# Bend it like… Bernoulli

#### By Colin White On May 13, 2014

In about a month’s time, 32 teams of players from around the world will converge in Brazil for the biggest event (at least in terms of viewership) in the history of sport. Two groups of eleven players will assemble in various combinations, creating 64 separate matches. The final match may draw a worldwide viewership of over a billion souls – a sizeable proportion of the planet’s population.

Clearly, a number of factors contribute to the success and enjoyment of a game of soccer, but arguably, the ball design is a key ingredient. And from a physics perspective, the soccer ball is one of the most interesting balls used in any sport. It is larger than most, lighter than most and has a strange geometrically-patterned profile which interacts with the airflow in a significant way. Furthermore, the fact that it is an inflated ball affects both the impact and spin characteristics. So, with reference to this excellent, lucid and comprehensive tome, let’s take a closer look at the science of the soccer ball and its trajectory.

In many ways, its aerodynamic characteristics are similar to those of other balls: only more so. It has the highest drag-to-weight ratio of all sports balls; its value is exceeded only by the shuttlecock (which is clearly not a ball) and the wiffleball (which doesn’t count because it’s a toy). Notwithstanding, the classic ball trajectory equations still apply, albeit without the benefit of the simplifying approximations and assumptions that are often available in the analysis of smaller, heavier balls. The equations of motion of the soccer ball (assuming it is subject to both drag and spin) are given by a series of six linked, non-linear differential equations with – surprise, surprise – no analytic solutions. However, computer solutions are relatively simple to obtain once the flight duration is partitioned into suitable time periods, and the differential equations can be treated as difference equations.

What we find is that, compared with many other sporting projectiles, the trajectory of a maximally kicked ball (i.e. a ball that has been given some wellie!) is rather special. All balls, or indeed any aerodynamic object, possess a property known as its Reynolds number. This governs how the air flows around the ball. Airflow can take on only one of two states: *laminar flow*, in which the air flows in bands like onion layers around the ball, and *turbulent* or *chaotic* flow, in which the air sets up little whirlwind-like eddies all over the surface of the ball. The air pressure on the surface of the ball is considerably lower in the latter, chaotic case and, as a consequence, the drag experienced by the ball will also reduce: the ball will travel faster. The switch between these two flow types is instantaneous and only dependent on the Reynolds number and the speed of the ball.

Now, in most sports, the projectile spends the vast majority of its flight in one or the other of these two flow-states, possibly only flipping state either just after launch, or just before the landing. However, in the case of the soccer ball, the Reynolds number is such that the chaotic to laminar flow flip is often observed *in* *mid-flight*. Typically, in, say, a long kick into a goal area, the ball will visibly decelerate and curve in on its approach as the airflow flips from chaotic to laminar flow. (Out of interest, in cricket, this effect is known as the late swing but is, in fact, a complete myth! Unlike the soccer ball, the Reynolds number and ball velocity for a cricket ball are such that such flow-state flips cannot happen in mid-flight. What cricketers refer to as a late swing is nothing more than an optical illusion.)

The fact that the soccer ball is inflated also affects its flight. Inflated balls have a higher moment of inertia than solid ones: they take more torque to set them spinning but will lose comparatively little angular velocity during the flight. So, when our soccer ball slows in flight as it approaches the goal area, and the airflow flips into laminar flow, it decelerates even more, even as its spin is approximately maintained over the duration of the flight. As a consequence of this, the ball is seen to curl in tightly towards the end of its trajectory.

Obviously, for this phenomenon to manifest, a considerable torque must be applied to the ball on launch, usually by means of a slice-kick. Again, the hollow inflated ball’s design is advantageous. When the boot impacts with the ball, the ball indents around the boot, effectively forming a mechanical ‘cog and sprocket’ grip. The efficacy of this grip can be seen in studies that compare a series of trajectories of wet and dry balls during penalty kicks. We often see players drying the ball on their shirts prior to placing the ball on the penalty spot. But does this make a difference? Well, yes it does, but, given the considerable reduction in the boot/ball coefficient of friction between the dry and wet balls, the difference in their respective trajectories is actually quite small. Typically, for a penalty kick distance of about 11 metres, the ball’s deviation between the wet and dry ball shots is about 20 cm (roughly one ball’s diameter).

While on the subject of impact, the inflated ball exhibits what we call the ‘trampoline effect’ on impact from the boot. The ball’s launch velocity will be about 1.2 times the velocity of the boot on the ball, at least for the maximal kick. For submaximal kicks, the ball’s velocity is derived from a regression equation that suggests an even great velocity multiplication factor than 1.2.

In summary, the modelling of soccer ball trajectories is not a simple science and probably more involved than the analysis of some other sports projectile trajectories. However, overall, the resulting models show accuracy, repeatability and robustness when compared with test kicks on actual balls carried out under laboratory conditions (using, for example, a robotic leg). As well as that excellent book alluded to earlier, you can also find working examples of a range of projectile models here, including a two-dimensional soccer ball deviation model. However, to reiterate and emphasise: the predictive qualities of the models are excellent; the science is solid.

Can you sense a ‘but’ coming?

For some, the word Jabulani is all that needs to be said. For the rest: a reminder. The last World Cup , in 2010, was engulfed by a controversy created by the chosen football, the Adidas Jabulani. Several key players hated it for a variety of reasons. Others loved it. It really seemed to be a case of the classic ‘Marmite’ scenario. The arguments spread, via the players, to the media pundits and, thence, to the fans. Soccer enthusiasts across the globe had an opinion on the Jabulani ball. It seemed that the Jabulani debate overwhelmed the actual outcome of the games. This became the real World Cup battle. Spectators were watching for the ball’s apparent flight peculiarities, rather than the game strategies, set pieces or elements of individual player’s skill. And yet, as stated, the science of soccer ball design and modelling was both rigorous and exact.

This year’s World Cup has its Adidas Brazuca ball, and regardless of the quality of the science, the same controversy will, without doubt, re-emerge. A follow-up to this post will attempt to unpick the arguments that underscore the apparent discrepancy between football science and football practice, and maybe resolve the dichotomy of opinion.

Let battle commence…

*Image: The University of Central Arkansas’ women’s team opens the 2010 at home to the University of Arkansas at Little Rock. Credit: University of Central Arkansas, reproduced under a Creative Commons licence.*

*NB: Despite the title of this post, the notion that the spin on the ball is due to the Bernoulli Principle is a misconception. It’s actually the Magnus Effect.*