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Differential equation for air resistance

  1. Sep 11, 2016 #1
    1. The problem statement, all variables and given/known data
    Solve the differential equation ##\displaystyle Cv^2 - mg = m\frac{d^2 y}{dt^2}##

    2. Relevant equations


    3. The attempt at a solution
    The problem is nonlinear, so we need to use unconventional methods. Specifically, if we can express the derivative of y with respect to v, then we might be able to integrate in order to find y.

    So ##\displaystyle \frac{dy}{dv} = \frac{dy}{dt}\frac{dt}{dv} = v \frac{dt}{dv} = \frac{v}{\frac{dv}{dt}}##

    But ##\displaystyle \frac{dv}{dt}## is given by ##\displaystyle \frac{C}{m}v^2 - g##, so
    ##\displaystyle \frac{dy}{dv} = \frac{mv}{Cv^2 - mg}##

    If we solve this, we get ##\displaystyle y = \frac{m}{2c} \ln{|1 - \frac{Cv^2}{mg}|}## where ##V_0 = 0##.

    Is this the correct solution?
     
  2. jcsd
  3. Sep 11, 2016 #2

    Ray Vickson

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    Check this solution by taking the derivative and seeing if the differential equation is satisfied. That is something you should always do, whenever it is possible.
     
  4. Sep 11, 2016 #3
    Actually, it does satisfy the original equation! So is my solution the correct one?
     
  5. Sep 11, 2016 #4
    bump. I need a definitive answer
     
  6. Sep 12, 2016 #5

    Math_QED

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    If it satisfies the equation, of course it is.
     
  7. Sep 12, 2016 #6

    Ray Vickson

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    Cannot give you one until you say what YOU regard as a solution. I would personally regard a formula such as ##v = f(t)## or ##t =h(v)## or ##y = F(t)## or ##t = H(y)## as a solution, so that if I were given ##t## I could compute ##v## and/or ##y##. So I myself would not say you were finished, but maybe your instructor has a different opinion.

    However, your relationship between ##y## and ##v## MUST BE correct if it satisfies the DE.
     
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