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Fluid Friction Question

  1. Sep 23, 2011 #1
    1. The problem statement, all variables and given/known data

    I'm trying to compare the velocities at 5 meters of three projectiles subject to air resistance: the first with an initial velocity of 121.632 m/s, the second with an initial velocity of 136.8m/s, and the third at 182.442 m/s.

    All three projectiles have a mass of 2.0x10^-4 kg (m), a cross sectional area of 2.81x10^-5 m^2 (A), and drag coefficient of .47 (Cd). Density of air is assumed to be 1.204 kg/m^3.

    2. Relevant equations

    The equation I have been primarily been using is the drag equation Fa = 0.5Dv^2CdA

    My initial plan was to use F = ma and Vf^2 = Vi^2 + 2a[itex]\Delta[/itex]x. However, I realized after doing these calculations that Fa changes with respect to velocity.

    3. The attempt at a solution
    This is my attempt to the solution of finding the velocity after 5 meters for 121.632 m/s.

    Fa = -(0.5)(1.204)(121.632 m/s)^2(.47)(2.81E-5) = -1.2E-1

    Which I then realized that only applied initially at launch.

    So I tried taking the derivative with respect to time.

    dFa = (.47)(2.81E-5)(dv/dt)

    I am fairly stuck at the moment. Where do I go from now? If there is not enough information, what information do I need and what hints would you give to experimentally gain this information?
     
  2. jcsd
  3. Sep 23, 2011 #2
    Here is my suggestion.

    Using the drag equation, Newton's second law and the fact that [itex]a = \frac{d v}{d t}[/itex] (where [itex]a[/itex] is the acceleration) you can find a differential equation for [itex]v[/itex]. Solving that, you can find [itex]v[/itex] at any time [itex]t[/itex]. Then, using the fact that [itex]v = \frac{d x}{d t}[/itex], you can find another differential equation for [itex]x[/itex]. Solving this, you can find the time at [itex]x = 5m[/itex]. Then you can plug in that time to the equation for [itex]v[/itex].

    Let me know if it works.
     
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