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Force dependent on velocity of particle

  1. Aug 31, 2009 #1
    1. The problem statement, all variables and given/known data

    A particle of mass m moves through a medium that resists its motions with a force of magnitude

    [tex]-mk(v^2+av)[/tex]

    where k and a are positive constants. If no other force acts, and the particle has an initial velocity v0, find the distance traveled after a time t.

    Show that the particle comes to rest as [tex]t \to \infty[/tex]

    2. Relevant equations

    [tex]F=m\frac{dv}{dt}[/tex]

    3. The attempt at a solution

    EOM: [tex]-k(v^2 + av) = \frac{dv}{dt}[/tex]

    [tex]dt=\frac{dv}{-k(v^2+av)}[/tex]

    [tex]\int \!dt=-\frac{1}{k} \int \! \frac{dv}{(v^2+av)}[/tex]


    ...Integrate in Mathematica...

    [tex]t-t_0 = \frac{Ln(a+v)-Ln(v)}{ak}[/tex]

    [tex]Exp(atk)=\frac{a+v}{v}[/tex]

    [tex]v(Exp(atk)-1)=a[/tex]

    [tex]v(t)=\frac{a}{Exp(atk)-1}[/tex]

    Set v = v0 at time t=0...

    [tex]v(0) = v_0 = \frac{a}{Exp(0)-1} = \frac{a}{0}[/tex]

    But this is not defined!

    Did I make a mistake? How do I set v = v0 if I get infinity?

    Thank you for your time and help.
     
  2. jcsd
  3. Sep 1, 2009 #2

    kuruman

    User Avatar
    Homework Helper
    Gold Member

    The problem is with your limits of integration. The left side (time) goes from 0 to t. That's fine. The right side must go from v0 (which is the velocity that matches time t = 0) to v (which is the velocity that matches time t).
     
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