Many fans will remember the free kick taken by the Brazilian Roberto Carlos in a tournament in France last summer. The ball was placed about 30 m from his opponents' goal and slightly to the right. Carlos hit the ball so far to the right that it initially cleared the wall of defenders by at least a metre and made a ball-boy, who stood metres from the goal, duck his head. Then, almost magically, the ball curved to the left and entered the top right-hand corner of the goal – to the amazement of players, the goalkeeper and the media alike.

Apparently, Carlos practised this kick all the time on the training ground. He intuitively knew how to curve the ball by hitting it at a particular velocity and with a particular spin. He probably did not, however, know the physics behind it all.

Aerodynamics of sports balls

The first explanation of the lateral deflection of a spinning object was credited by Lord Rayleigh to work done by the German physicist Gustav Magnus in 1852. Magnus had actually been trying to determine why spinning shells and bullets deflect to one side, but his explanation applies equally well to balls. Indeed, the fundamental mechanism of a curving ball in football is almost the same as in other sports such as baseball, golf, cricket and tennis.

Consider a ball that is spinning about an axis perpendicular to the flow of air across it (see left). The air travels faster relative to the centre of the ball where the periphery of the ball is moving in the same direction as the airflow. This reduces the pressure, according to Bernouilli's principle. The opposite effect happens on the other side of the ball, where the air travels slower relative to the centre of the ball. There is therefore an imbalance in the forces and the ball deflects – or, as Sir J J Thomson put it in 1910, "the ball follows its nose". This lateral deflection of a ball in flight is generally known as the "Magnus effect".

The forces on a spinning ball that is flying through the air are generally divided into two types: a lift force and a drag force. The lift force is the upwards or sidewards force that is responsible for the Magnus effect. The drag force acts in the opposite direction to the path of the ball.

Let us calculate the forces at work in a well taken free kick. Assuming that the velocity of the ball is 25–30 ms–1 (about 70 mph) and that the spin is about 8–10 revolutions per second, then the lift force turns out to be about 3.5 N. The regulations state that a professional football must have a mass of 410–450 g, which means that it accelerates by about 8 ms–2. And since the ball would be in flight for 1 s over its 30 m trajectory, the lift force could make the ball deviate by as much as 4 m from its normal straight-line course. Enough to trouble any goalkeeper!

The drag force, FD, on a ball increases with the square of the velocity, v, assuming that the density, r, of the ball and its cross-sectional area, A, remain unchanged: FD = CDrAv2/2. It appears, however, that the "drag coefficient", CD, also depends on the velocity of the ball. For example, if we plot the drag coefficient against Reynold's number – a non-dimensional parameter equal to rv D /μ, where D is the diameter of the ball and μ is the kinematic viscosity of the air – we find that the drag coefficient drops suddenly when the airflow at the surface of the ball changes from being smooth and laminar to being turbulent (see right).

When the airflow is laminar and the drag coefficient is high, the boundary layer of air on the surface of the ball "separates" relatively early as it flows over the ball, producing vortices in its wake. However, when the airflow is turbulent, the boundary layer sticks to the ball for longer. This produces late separation and a small drag.

The Reynold's number at which the drag coefficient drops therefore depends on the surface roughness of the ball. For example, golf balls, which are heavily dimpled, have quite a high surface roughness and the drag coefficient drops at a relatively low Reynold's number (~2 × 104). A football, however, is smoother than a golf ball and the critical transition is reached at a much higher Reynold's number (~4 × 105).

The upshot of all of this is that a slow-moving football experiences a relatively high retarding force. But if you can hit the ball fast enough so that the airflow over it is turbulent, the ball experiences a small retarding force (see right). A fast-moving football is therefore double trouble for a goalkeeper hoping to make a save – not only is the ball moving at high speed, it also does not slow down as much as might be expected. Perhaps the best goalkeepers intuitively understand more physics than they realize.

In 1976 Peter Bearman and colleagues from Imperial College, London, carried out a classic series of experiments on golf balls. They found that increasing the spin on a ball produced a higher lift coefficient and hence a bigger Magnus force. However, increasing the velocity at a given spin reduced the lift coefficient. What this means for a football is that a slow-moving ball with a lot of spin will have a larger sideways force than a fast-moving ball with the same spin. So as a ball slows down at the end of its trajectory, the curve becomes more pronounced.

Roberto Carlos revisited

How does all of this explain the free kick taken by Roberto Carlos? Although we cannot be entirely sure, the following is probably a fair explanation of what went on.

Carlos kicked the ball with the outside of his left foot to make it spin anticlockwise as he looked down onto it. Conditions were dry, so the amount of spin he gave the ball was high, perhaps over 10 revolutions per second. Kicking it with the outside of his foot allowed him to hit the ball hard, at probably over 30 ms–1 (70 mph). The flow of air over the surface of the ball was turbulent, which gave the ball a relatively low amount of drag. Some way into its path – perhaps around the 10 m mark (or at about the position of the wall of defenders) – the ball's velocity dropped such that it entered the laminar flow regime. This substantially increased the drag on the ball, which made it slow down even more. This enabled the sideways Magnus force, which was bending the ball towards the goal, to come even more into effect. Assuming that the amount of spin had not decayed too much, then the drag coefficient increased. This introduced an even larger sideways force and caused the ball to bend further. Finally, as the ball slowed, the bend became more exaggerated still (possibly due to the increase in the lift coefficient) until it hit the back of the net – much to the delight of the physicists in the crowd.

Current research into football motion

There is more to football research than simply studying the motion of the ball in flight. Researchers are also interested in finding out how a footballer actually kicks a ball. For example, Stanley Plagenhof of the University of Massachusetts in the US has studied the kinematics of kicking – in other words, ignoring the forces involved. Other researchers, such as Elizabeth Roberts and co-workers at the University of Wisconsin, have done dynamic analyses of kicking, taking the forces involved into account.

These experimental approaches have produced some excellent results, although many challenges still remain. One of the most critical problems is the difficulty of measuring the physical motion of humans, partly because their movements are so unpredictable. However, recent advances in analysing motion with computers have attracted much attention in sports science, and, with the help of new scientific methods, it is now possible to make reasonably accurate measurements of human motion.

For example, two of the authors (TA and TA) and a research team at Yamagata University in Japan have used a computational scientific approach coupled with the more conventional dynamical methods to simulate the way players kick a ball. These simulations have enabled the creation of "virtual" soccer players of various types – from beginners and young children to professionals – to play in virtual space and time on the computer. Sports equipment manufacturers, such as the ASICS Corporation, who are sponsoring the Yamagata project, are also interested in the work. They hope to use the results to design safer and higher performance sports equipment that can be made faster and more economically than existing products.

The movement of players was followed using high-speed video at 4500 frames per second, and the impact of the foot on the ball was then studied with finite-element analysis. The initial experiments proved what most footballers know: if you strike the ball straight on with your instep so that the foot hits the ball in line with the ball's centre of gravity, then the ball shoots off in a straight line. However, if you kick the ball with the front of your foot and with the angle between your leg and foot at 90° (see left), it will curve in flight. In this case, the impact is off-centre. This causes the applied force to act as a torque, which therefore gives the ball a spin.

The experimental results also showed that the spin picked up by the ball is closely related to the coefficient of friction between the foot and the ball, and to the offset distance of the foot from the ball's centre of gravity. A finite-element model of the impact of the foot on the ball, written with DYTRAN and PATRAN software from the MacNeal Schwendler Corporation, was used to numerically analyse these events. This study showed that an increase in the coefficient of friction between the ball and the foot caused the ball to acquire more spin. There was also more spin if the offset position was further from the centre of gravity. Two other interesting effects were observed. First, if the offset distance increased, then the foot touched the ball for a shorter time and over a smaller area, which caused both the spin and the velocity of the ball to decrease. There is therefore an optimum place to hit the ball if you want maximum spin: if you hit the ball too close or too far from the centre of gravity, it will not acquire any spin at all.

The other interesting effect was that even if the coefficient of friction is zero, the ball still gains some spin if you kick it with an offset from its centre of gravity . Although in this case there is no peripheral force parallel to the circumference of the ball (since the coefficient of friction is zero), the ball nevertheless deforms towards its centre, which causes some force to act around the centre of gravity. It is therefore possible to spin a football on a rainy day, although the spin will be much less than if conditions were dry.

Of course, the analysis has several limitations. The air outside the ball was ignored, and it was assumed that the air inside the ball behaved according to a compressive, viscous fluid-flow model. Ideally, the air both inside and outside the ball should be included, and the viscosities modelled using Navier-Stokes equations. It was also assumed that the foot was homogeneous, when it is obvious that a real foot is much more complicated than this. Although it would be impossible to create a perfect model that took every factor into account, this model does include the most important features.

Looking to the future, two of us (TA and TA) also plan to investigate the effect of different types of footwear on the kicking of a ball. Meanwhile, ASICS is combining the Yamagata finite-element simulations with biomechanics, physiology and materials science to design new types of football boots. Ultimately, however, it is the footballer who makes the difference – and without ability, technology is worthless.

The final whistle

So what can we learn from Roberto Carlos? If you kick the ball hard enough for the airflow over the surface to become turbulent, then the drag force remains small and the ball will really fly. If you want the ball to curve, give it lots of spin by hitting it off-centre. This is easier on a dry day than on a wet day, but can still be done regardless of conditions. The ball will curve most when it slows down into the laminar flow regime, so you need to practise to make sure that this transition occurs in the right place – for example, just after the ball has passed a defensive wall. If conditions are wet, you can still get spin, but you would be better off drying the ball (and your boots).

Nearly 90 years ago J J Thomson gave a lecture at the Royal Institution in London on the dynamics of golf balls. He is quoted as saying the following: "If we could accept the explanations of the behaviour of the ball given by many contributors to the very voluminous literature which has collected around the game...I should have to bring before you this evening a new dynamics, and announce that matter, when made up into [golf] balls obeys laws of an entirely different character from those governing its action when in any other conditions." In football, at least, we can be sure that things have moved on.