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Modeling Quadratic Air Resistance in 2-D using Mathematica

  1. Apr 18, 2015 #1
    1. The problem statement, all variables and given/known data

    I am working on a little project in which I analyze a video I found online and try to determine if it is real of a hoax. The video I am analyzing includes a man throwing a football off of a football stadium (at the very top) and making it into a basketball hoop at field level. I have looked up everything I believe I need to try to set up the problem, and here is what I have:

    1. The football will be modeled as a Prolate spheroid with an elliptical cross section of area pi*a*b where a and b are radii (a>b). I have determined that a = 0.2794m and b=0.0859m, making the cross sectional area about 0.078m2.
    2. As I am modeling this using quadratic air resistance, given my fquad = F - cv2, where c is determined by the density of the air (which is about 1.225 kg/m3 and the cross-sectional area. That is, with these values, c = 0.09559 kg/m.
    3. the mass of the football is about 14 ounces, which is about 0.397 kg.
    4. I am only considering the x and y equations of motion, as I am assuming the football doesn't move much in or out of the page, just up and down in y and forwards in x.
    5. Then, the equations of motion are:
    mx'' = mv' = -c*sqrt(vx2+vy2)*vx.
    my'' = my' = -mg-c*sqrt(vx2+vy2)*vy.

    Initial Conditions: vx(t=0) = 25 mph = 11.176 m/s. vy(t=0) = 0. x(t=0) = y(t=0) = 0.

    I've never used mathematica before, but the mechanics book I am using (Taylor's Classical Mechanics) suggests that the NDSolve

    2. Relevant equations


    3. The attempt at a solution

    As I've said, I've never used Mathematica, but using a few guides, my code (which is wrong) to start, is:

    NDSolve[{x''[y, t] == -c/m * sqrt (x'[t]^2 + y'[t]^2) x'[t],
    y''[x, t] == -g - c/m * sqrt (x'[t]^2 + y'[t]^2) y'[t] , x[0] == 0,
    y[0] == 0, x'[0] == 11.176, y'[0] == 0}, x, y, {t, 0, 10}]

    with variables defined:

    m = 0.396893;
    c = 0.078;
    g = 9.8;

    Trying to run this, I get the error: "The length of the derivative operator Derivative[2] in x''[y,t] is not the same as the number of arguments.

    I am extremely lost as to how to solve these differential equations with a set of initial conditions.

    Any help is greatly appreciated. Thank you.
     
  2. jcsd
  3. Apr 18, 2015 #2
    You are working with parametric equations each as a function of time, but you have defined the second derivative as a function of two variables; it should just read x''[t]==.... I also think you need curly brackets around the x-y pair before you give the time interval (i.e. y'[0]==0, {x,y}, {t,0,2}]

    Other than that it looks good
     
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