*Mathematica* Files for AJP
Paper

Know_CL_Range_AJP This
*Mathematica *
file determines *C*_{D} using an
experimentally-determined range. The user also supplies *C*_{L}, which we
take to be zero for no-spin launches. We used this file to determine that
*C*_{D} = 0.2 for the 2D trajectory shown
in Figure 12 in the AJP paper. This file contains numerous
comments.

Sample_Excel_Trajectory_Analysis
This *Excel* file determines *C*_{D} and *C*_{L} using Camera 1 data. The idea with
this file is to use a computational tool simpler than
*Mathematica*. We fit our data using quadratics, which assumes a
*constant* net force. The ball does *not*, of course, have a
constant net force on it. During the time Camera 1 records, however, the
net force on the soccer ball does not change too much. By smoothing the
data, we can use equations (14) and (15) in our AJP paper to obtain estimates of
*C*_{D} and *C*_{L},
respectively. The *Mathematica* file below shows that the
*C*_{D} obtained in the *Excel* file is reasonable due to
the fact that the *C*_{L} obtained in
*Mathematica* is not terribly sentistive to our choice of
*C*_{D}.

Trajectory_Coefficients_Camera_1_KnowCD_AJP This *Mathematica* file determines *C*_{L} using Camera
1 data and a user-supplied *C*_{D}. The assumption is that
both *C*_{D} and *C*_{L} are constant during
the short time Camera 1 records the launch of the soccer ball. We set this
up so that the data are equally-spaced in time, and the data set is of the form
{*t*, *x*(*t*), *y*(*t*),
*z*(*t*)}. All *y* values are 0; the reason we
include *y* values is for future extensions to our work. This file
contains numerous comments. Click here for a the
data file we used to make the graph in the *Mathematica* file. We shift the origin to match
the measured launched position (shifting the orgin does *not*, of
course, change the physics!). It corresponds to the
Camera 1 data in the lower-left corner of Figure 14 in the AJP paper. Note
that this file may be modified to include *C*_{D} as a free
parameter as well. Of course, the more free parameters, the more
challenging it is to optimize them all. Our strategy is to find
*C*_{D} from the first *Mathematica* file and then determine
*C*_{L} once we know *C*_{D}
.

Know_CD_CL_Range_AJP This
*Mathematica* file determines *C*_{S} using an experimentally-determined
range. The user supplies *C*_{D} and
*C*_{L};_{ }the latter we take to be zero for pure side-spin
launches. This code solves the 3D trajectory problem; it contains
numerous comments. Our goal in the future is to obtain experimental data
for the full three-dimensional trajectory, and then use that data to determine
*C*_{S} in the same way we found *C*_{L} from the
Camera 1 data. Assuming a constant *C*_{S}, however, gives us a
*Mathematica*-produced
value of *C*_{S} that is
consistent with Figure 7 in the AJP paper.