'Impossible' Soccer Goal Explained by New Twist on Curveball Physics

France's soccer match with Brazil on 3 June 1997 was especially memorable. In the final moments of the opening round of Tournoi de France—a tune-up for the World Cup the following year—Brazilian player Roberto Carlos scored what has become known as the impossible goal. Standing 35 meters from the net, he kicked the soccer ball so far to the right that the goalkeeper, Fabien Barthez, didn't even move to block it. But in midair, the ball suddenly and dramatically arced down and to the left, landing just inside the net and scoring the point that ended the match in a tie.

How could a ball apparently flying so far off the mark—and sending a ball boy who was nowhere near the goal ducking for cover—suddenly make such an eye-popping change of course? To find out, researchers at the École Polytechnique in Palaiseau, France, and the École Supérieure de Physique et de Chimie Industrielles in Paris, expanded their experiments on the behavior of projectiles in fluids to include the physics of sports balls. Instead of swinging on a golf course or kicking on a soccer field, they used a hand-held slingshot to fire tiny plastic balls into a tank of water. The water greatly simplified the logistics: It affected the projectiles the same way air affects larger balls, but on a scale of centimeters.

Using high-speed photography and computer models, the team found that the tiny balls behaved surprisingly like Carlos's impossible kick. When released, they initially zipped through the water in a straight line. But within a few milliseconds, they arced off course.

Part of this behavior is due to a well-known phenomenon called the Magnus effect. The slingshot shoots the ball forward but also sends it spinning on an axis perpendicular to its direction of motion. As the ball speeds through the fluid, its spinning surface drags the fluid with it, speeding up the flow of fluid past one side of the sphere and slowing it down on the other. Because a fluid's pressure decreases as its speed increases, that speed difference creates a push toward the side of the faster-moving fluid and makes the ball move in a gentle arc. The effect explains baseball's curveballs.

But resistance from the surrounding medium also slows the ball down without greatly affecting its spin or the Magnus effect. As a result, the curving of the path gets tighter and tighter, eventually forming a spiral instead of a simple up-and-down arc.

Based on the water-tank experiments, the researchers created an equation to describe precisely what happened to the plastic balls. By plugging in a variety of factors, including the original heading, density, and velocity, they could predict the trajectory of any spherical object shot into the water, they report online today in the New Journal of Physics.

Applied to Carlos's kick, the experiments showed that he "could only make this wonderful shot because he was far enough from the goal," explains physicist and co-author David Quéré. Essentially, Carlos's distance and extremely strong kick—about 130 kilometers per hour—gave the ball a high enough velocity to propel it away from the goal before the Magnus effect took over and sent it spinning into the net. The effect is not seen as dramatically in other sports, such as baseball and tennis, because the distance between pitcher and batter or between the two tennis players is much shorter. The main point of the study, Quéré says, is that "rotation can be used to [pilot] the trajectory of a projectile in a smart and predictable yet surprising way."

It's a powerful and relevant paper that demonstrates how fluid mechanics can be used as a descriptive and predictive tool, says fluid dynamicist Andrew Mackowski of Cornell University. "The existence of a spiral path is new," he says, "and their explanation of the phenomenon is plausible."

*This story has been corrected to resolve an ambiguity in the description of the Magnus force.