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Horizontal Motion with Quadratic Drag

  1. Mar 15, 2010 #1
    1. The problem statement, all variables and given/known data

    Consider an object that is coasting horizontally (positive x direction) subject to a drag force f = -bv - cv^2 . Write down Newton's second law for this object and solve for v by separating variables. Sketc the behaviour of v as a function of t. Explain the time dependence for t large. (Which force term is dominant when t is large?)

    2. Relevant equations

    3. The attempt at a solution

    I start with Newton's second law:

    m(dv/dt) = -bv -cv^2

    m(dv/dt) = -v(cv + b)

    (-m/v)(dv/dt) -cv = b

    dv/v + (cv/m)dt = (-b/m)dt

    Then... do I integrate with respect to v for the dv terms and t for the dt terms? I can't figure out what this will give me with the second term on the right...
  2. jcsd
  3. Mar 15, 2010 #2
    You need all the 'v' terms together with 'dv'. But you subtracted 'cv' in line 3 which is a mistake. Instead divide the whole equation in line 2 by 'v(cv+b)'. Then solve for it.
  4. Mar 15, 2010 #3
    I get to the point where all my v terms are on one side so that I have:

    (m dv)/[v(cv+b)] = -dt

    From here when I try and integrate with respect to v on the left and t on the right, I'm doing a definite integral setting the limits between v and v0 and t and t0 so I get:

    m[ln(cv^2 + bv) - ln(cv0^2 - bv0^2)] = -t

    ln(cv^2 + bv) = -t/m + ln(cv0^2 - bv0^2)

    Then if I take the exponential of both sides I still end up in a situation where I can't solve for v because there will be v on both sides... i.e

    cv^2 + bv = e^(-t/m) + cv0^2 - bv0^2

    How can I solve this for v?
  5. Mar 15, 2010 #4
    You need to check your integration, it is incorrect.
  6. Mar 16, 2010 #5
    How exactly do I integrate m(dv)/[v(cv+b)] ? The way I do it I consistently get natural log expressions which I am guessing is wrong because the function should tend towards 0 when I plot it...
  7. Mar 16, 2010 #6


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    Gold Member


    To integrate your expression properly, try completing the square on the denominator and then making an appropriate substitution.
  8. Mar 16, 2010 #7
    Another trick is to break the fraction up into to fractions that look like this:

    [tex]\frac{1}{v(cv+b)} = \frac{A}{v} + \frac{B}{cv+b}[/tex]

    Then solve for A and B. Integrating the two fractions separately will be much easier.
  9. Mar 16, 2010 #8
    Ah that clears things up. I see where I went wrong there, thanks for your help!
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