Modeling Soccer Ball Flight Path with Drag and Magnus Forces | Extended Essay"

[Grantismo]

Homework Statement

For my Extended Essay I am modeling the flight path of a soccer ball under various conditions. Currently, I have a model for an extremely basic soccer kick with no spin and only gravitational forces, with no drag. My goal is to model the kick firstly with the addition of Drag forces and finally with the addition of the Magnus force.

Homework Equations

Basic Projectile Flight in a vacuum on Earth(v=km/hr):
x=(5v/18)Cos[tex]\vartheta[/tex]Sin[tex]\beta[/tex]t,
y=(5v/18)Cos[tex]\vartheta[/tex]Cos[tex]\beta[/tex]t
z=-4.9t[tex]^{2}[/tex] + (10 v/36)Sin[tex]\vartheta[/tex]t

Drag Force:
F=-(1/2)pv[tex]^{2}[/tex]CdA
(on an additional note, what units should be used for these variables?)

The Attempt at a Solution

Initially I attempted to model the situation like this:
x=-(1/4m)p((5v/18)Cos[tex]\vartheta[/tex]Sin[tex]\beta[/tex])[tex]^{2}[/tex]CdAt[tex]^{2}[/tex]+(5v/18)Cos[tex]\vartheta[/tex]Sin[tex]\beta[/tex]t,
y=-(1/4m)p((5v/18)Cos[tex]\vartheta[/tex]Cos[tex]\beta[/tex])[tex]^{2}[/tex]CdAt[tex]^{2}[/tex]+(5v/18)Cos[tex]\vartheta[/tex]Cos[tex]\beta[/tex]t
z=((-(1/4m)p(-9.8t+(5v/18)Sin[tex]\vartheta[/tex])[tex]^{2}[/tex]CdA)-4.9)t[tex]^{2}[/tex] + (5v/18)Sin[tex]\vartheta[/tex]t

As I was looking for an appropriate drag coefficient for a soccer ball, I learned about the phenomenon known as Drag Crisis, where as the Reynolds number increases, the drag coefficient drops. Hence, I would need some sort of mathematical representation of this drop as a function of velocity in order to accurately model the flight path. I am simply asking for some help to this effect, whether it is resources or an actual function. Any help would be appreciated.

I am modeling this situation using Mathematica, so if anyone wants the code or actual file, I would be more than happy to provide it.

 

[chanvincent]

You know [tex] \vec{v} [/tex] and [tex] F_{drag} [/tex] is not a constant during the flight, do you?

There are two major conceptual errors I notice in your solution,
First, you separated the x, y, and z component. The drag force ([tex] F_{drag} [/tex]) is not a linear combination to [tex] \vec{v}_x, \vec{v}_y, \mbox{ and } \vec{v}_z [/tex]. You have no reason to separate them into 3 independent component.
Secondly, you didn't address the fact that the direction and magnitude of F is changing with respect to time t.