Ever wondered how Roberto Carlos or David Beckham could fool goalkeepers by putting a spin on the ball? Maybe not, since they are so good, but have you wondered why the spin actually changes the trajectory of the ball? This example models a free kick in soccer and examines what happens.

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All that a free kick model needs is essentially a ball that can move and rotate. The tricky part is to assign the correct forces that act on the ball, depending on velocity and spin. Three forces affect the ball: *gravity* (downward force), *drag* (force opposing the velocity vector), and *the Magnus force* (depending on velocity, angular velocity, and spin axis).

The Magnus force is dependent on the spin and velocity. All other variables contributing to the Magnus force can be put into a so-called "lift coefficient", *Cl*. This lift coefficient is difficult to calculate analytically and is often measured from experiments. The left plot below shows experimental data of the lift coefficient to spin ratio (= , rotational velocity times radius divided by velocity) and the function that is used to approximate *Cl*. The drag force is dependent on the velocity and the Reynolds number. The right plot shows experimental data of the drag coefficient compared to the Reynolds number, with the approximated function for *Cd* used in the model.

What initial velocity and spin are required to score a goal directly from a corner kick?

Visualize the forces automatically

The initial velocity and spin required to score can be calculated analytically, or alternatively (the more convenient way), be tested by making simulations and analyzing the resulting animation to see which initial parameter needs to be tuned. One example of initial conditions to score from the corner position is initVel = {35, 8, 12} m/s and initAngVel = {0, 50, 10} rad/s. See the video for a nice corner goal!

If you find that simulating one kick at a time is too slow, Mathematica can help you out using *WSMLink* and parameter sweep. In the graphic above, 300 simulations with different initial velocities were done at the same time. Out of the 300 hundred trajectories, 45 scored and 255 missed. Imagine if Beckham had access to SystemModeler and Mathematica in his glory days, how he could have optimized his free kicks!

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