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Newtonian Mechanics - Particle in Motion with Air Resistance

  1. Jan 24, 2013 #1
    1. The problem statement, all variables and given/known data
    A particle of mass m slides down an inclined plane under the influence of gravity. If the motion is resisted by a force f = kmv^2, show that the time required to move a distance d after starting from rest is

    t = [arccosh(e^(kd))]/√(kgsin(θ)

    where θ is the angle of inclination of the plane.


    2. Relevant equations
    F = ma
    F_g = mgsin(θ)
    Resistance = -kmv^2
    Motion of particle => ma = mgsin(θ) - kmv^2


    3. The attempt at a solution
    My attempt was to set up the equation for the motion (ma = mgsin(θ) - kmv^2) and use differential equations to solve. After dividing by mass, I had:
    dv/dt = gsin(θ) - kv^2
    which I then divided by k, and substituted (g/k)sinθ for C^2 giving

    dv/kdt = C^2 - v^2

    After collecting terms and integrating I came to

    t = arctan(v/√((g/k)sinθ))/√(kgsinθ)

    I thought I was on the right track as I have the numerator correct, but I do not know where to go from here, or if this is even correct so far. Any help would be much appreciated. Also, I know this question has been asked before, but the answer given didn't make sense to me so it doesn't help.
     
  2. jcsd
  3. Jan 24, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    Try writing the acceleration as ##a = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v \frac{dv}{dx}## where ##x## is distance along plane. Separate variables and integrate to find ##v## as a function of ##x##. Then, writing ##v = \frac{dx}{dt}## you can separate variables again and integrate to find ##t## as a function of ##x##.

    [EDIT: Actually, your way works too. However, I think there is a typo in your final expression for t. Did you mean to write arctanh rather than arctan? If you solve your (corrected) expression for ##v## and then let ##v = \frac{dx}{dt}## you can separate variables and integrate to find ##t## as a function of ##x##]
     
    Last edited: Jan 24, 2013
  4. Jan 25, 2013 #3
    Yes, it was supposed to be arctanh (and also denominator was correct not numerator). I just finished this, thank you very much for your assistance.
     
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