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Solve relativistic differential force equation for velocity

  1. Feb 6, 2014 #1
    1. The problem statement, all variables and given/known data
    I proved that a relativistic 1D force is
    F = [itex]\gamma[/itex]3*m*dVx/dt = m * dVx/dt * 1/ (1 - (v/c)2)3/2

    Then, "This is a separable differential equation that can be solved using a trig
    substitution. Use this (or some other technique that works) to show that the velocity is given by
    v(t) = [itex]\frac{a*t}{\sqrt{1 + \frac{at}{c}2}}[/itex]

    2. Relevant equations

    a = [itex]\frac{dVx}{dt}[/itex] * [itex]\frac{1}{(1-\frac{v}{c}3/2}[/itex]
    β = [itex]\frac{v}{c}[/itex] = sinΘ
    cosΘ = [itex]\sqrt{1 - β2}[/itex]

    3. The attempt at a solution
    dβ = cosθdθ
    a(t) = [itex]\frac{c*cosθdθ}{cos2θ}[/itex] = [itex]\frac{cdθ}{cosθ}[/itex]
    I don't really know what to do from here to arive at the answer
     
    Last edited: Feb 6, 2014
  2. jcsd
  3. Feb 6, 2014 #2

    LCKurtz

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    I think I fixed the Tex in your first equation. Don't use the sup and /sup tags in a tex expression. If you want an exponent of 3/2 just use ^{3/2} in the Tex. You also don't need the * for multiplication. You can use \cdot if you really want a multiplication sign. I will leave it to you to fix the rest if you desire. Also you can preview your posts before posting to see if the Tex is working.
     
    Last edited: Feb 6, 2014
  4. Feb 6, 2014 #3

    vela

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    I take it you're using ##V_x## and ##v## to represent the same thing. Don't do that. Pick one. It also looks like you're supposed to assume the acceleration ##a## is constant.

    After the substitution, you have
    $$a\,dt = \frac{c \cos\theta \, d\theta}{(1-\sin^2\theta)^{3/2}}.$$ If you simplify that, you don't get what you got. Your last line, in particular, doesn't make sense. You shouldn't have a ##d\theta## all alone in the equation.
     
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