When Brazilian defender Roberto Carlos struck a powerful free-kick from about 30 meters out in a 1997 international match against France, he could not have known that scientists would still be discussing his feat more than a dozen years later. Indeed, he could not even have known that the ball would improbably find the back of the net. But find the net it did, swinging well wide of a wall of French defenders, hooking viciously to the left, and glancing off the inside of the goalpost. The French goalkeeper could only turn and watch in apparent disbelief as the ball came to rest in his goal.

It seems impossible that someone could blast the ball past a world-class goalkeeper from such a great distance without eliciting from the keeper so much as a dive in the ball's direction. But the distance of the kick may have played a role in its deceptive flight path, according to a study—by a team of French scientists, no less—published September 2 in the New Journal of Physics. With sufficient distance, the group has found, spinning balls suddenly change trajectory, from a smooth arc to a sharp inward curl. "Provided that the shot is powerful enough, another characteristic of Roberto Carlos' abilities, the ball trajectory brutally bends toward the net, at a velocity still large enough to surprise the keeper," the study's authors reported.

The researchers, from ESPCI ParisTech and the Polytechnic School in Palaiseau, experimented with small plastic balls launched by a slingshot into a pool of water to investigate how drag affects a spinning ball traveling through a medium such as water or air.

What they found was that spinning balls follow a kind of circular arc as they slow from drag forces and are pushed from a linear path by the so-called Magnus effect, which gives spinning bodies such as Major League curveballs their arcing trajectory. But the ball's velocity decreases more quickly than its spin rate, and at a certain point the spin becomes dominant in guiding the path of the ball, producing a sudden and dramatic curvature in the ball's trajectory [see photograph above]. (The basic concept is not entirely new; a 1998 Physics World article on soccer physics invoked golf-ball research from the 1970s to describe the phenomenon.) The point at which that spiraling curl takes place depends on a number of factors, including the size, density, velocity and spin of the ball, as well as the density of the surrounding fluid, whether air or water.

For a well-struck soccer ball, the researchers estimate, one might expect a gentle arc followed by a sharp hook at about 50 meters—in rough agreement with the distance of Roberto Carlos's free kick. In other words, if a soccer player has the strength to drive a ball halfway down the field with plenty of velocity and spin, he or she can expect to benefit from an unexpected curve late in the ball's trajectory.

The news is not as encouraging for aspiring baseball pitchers: Although the Magnus effect works wonders for curveballs, the late-break motion described in the new study would not take effect on a hard-thrown baseball for 160 meters—roughly 10 times the distance between the pitcher's mound and home plate.

Image credit: New Journal of Physics