1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Nonlinear Retarding force

  1. Feb 15, 2007 #1
    1. The problem statement, all variables and given/known data
    A particle moves in a medium under the influence of a retarding force equal to [tex]mk(v^3 + a^2 v)[/tex], where k and a are constants. Show that for any value of the initial speed the particle will never move a distance greater than pi/2ka and that the particle comes to rest only for [tex]t \rightarrow \infty[/tex]

    2. Relevant equations
    Legrangian seems overkill, so I used Newton's.

    3. The attempt at a solution
    [tex]\frac{dv}{dt} = k (v^3 + a^2 v)[/tex]

    then seperate the ODE and integrate

    [tex] \frac{lnv}{a^2} - \frac{ln(a^2 + v^2)}{2a^2} = kt + C[/tex]

    multiply by 2a^2

    [tex] 2lnv - ln(a^2 + v^2) = 2kta^2 + C[/tex]

    use log properties and combine the natural logs

    [tex]ln ( \frac{v^2}{a^2 + v^2} ) = 2kta^2 + C[/tex]

    exponentiate and carry constant down

    [tex]\frac{v^2}{a^2 + v^2} = Ce^{2kta^2}[/tex]

    add and subtract a^2 in numerator to simplify, and then subtract the one

    [tex] \frac{-a^2}{v^2+a^2} = Ce^{2kta^2}[/tex]

    use algebra to isolate v

    [tex] v^2 = a^2 (1 - \frac{1}{1-Ce^{2kta^2}})[/tex]

    Now I am at the point where I can solve for the integration constant, but I don't see it giving me anything close to revealing a max distance of pi/2ka. Also, in my solution as t approaches infinity v = a, and not zero. Maybe I made an algebraic mistake?
    Last edited: Feb 16, 2007
  2. jcsd
  3. Feb 16, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    The error is clearly in the integration of the ODE.
  4. Feb 16, 2007 #3


    User Avatar
    Science Advisor

    Presumably, though you didn't show it, you used partial fractions to write
    [tex]\frac{dx}{k(v^3+ a^2v)}= \frac{A}{v}+ \frac{Bv+C}{v^2+ a^2}[/tex]

    How did you integrate [tex]\frac{Bv+C}{v^2+ a^2}[/tex]?
  5. Feb 16, 2007 #4
    I used mathematica, though partial fractions is the obviously the way to do it by hand.
  6. Feb 16, 2007 #5


    User Avatar
    Science Advisor
    Homework Helper

    The first error is integrating the wrong ODE.

    [tex]\frac{dv}{dt} = - k (v^3 + a^2 v)[/tex] would be better, since the form of the answer implies k is positive.
  7. Feb 16, 2007 #6
    Yep, that was actually the problem. I went through it again and figured out what I did wrong. You end up with v = +/- a/sqrt(-1-Ce^(2kta^2).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook