Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding a solution for Relativistic Acceleration

  1. Jun 8, 2012 #1
    I tried all my mathcad software such as Maple and I can't seem to find a solution to this time based differential equation.

    m}\left%20(%201%20-%20\frac{\left%20[%20f(t)%20\right%20]^{2}}{c^{2}}%20\right%20)^{\frac{3}{2}}.gif
     
  2. jcsd
  3. Jun 8, 2012 #2

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    Solving differential equations is often a matter of guessing and checking.
    This equation comes up in Rindler coordinates, so I actually know the solution.

    Let g = F/m.
    Switch to a new independent variable T that is related to t through
    t = c/g sinh(gT/c).

    Then try the solution:

    f = c tanh(gT/c)

    In terms of the original t, we can rewrite f as

    f = c (gt/c)/√(1 + (gt/c)2)

    or

    f = c (ct)/√((ct)2 + c2/g)
     
  4. Jun 8, 2012 #3

    Bill_K

    User Avatar
    Science Advisor

    Michio, It's important part of everyone's physics education to pick up some standard techniques for solving DEs, so you don't have to depend entirely on the math software to do it for you, which often does an imperfect job. And this equation is a very easy one to solve. Setting the constant parameters to one to simplify the discussion,

    df/dt = (1 - f2)3/2

    Separate the variables (t on one side, f on the other) and integrate immediately:

    t = ∫(1 - f2)-3/2 df

    To do the integral, the factor 1 - f2 suggests making a trig substitution. Let f = sin θ.

    t = ∫(cos θ)-3 cos θ dθ = ∫ sec2 θ dθ = tan θ

    So we have f = sin θ, t = tan θ, giving us an algebraic relationship and the solution:

    t = f/√(1 - f2) or f = t/√(1 + t2)
     
  5. Jun 8, 2012 #4

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    Interestingly, it works to try the substitution f = sin(θ), then you get t = tan(θ), but it also works to try the substitution f = tanh(θ), then you get t = sinh(θ). I like the latter better, because that gets you to the Rindler coordinates.
     
  6. Jun 8, 2012 #5

    Bill_K

    User Avatar
    Science Advisor

    sin θ = tanh χ
    cos θ = sech χ
    tan θ = sinh χ

    is known as the Gudermannian transformation and written θ = gd χ. See "Gudermannian Function" in Wikipedia.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding a solution for Relativistic Acceleration
Loading...