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Time of motion, natural logarithm help

  1. Nov 24, 2009 #1
    This equation specifies displacement x in terms of motion time t (starting from rest).

    [tex]x = \tau g\left(t + \tau e^{-t/\tau} -\tau\right)[/tex]

    where tau = m/b is the system time constant of a mass m suspended from a mechanical braking device with rectilinear damping constant b and g is standard gravity.

    Can anyone help me find t as a function of displacement x? I'm having trouble with the logarithm algebra. I'm new to the forum, so any tips are appreciated.
  2. jcsd
  3. Nov 24, 2009 #2


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    This is a trancendental equation for "t", you won't be able to find an exact solution.
  4. Nov 24, 2009 #3
    Thanks for that help.

    I'm trying to predict the time down a ramp of known length xL as shown in the diagram. I'd like a closed form approximation to write in a code statement. This will set the transient analysis time based on the input data inside a computer simulation.

    The differential equation solved for acceleration looks like this:

    [tex]\frac{dv}{dt} = \frac{mg sin\theta - bv}{m}[/tex]

    Any ideas appreciated.

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    Last edited: Nov 24, 2009
  5. Dec 2, 2009 #4
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