Original URL: http://www.theregister.co.uk/2010/09/02/exponential_spiral/

## Boffins explain greatest ever free kick

Roberto Carlos and his amazing exponential spiral

Posted in Science, 2nd September 2010 07:50 GMT

Scientists have agreeably concluded that Roberto Carlos's 1997 free kick against France - a seemingly impossible blast into the back of the net from 115ft - was not the fluke some have claimed.

While hapless French keeper Fabien Barthez might have taken solace from the thought that perhaps a gust of wind had accounted for the ball's improbable trajectory, the fact is that it was all down to Carlos and his amazing exponential spiral.

That's according to a team of researchers from the École Polytechnique in Palaiseau, near Paris, who demonstrated that while the ball's trajectory initially conformed to the Magnus effect, which gives a spinning sphere a curved path, the distance of this particular free kick added a surprise ending.

By firing plastic balls into a tank of water, the boffins showed that if a football's spin speed remains constant, "its trajectory follows an exponential spiral", with the bend "surprisingly" increasing as drag slows the ball down.

In the final phase, where both velocity and spin decrease, the ball tends to follow a straight line before hitting the back of the net.

The distance here is critical. Researchers Christophe Clanet and David Quéré explained: "When shot from a large enough distance, and with enough power to keep an appreciable velocity as approaching the goal, the ball can have an unexpected trajectory.

"Carlos' kick started with a classical circular trajectory but suddenly bent in a spectacular way and came back to the goal, although it looked out of the target a small moment earlier.

"People often noticed that Carlos' free kick had been shot from a remarkably long distance, we show in our paper that this is not a coincidence, but a necessary condition for generating a spiral trajectory."

The full explanation of the Roberto Carlos exponential spiral can be found here in the latest edition of *New Journal of Physics*. ®