    In addition, check that in the non-relativistic limit, ([tex]c \rightarrow \infty[/tex]), [tex]x(t)[/tex] is what you expect for motion in a constant field and check that [tex]y(t)[/tex] is what you expect for motion in a constant field when the force is orthogonal to the y direction.

    HINT: You will need the Taylor expansion of the functions [tex]\sqrt{1 + x}[/tex] and [tex]\ln(1 + x)[/tex]."

    The 2nd part seems easy, but I'm simply not sure how to find [tex]x(t)[/tex] or [tex]y(t)[/tex] in the first place.

    My work so far:
    [tex]\gamma = 1/\sqrt(1 - v^2/c^2)\\
    \gamma_v_0 = 1/\sqrt(1 - {v_0}^2/c^2)[/tex]
    [tex]p_0 = (0, p_0, 0) = m*\gamma_v0*(0, v_0, 0)[/tex]
    [tex]F_0 = (F_0, 0, 0)[/tex]
    [tex]F = dP/dt, \mbox{so } P - p_0 = F*t[/tex]
    [tex]P = F_0*t + p_0 = m*\gamma*v[/tex]
    [tex]v = (v_x, v_y, v_z), v_z = 0[/tex]
    [tex]m*\gamma*v_x = F_0*t[/tex]
    [tex]m*\gamma*v_y = p_0 = m*\gamma_v0*v_0[/tex]

    I'm not exactly sure how to proceed from here since I can't really isolate [tex]v_x[/tex] or [tex]v_y[/tex] because the gamma term contains only the magnitude of [tex]v[/tex]. Should I use [tex]\|v\| = \sqrt{v_x^2 + v_y^2}[/tex] and then work through some really terrible algebra? Or is this even the right way to approach this problem?

    Edit: I've actually done the terrible alegbra using [tex]\|v\| = \sqrt{v_x^2 + v_y^2}[/tex], but the equations I end up with are ridiculous. Can anyone suggest a different method?
    Last edited: Oct 6, 2005
  2. jcsd
  3. Oct 12, 2005 #2


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    Staff Emeritus
    Science Advisor

    The 4-momentum of the particle, which I'll call P, is supposed to be a 4-vector.

    Write down the four components of P:

    E (the energy)
    Px (the momentum in the x direction)
    Py (the momentum in the y direction)
    Pz (the momentum in the z direction, which is very easy, it's zero)

    You have something that you call P, but it doesn't appear to be a 4-vector.
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