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Transcendental Retarding Force

  1. Jan 27, 2008 #1
    1. The problem statement, all variables and given/known data
    A boat with initial speed v[o] is launched, and experiences a retarding force of F = -ae^Bv, where a=alpha=constant and b=beta=constant

    Find v(t)
    Find Time and Distance for the boat to stop

    2. Relevant equations

    3. The attempt at a solution
    the second part for tmie and distance, I am attempting to solve using mathcad with a perturbative approach incrementing vt o find dv/dt, reinserting v-dv/dt until dv/dt approaches closely to zero. This makes sense, however, only with known values for alpha and beta. I can't really think of a way to formulate v(t).
  2. jcsd
  3. Jan 27, 2008 #2
    I retried the problem, and was able to do some integration by treating it as a differentials and deriving a seperable ODE. However, my answers seem very odd. Here's what I did.

    m*dv/dt = -[tex]\alpha[/tex]*e^[tex]\beta[/tex]*v

    dv[tex]/[/tex]e^[tex]\beta[/tex]*v = -[tex]\alpha[/tex]*dt[tex]/[/tex]m

    letting u = e^-Bv
    du = -B*e^-Bv

    [tex]\int[/tex]1/-B * du = [tex]\int[/tex]-A/m dt

    going through all the integration...

    we get


    with v(0) = v0

    constant = (m/a*B)*e^-Bv0

    so... let && = A*B/m

    v(t) = ln(&&*t + e^(-B*v0))/-b

    which is weird to me, because the higher we set the initial velocity, the slower the speed is after a set t. Someone help!
  4. Jan 27, 2008 #3
    It isn't wierd, because [tex]\beta[/tex] must be negative, or else the boat would move with negative velocity! :smile:
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