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Physics type math problem

  1. Jul 18, 2007 #1
    Ok, theres a differential equation that is:

    dv/dt = 9.8 - (v/5), v(0) = 0

    to represent a falling object. So the solution ends up being

    v = 49(1 - e^t/5)

    and the equilibrium solution (terminal velocity) is v = 49.

    Now I have a problem that says "find the time that must elapse for the object to reach 98% of its limiting velocity"

    To do this, I am doing the following:

    terminal velocity * 98% = the equation...
    49 * 0.98 = 49(1 - e^t/5)

    div by 49...
    0.98 = 1 - e^t/5

    move 1 over and multiply both sides by -1...
    1 - 0.98 = e^t/5

    use ln...
    ln(1 - 0.98) = t/5

    mult by 5...

    5 * ln(1 - 0.98) = t

    so:
    t = -19.56


    now... in the book's solution, it states:
    T = 5 * ln(50) ~= 19.56

    So, where is this 50 coming from? And my answer looks to be right except for the sign, where did I go wrong?
     
  2. jcsd
  3. Jul 18, 2007 #2

    StatusX

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    Homework Helper

    It should be v = 49(1 - e^(-t/5)). Your solution has the velocity exponentially growing.
     
  4. Jul 18, 2007 #3
    thanks! I knew it would be something stupid that I missed. I still don't get where they got the "5 * ln(50)" from though...
     
  5. Jul 18, 2007 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    0.98 = 1 - e-t/5

    e-t/5= 1-0.98= 0.02= 2/100= 1/50
    That's where the "50" is from.

    Now -t/5= ln(1/50)= -ln(50) so t= 5 ln(50).
     
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