I was wondering how difficult it would be to replicate Roberto Carlos's free kick from 1997. I
made a mathematical model in MATLAB and determined the range of hit points on the ball that produce
an on target shot the results that I got explain why this shot is almost never attempted
with all the theoretical knowledge I thought I would attempt the shot myself
I used Newtonian physics to determine the trajectory of the ball. Newtonian physics
accurately predicts pretty much anything around you from rockets to sprinklers so the trajectory
of a flying ball should also be possible to predict. There are several forces acting on a
flying ball there is of course the gravitational force always acting vertically downwards.
There is also the drag force which is present whenever there is a body moving through a fluid.
The drag effects become prominent at high velocities and cannot be ignored for this shot.
The drag force always opposes direction of motion and slows down the object. Lastly we have
the magnus force which acts perpendicular to the direction of motion. The Magnus force is the one
force we are interested in the most because it is responsible for the curved trajectory of the ball.
This force is highly dependent on the spin and velocity of the ball this model accounts for
the magnus force due to rotations about the z and y-axis. The rotation above the x-axis
is deemed negligible. Summing up the forces gives us the three components of the acceleration.
Newtonian physics models a force as a vector acting at a single point. Obviously in reality
the contact area between the ball and the foot is much larger however the resulting force still acts
through a single point. The spin of the ball is the parameter that affects the trajectory the
most; the spin depends on the point and the angle at which the ball is hit. If the kick induces a
topspin then the magnus force will point downwards which could produce a knuckleball effect if the
shot hits the ball lower and induces a bottom spin then the ball could end up several meters above
the goal which is something we often see with free kick takers I combined all of the equations into
MATLAB and used the modified Euler's method with a small enough time-step to numerically solve the
resulting differential equation which gave me the position of the ball as a function of time
the MATLAB model is only dependent on the power of the shot and the exit conditions of the ball this
allowed me to figure out the exact point where the ball needs to be hit to produce an on target shot.
With the solution of the equations available I attempted to model Roberto Carlos's free kick
to prove the validity of the equations we know that the ball's velocity is measured at around
137 kilometers per hour and the exit angle is at around 13 degrees. We need to determine the
contact point and the vertical angle to have a complete model to do that I made a Monte Carlo
simulation that simulates 500 shots with the ball being hit at 500 random points. The black points
show shots that miss the target, the red shots hit the wall and the green shots are on target
Once I got a better idea of where the contact point should be, I focused the randomized shots
in that area of the ball and observed the effects. Even though the kick points are
fairly close to one another, the trajectories can still differ by as much as 30 meters.
The ball diagram clearly establishes a transition line from shots that hit the
wall and off target's shots the only kick plays which produces an on target shot
is somewhere between the red and black X's. Now I further narrowed down the focus point
and ran another 150 randomized shots with a much smaller standard deviation
this simulation started to produce a lot more on target shots, zooming in to the focus area
we can see that the only place which produces an on target shot is about 4 millimeters wide.
Clearly this shot requires insane accuracy and technique hitting the ball only a few
millimeters to the left or downward could send the ball to the sideline and embarrass the taker.
If the exit velocity changes then the whole situation is different. If Carlos had kicked
the ball 10% harder or a few millimeters to the left then the ball would have ended up
at a completely different place. So no wonder why we barely see the shots successfully taken.
The last remaining part is to validate the model with Carlos's free kick.
The shape of the mathematical trajectory matches very well with the observed real trajectory;
it is quite amazing how pure mathematics and physics could reproduce real-world observations
With all the theoretical knowledge, it was time to test whether I could reproduce
a similar shot. The specific solution obtained in this simulation is only valid if you could
kick the ball as hard as Roberto Carlos so I went out and measured the power of my shot so that I
could calculate the exact point where I need to kick the ball to reproduce a similar trajectory.
With the numbers available I went back to the MATLAB model and looked for a solution. The
first and obvious problem with this was the power of my shot is less than half of Carlos's shot.
This limited the hit points to the lower part of the ball which produces a large hit angle
resulting in a parabolic trajectory. The longer trajectory increase the travel time to 2.4 seconds
which is not quick enough to surprise even the worst of goalkeepers.
A lower exit angle reduces the travel time but prevents the ball from making it to the goal.
The obvious conclusion is that my shot is just not powerful enough to score from 35 meters
That was all for this video, thanks for watching and see you soon!