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Homework Help: Model for single lane of traffic

  1. Jul 11, 2011 #1
    A model for a single lane of traffic is given by the following pde

    p dv/dx + v dp/dx + dp/dt = 0

    Where:

    v = kx/p



    Show that

    dp/dt = -k



    Here is my attempt

    v = kx/p

    dv/dx = k/p

    p= kx/v

    dp/dx = k/v

    Substituting into original pde:

    p (k/p) + v(k/v) + dp/dt=0

    Clearly made an error, any help would be great
     
  2. jcsd
  3. Jul 11, 2011 #2
    p dv/dx + v dp/dx + dp/dt = 0

    Where:

    v = kx/p

    Show that

    dp/dt = -k

    p dv/dx + v dp/dx + dp/dt = 0 can be written as

    d(pv)/dx + dp/dt = 0 ----------- 1

    v = kx/p => pv = kx -> put in 1

    we get dp/dt = - k

    method you used is wrong because you cant partially differentiate v w.r.t x in an equation which contains p (also a function of x) -. while differentiating you treated p as a constant w.r.t to x..
     
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