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Solve the differential equation F=F0+kv

  1. Sep 25, 2016 #1
    1. The problem statement, all variables and given/known data

    Find the velocity of v as a function of displacement x for a particle of mass m which starts from rest at x=0 and subject to the following force:

    [tex]F=F_0+kv[/tex]

    You could say mv = F0*t + kx, but the answer in the back of the book is an equation that is only in terms of x and v, not t. The answer in the back of the book involves ln.

    2. Relevant equations

    Maybe this:

    [tex]\ddot {x}= \frac{d \dot{x}}{dx}[/tex]

    3. The attempt at a solution

    [tex]m\ddot{x}=F_0 +k\dot{x}[/tex]
    [tex]m\dot{x} \frac{d\dot{x}}{dx}=F_0 +k\dot{x}[/tex]
    [tex]m\dot{x} d\dot{x} = F_0dx + k\dot{x} dx[/tex]
    [tex]\frac{1}{2}m\dot{x}^2 = F_0*x + ???[/tex]
     
    Last edited: Sep 25, 2016
  2. jcsd
  3. Sep 25, 2016 #2

    Orodruin

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    The second equation you wrote down in your attempted solution is separable, but in order to separate it the side with dx should not depend on ##\dot x##.
     
  4. Sep 27, 2016 #3

    rude man

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    Is this eq'n dimensionally correct?
     
  5. Sep 27, 2016 #4

    Orodruin

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    Well, he is not actually using that equation. He is using ##\ddot x = \dot x \, d\dot x/dx##.
     
  6. Sep 28, 2016 #5

    rude man

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    Assuming F0 and k are constants, how about a substitution of variables to reduce the 2nd order linear ODE into a 1st, then taking orodruin's hint to employ separation of variables to solve the new equation?
     
  7. Sep 28, 2016 #6

    rude man

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    Right, but he wrote it down & should learn to check for dimensional consistency, a powerful error-detecting tool that doesn't seem to be sufficiently emphasized in our classrooms.
     
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