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Position as a function of speed

  1. Nov 21, 2009 #1
    1. The problem statement, all variables and given/known data
    There's an object with mass [tex]m[/tex] in movement in the horizontal axes. There's a force [tex]\textbf{P}[/tex] of constant power acting on the object. Another force is the air drag which has the magnitude of [tex]\beta m v^{2}[/tex]. I need to find the position [tex]x[/tex] as a function of the speed [tex]v[/tex].


    2. Relevant equations
    [tex]\textbf{P} = \vec{F} \cdot \vec{v} = Fv[/tex] because the vectors are parallel

    [tex]\Rightarrow F = \frac{\textbf{P}}{v}[/tex]

    [tex]\left|\vec{f}\right| = \beta m v^{2}[/tex]

    [tex]F = ma[/tex]


    3. The attempt at a solution

    With these equation I plug them in [tex]F = ma[/tex] and I get [tex]\frac{\textbf{P}}{mv}-\beta v^{2}=\frac{dv}{dt}[/tex] then by multiplying by [tex]\frac{dx}{dx}[/tex] I got [tex]\frac{\textbf{P}}{mv}-\beta v^{2}=v\frac{dv}{dx}[/tex].

    Then I don't know how to solve this. Or perhaps there's an easier way to this problem?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 21, 2009 #2

    rl.bhat

    User Avatar
    Homework Helper

    P/mv - βv^2 = v*dv/dx.
    Multiply by v on both the side. You get
    P/m -β*v^3 = v^2*dv/dx.
    So
    dx = v^2*dv/( P/m -β*v^3 )
    Now find the integration to find x.
     
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