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Quadratic drag equation by partial fractions

  1. Sep 23, 2004 #1
    hi, i am trying to show that

    dv/(1- (v^2/v_ter^2)) = g*dt which after integrating is

    v=v_ter*tanh(g*t/v_ter) (motion with quadratic drag) can also be obtained by using natural logs.

    so far i have this:

    letting u = v/v_ter

    i can use partial fractions to get

    du/(1-u^2) = 1/2 *(1/(1+u) + 1/(1-u)) *du

    then using my limits of integration as 0 to u , i get

    1/2* [ln(1+u) + ln(1-u)] = g*dt

    then integrating the other side i get as my final equation

    1/2 *[ ln(1+v/v_ter) + ln(1-v/v_ter)] = g*t

    but when i tried to plug numbers into each equation the numbers didnt match.

    does anyone know what i may have done wrong?
  2. jcsd
  3. Sep 23, 2004 #2

    Dr Transport

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    Science Advisor
    Gold Member

    What is [tex] \int \frac{1}{1- \frac{v^{2}}{v_{0}}} dv [/tex], look it up in an integral table, you shouldn't have to resort to partial fractions, I suspect that it will be arctanh().........
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