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Retarding Force Problem

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  1. Sep 27, 2014 #1
    The Problem:
    A particle moves in a medium influenced by a retarding force mk(v^3+va^2), where k and a are constants.
    Show that
    for any initial velocity the particle will never move more than (pi)/(2ka)
    and
    that it comes to rest only for t -> infinity

    Attempt to solution:

    I know I have to integrate the retarding force [F=-mk(v^3+va^2))] so that I can solve for v. Later, I have to solve for the constant C of the indefinite integral. Afterwards, I have to integrate to find the position.

    To show the first part, I believe I have to find the value of the position as t-> infinity.
    To show the second part, I believe I have to find the value of the velocity as t->infinity.

    My problem: I cannot solve for v

    I have solved the integral of the retarding force using trig substitution (v=acot(theta)) (dv=-acsc^2(theta)d(theta))
    My final answer is:

    t=(-1/ka^2)(ln(v/(sqrt(v^2+a^2))+C

    If I use the quadratic equation to solve for v, I get imaginary numbers....
    Am I making a mistake in my calculus or algebra?
    or
    Am I missing a physical concept?
     
  2. jcsd
  3. Sep 27, 2014 #2

    Simon Bridge

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    Draw a free body diagram for the particle in motion.
    Apply ##F=m\dot v##.
     
  4. Sep 28, 2014 #3
    After the free body diagram this is what I obtained.
    Since the only force acting on it is the retarding force, it equals: F= -mk(v^3+va^2) = m(dv/dt)
    This is a velocity dependent force. Therefore, I integrated the force as dt = -dv/(k(v^3+va^2)
    then, I used trig substitution for v = acot(x) and dv=-csc^2(x)
    And after integrating I obtain:

    t= (-1/ka2)(ln(v/(sqrt(v^2 +a^2)) + C

    Is this correct so far?
     
  5. Sep 28, 2014 #4

    haruspex

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    Have you checked that by differentiating to get the ODE?
     
  6. Sep 29, 2014 #5
    Yes, I checked. I realize also what was my problem. Thank you all
     
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