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Homework Help: Velocity as a function of distance [v(x)]

  1. May 3, 2012 #1
    1. The problem statement, all variables and given/known data
    a body with mass M moves across a plane with friction

    friction constant:
    [itex]\mu = \lambda x^2[/itex]

    the body starts at x=0
    with velocity v0

    find at what x
    the body stops
    and what was the velocity half way there.

    2. Relevant equations

    [itex]v^2=v_0^2+2a\Delta x[/itex]

    3. The attempt at a solution

    [itex]F(x)=mg\mu = mg\lambda x^2[/itex]
    [itex]a(x)=g \lambda x^2[/itex]

    so in the equation [itex]v^2=v_0^2+2a\Delta x[/itex]
    I get
    [itex]v^2=v_0^2+2g\lambda x^3[/itex]

    the Question is, can I use this equation? the acceleration is not constant and this equation
    depend on the fact that [itex]x=v_0t+ \frac{a}{2}t^2[/itex]
    and [itex]v=v_0+at[/itex]
    (and it's not true for non-constant acceleration)

    if I cant, how can I integrate the acceleration?
    or how do I get v(x)?


    I used [itex] a= v\frac{dv}{dx}[/itex]

    [itex]\int_{v_0}^{v(x)}{vdv} = g\lambda \int_{0}^{x}{x^2}[/itex]

    [itex]\frac{1}{2} ( v(x)^2- v_0^2) =\frac{1}{3} g\lambda x^3[/itex]

    [itex]v(x)^2=v_0^2+\frac{2}{3}g\lambda x^3[/itex]

    does that seem right?
    Last edited: May 3, 2012
  2. jcsd
  3. May 3, 2012 #2


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    Gold Member

    I don't see anything wrong with your edited solution.
  4. May 4, 2012 #3


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    Homework Helper

    Does the friction increase speed?
    Remember that velocity, acceleration and force are all vectors. You need to use proper signs with them.

  5. May 4, 2012 #4
    ah, of course... it's with a minus :)
    on paper I actually did it with a minus. Thanks for pointing it out though!
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