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Ordinary differential equations - boat example

  1. Sep 21, 2010 #1
    URGENT! ordinary differential equations - boat example

    1. The problem statement, all variables and given/known data

    A boat of mass m is travelling with the velocity v0. At t=0 the power is shut off. Assuming water resistance proportional to v^n, where n is a constant and v is the instantaneous velocity, find v as a function of the distance travelled. (Note that you need to consider the two cases).

    Please help asap!
    Thanks

    2. Relevant equations

    force balance equations

    3. The attempt at a solution

    I've gotten my ODE to be: m*dv/dt = -kv^n with the initial condition v0 at t = 0.
    It says find v as a function of the distance travelled.
    So i went on to say dv/dt= (dv/dx)*(dx/dt)= v*(dv/dx) so now x is the independent variable not t.

    so, m*v*(dv/dx) = -kv^n
    v*(dv/dx)= -(k/m)*v^n
    v*(dv/dx)= -av^n where a =k/m
    v dv = -av^n dx
    integral (v/(v^n)) dv = integral (-a) dx
    integral (v^(1-n)) dv= -ax + c
    (v^(2-n))/(2-n) = -ax + c (don't know if the integral on the LHS is correct?)

    but x=o at t=0 so v(t=0)=v(x=0)=v0

    so c= (v0^(2-n))/(2-n)
    therefore,
    (v^(2-n))/(2-n) = -ax + (v0^(2-n))/(2-n)
    v^(2-n) = -(2-n)ax + v0^(2-n)
    v^(2-n) = (n-2)ax + v0^(2-n)
    therefore, v= [(n-2)ax + v0^(2-n)]^(1/(2-n))

    but a =k/m
    so v(x) = [(n-2)kx/m + v0^(2-n)]^(1/(2-n))

    does this seem right at all because it seems a little messy?? Also it says to note that you need to consider the two cases. What are the two cases as I've only considered one???
     
  2. jcsd
  3. Sep 21, 2010 #2

    lanedance

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    Homework Helper

    Re: URGENT! ordinary differential equations - boat example

    haven't been through the working, but few standrad checks are
    - see if it behaves as you expect, ie monotonically decreasing
    - substitute back into the equation and check it satsifies the orginal DE

    for the 2 cases have a look at the integration step, and consider n = 2
     
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