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Expressing velocity as a function of the distance (not time elapsed)

  1. Jul 31, 2005 #1
    Hi, I encountered the great difficulty solving the following question.
    (The reason why its is hard is because I have to express the velocity as a function of the distance of the object has traveled in stead of as a function of the time elapsed.)

    Starting from rest at t=0, an 8 lb. object is pulled along a surface with a force that is equal to twice the distance (in feet) of the object from its starting point x=0. The coefficient of sliding friction is 1/4.
    The air resistance is numerically equal to one-eight the square of the velocity (in feet/second).
    What is the velocity formula of the object as a function of the distance of the object has traveled?

    So, weight = 8lb.
    [tex] m=\frac{8}{32}=\frac{1}{4}[/tex]
    [tex]Air Resistance = \frac{1}{8}v^2[/tex]
    [tex]Friction = \frac{-1}{4}8 = -2[/tex]
    Pulling force = 2x

    First I tried to write the velocity formula as a function of the time elapsed. The I got

    [tex]\frac{1}{4}\frac{dv}{dt}=2x - 2-\frac{1}{8}v^2[/tex]
    [tex]2\frac{dv}{dt}=16x - 16-v^2[/tex]
    Integrate, I had
    Applying the I.C., v(0)=0
    c = 0
    so, [tex]tan^-^1(0.25v)=\frac{-8xt}{0.004363}[/tex]
    I don't know how to solve this equation for v from here.

    Besides, the answer provided for this problem was [tex]v=\sqrt{16x-32+32e^-^x}[/tex]
    which gives me the idea that my approach is off and should be a way to obtain the desired formula without obtaining the formula as a function of time elapsed first...

    What is the best way to attack this problem?
  2. jcsd
  3. Jul 31, 2005 #2
    begin with your correct equation:

    [tex]2\frac{dv}{dt} \ = \ 16x - 16-v^2[/tex]

    rearrange terms:

    [tex]\frac{dv}{dt} + \frac{v^2}{2} \ = \ 8x - 8 [/tex]

    then use:

    [tex]\frac{dv}{dt} \ = \ \frac{dv}{dx} \cdot \frac{dx}{dt} \ = \ \frac{dv}{dx} \cdot v [/tex]

    so that placing this result into the diff eq:

    [tex]v\frac{dv}{dx} + \frac{v^2}{2} \ = \ 8x - 8 [/tex]


    [tex]2v\frac{dv}{dx} + v^2 \ = \ 16x - 16 [/tex]

    [tex]\frac{d(v^{2})}{dx} + v^2 \ = \ 16x - 16 [/tex]

    now let y=v2 and solve for y(x) using standard techniques.
    Last edited: Jul 31, 2005
  4. Jul 31, 2005 #3
    this is correct

  5. Jul 31, 2005 #4
    Thanks for helping me again geosonel. I got it!
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