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Variable mass dust particle

  1. Apr 19, 2014 #1
    1. The problem statement, all variables and given/known data

    At time t=0 a dust particle of mass m_0 starts to fall from rest through a cloud. Its mass grows exponentially with the distance fallen, so that after falling through a distance x its mass is m_0exp[αx] where α is constant. Show that at time t the velocity of the particle is given by:

    v=sqrt(g/α)tanh(t(sqrt(αg))

    2. Relevant equations

    Variable mass equation:

    mg= mv'+vm'

    3. The attempt at a solution

    Using the variable mass equation i've fiddled around with it and gotten out a differential equation in terms of v, which is:

    dv/dt=-αv^2+g

    Which after filling in the answer im looking for is the correct equation. However solving it properly by hand seems harder as its a non-linear ODE. I was just wondering is there any other way of going about this problem or if not, can anyone give me a tip on how to solve the ODE properly.

    Thank you for your time
     
    Last edited: Apr 19, 2014
  2. jcsd
  3. Apr 19, 2014 #2

    TSny

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    Hello, Matt_993. Welcome to PF!

    Did you mistype this? What does the symbol y stand for?

    Once you get the equation written in terms of just 2 variables, you might try the technique of separation of variables.
     
  4. Apr 19, 2014 #3
    Yea the y should've been a v, I've changed it. But I can't seem to be able to separate the variables in the equation because of g, is there another way?
     
  5. Apr 19, 2014 #4

    TSny

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    Divide both sides of the equation by the entire right hand side.
     
  6. Apr 19, 2014 #5
    How very stupid of me, can't believe I missed that.

    That works, thanks for the help :)
     
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