The Science of Soccer

John Wesson

• Publisher: CRC Press
• Year: 2019
• ISBN:
  9781000732450
• Edition: 2
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Updated and revised throughout, this new edition of The Science of Soccer applies scientific analysis to football, giving us the answers to questions like "what's the chance of a team that wins the Premiership also winning the Cup? Can you predict how many goals will be scored? What's the best height for footballers? Is the team that wins the league the best team?"

Starting with a qualitative description of the basic physics that relate to the ball and its bounce, the author then moves through kicks and throws, to a simple account of the more complex physics of a ball in flight. Fulfilling your scientific curiosity, this book uncovers aspects of the game that are not normally discussed. It includes a look at game theory, how the rules affect the flow and enjoyment of the game, unusual statistics about players, and an insight into the economics of the game.

For those with a more mathematical interest in the physics, the final chapter provides a readable account of the theory behind the beautiful game.

Features:

• Accessible to anyone interested in understanding more about the science behind the sport
• Updated throughout, with new content on transfer fees, wages, and the top goal-scorers
• Discusses topics not explored in current literature, including rudimentary game theory

Chapter 1

1

The ball and the bounce

The ball

Ball-like objects must have been kicked competitively for thousands of years. It doesn’t require much imagination to picture a boy kicking a stone and being challenged for possession by his friends. However the success of ‘soccer’ was dependent on the introduction of the modern ball with its well-chosen size, weight and bounce characteristics.

When soccer was invented in the nineteenth century the ball consisted of an ox or pig bladder encased in leather. The bladder was pumped through a gap in the leather casing, and when the ball was fully pumped this gap was closed with lacing. While this structure was a great advance, a good shape was dependent on careful manufacture and was often lost with use. The animal bladder was soon replaced by a rubber ‘bladder’ but the use of leather persisted until the 1960s.

The principal deficiency of leather as a casing material was that it absorbed water. When this was combined with its tendency to collect mud the weight of the ball could be doubled. Many of us can recollect the sight of such a ball with its exposed lacing hurtling toward us and expecting to be headed.

The period up to the late 1980s saw the introduction of multi-layer casing and the development of a totally synthetic ball. Synthetic fibre layers are covered with a smooth polymer surface material and the ball is inflated with a latex bladder. This ball resists the retention of water and reliably maintains its shape.

The casing of high-quality balls is made up of panels. These panels, which can have a variety of shapes, are stitched together through pre-punched stitch holes using threads which are waxed for improved water resistance. This can require up to 2000 stitches. The lacing is long gone, the ball now being pumped through a tiny hole in the casing. Such balls are close to ideal.

The general requirements for the ball are fairly obvious. The ball mustn’t be too heavy to kick, or so light that it is blown about, or will not carry. It shouldn’t be too large to manoeuvre or too small to control, and the best diameter, fixed in 1872, turned out to be about the size of the foot. The optimisation took place by trial and error and the present ball is defined quite closely by the laws of the game.

The laws state that ‘The circumference shall not be more than 28 inches and not less than 27 inches. The weight of the ball shall be not more than 16 ounces and not less than 14 ounces. The pressure shall be equal to 0.6 to 1.1 atmosphere.’ Since 1 atmosphere is 14.7 pounds per square inch this pressure range corresponds to 8.8 to 16.2 pounds per square inch. (The usually quoted 8.5 to 15.6 pounds per square inch results from the use of an inaccurate conversion factor.)

From a scientific point of view the requirement that the pressure should be so low is amusing. Any attempt to reduce the pressure in the ball below one atmosphere would make it collapse. Even at a pressure of 1.1 atmosphere the ball would be a rather floppy object. What the rule really calls for, of course, is a pressure difference between the inside and the outside of the ball, the pressure inside being equal to 1.6 to 2.1 atmosphere.

Calculation of the ball’s behaviour involves the mass of the ball. For our purposes mass is simply related to weight. The weight of an object of given mass is just the force exerted on that mass by gravity. The names used for the two quantities are rather confusing, a mass of one pound being said to have a weight of one pound. However, this need not trouble us; suffice it to say that the football has a mass of between 0.875 and 1.0 pound or 0.40 and 0.45 kilogram.

Although it will not enter our analysis of the behaviour of the ball, it is of interest to know how the pressure operates. The air in the atmosphere consists of very small particles called molecules. A hundred thousand air molecules placed sided by side would measure the same as the diameter of a human hair. In reality the molecules are randomly distributed in space. The number of molecules is enormous, there being 400 million million million (4 × 10²⁰) molecules in each inch cube. Nevertheless most of the space is empty; the molecules occupying about a thousandth of the volume.

The molecules are not stationary. They move with a speed greater than that of a jumbo jet. The individual molecules move in random directions with speeds around a thousand miles per hour. As a result of this motion the molecules are continually colliding with each other. The molecules which are adjacent to the casing of the ball also collide with the casing and it is this bombardment of the casing which provides the pressure on its surface and gives the ball its stiffness.

The air molecules inside the ball have the same speed as those outside, and the extra pressure inside the ball arises because there are more molecules in a given volume. This was the purpose of pumping the ball – to introduce the extra molecules. Thus the outward pressure on the casing of the ball comes from the larger number of molecules impinging on the inner surface as compared with the number on the outer surface.

The bounce

The bounce seems so natural that the need for an explanation might not be apparent. When solid balls bounce it is the elasticity of the material of the ball which allows the bounce. This applies for example to golf and squash balls. But the casing of a football provides practically no elasticity. If an unpumped ball is dropped it stays dead on the ground.

Figure 1.1. Sequence of states of the ball during the bounce.

It is the higher-pressure air in the ball which gives it its elasticity and produces the bounce. It also makes the ball responsive to the kick. The ball actually bounces from the foot, and this allows a well-struck ball to travel at a speed of over 80 miles per hour. Furthermore, a headed ball obviously depends upon a bounce from the forehead. We shall examine these subjects later, but first let us look at a simpler matter, the bounce itself.

We shall analyse the mechanics of the bounce to see what forces are involved and will find that the duration of the bounce is determined simply by the three rules specifying the size, weight and pressure. The basic geometry of the bounce is illustrated in figure 1.1. The individual drawings show the state of the ball during a vertical bounce. After the ball makes contact with the ground an increasing area of the casing is flattened against the ground until the ball is brought to rest. The velocity of the ball is then reversed. As the ball rises the contact area reduces and finally the ball leaves the ground.

It might be expected that the pressure changes arising from the deformation of the ball are important for the bounce, but this is not so. To clarify this we will first examine the pressure changes which do occur.

Pressure changes

It is obvious that before contact with the ground the air pressure is uniform throughout the ball. When contact occurs and the bottom of the ball is flattened, the deformation increases the pressure around the flattened region. However, this pressure increase is rapidly redistributed over the whole of the ball. The speed with which this redistribution occurs is the speed of sound, around 770 miles per hour. This means that sound travels across the ball in about a thousandth of a second and this is fast enough to maintain an almost equal pressure throughout the ball during the bounce.

Although the pressure remains essentially uniform inside the ball the pressure itself will actually increase. This is because the flattening at the bottom of the ball reduces the volume occupied by the air, in other words the air is compressed. The resulting pressure increase depends on the speed of the ball before the bounce. A ball reaching the ground at 20 miles per hour is deformed by about an inch and this gives a pressure increase of only 5%. Such small pressure changes inside the ball can be neglected in understanding the mechanism of the bounce. So what does cause the bounce and what is the timescale?

Mechanism of the bounce

While the ball is undeformed the pressure on any part of the inner surface is balanced by an equal pressure on the opposite facing part of the surface as illustrated in figure 1.2. Consequently, as expected, there is no resultant force on the ball. However, when the ball is in contact with the ground additional forces comes into play. The casing exerts a pressure on the ground and, from Newton’s third law, the ground exerts an equal and opposite pressure on the casing. There are two ways of viewing the resultant forces.

In the first, and more intuitive, we say that it is the upward force from the ground which first slows the ball and then accelerates it upwards, producing the bounce. In this description the air pressure force on the deformed casing is still balanced by the pressure on the opposite surface, as shown in figure 1.3(a). In the second description we say that there is no resultant force acting on the casing in contact with the ground, the excess air pressure inside the ball balancing the reaction force from the ground. The force which now causes the bounce is that of the unbalanced air pressure on that part of the casing opposite to the contact area, as illustrated in figure 1.3(b). These two descriptions are equally valid.

Figure 1.2. Pressure forces ...