# What can the speed of a bullet create

## The "Coca-Cola formula" of a football

Without air resistance, there are symmetrical flight curves and quite unrealistic flight distances of shots. And such idealized considerations were only valid for very low speeds anyway. But footballs describe anything but beautiful symmetrical trajectory parabolas, they look much more complicated. The reason for this is the air resistance, which is quite large in a football and has a very strong influence on the trajectory of the round leather. What does the air resistance of a football depend on and how does it influence its flight characteristics?

Moving ball in a liquid

The cause of the air resistance are vortices that arise when the air flows around the ball in flight. The figure shows such eddies that form behind a sphere when it moves through a liquid.

A similar picture would result if a gas like air flows around a sphere. At first, the air flows around the sphere, only to then detach itself from its surface about halfway. This detachment is obviously the cause of the vortex formation and thus the air resistance. It can be seen again schematically in the figure on the left.

Explanation for the creation of a vortex

Now you can ask yourself why the air doesn't just flow evenly around the ball without forming eddies? The reason is very simple: It does not make it because air also has a certain "tenacity" (the physical term would be "viscosity"), which slows down the flow due to internal friction, as one would expect from the tenacity of honey knows. The viscosity of the air is, of course, much less than that of honey, but it is not zero. The air that flows directly around the surface of the football will eventually detach from this surface and form eddies. The whole process takes place in a layer that is only a few millimeters thick, the so-called "boundary layer". It is now important that this process happens even if the surface of the ball is perfectly smooth. In fact, it's greatest even on a perfectly smooth football, as we'll see! Due to the detachment of individual air vortices in the boundary layer, so-called “wake vortices” arise behind every ball, including every football that flies through the air, which draw energy from the ball and thus slow down its movement.

Air resistance of a ball

It is now certainly easy to see that the degree of vortex formation on the surface depends on the speed of the air that flows around the sphere. But this is precisely the speed at which our soccer ball flies through the air. The faster the ball moves, the sooner the air separates from the surface of the ball and the more eddies should arise. A very complex consideration shows that the air resistance of a ball is the square of the speed v depends, that is, if the speed doubles then the air resistance quadruples. A detailed analysis gives the following expression for the force exerted by air resistance on the ball:

\$\$ F _ {\ textrm {Luftw.}} = 0.5 \ rho c_w A v ^ 2 \$\$

It is ρ = 1.2 g / l the density of the air and cW. is the so-called "drag coefficient", which indicates the streamlining of a body moving through the air. The following applies to a sphere cW. ≈ 0.3-0.5. The size A. is the cross-sectional area of ​​a soccer ball with the known radius of R. = 11 cm of a FIFA sports device easily by means of A. = πR.² can be calculated. The direction of the force of the air resistance must, however, still be specified for a complete description. It always works directly opposite to the direction of movement of the football. In contrast to gravity, which always acts in the same direction, namely vertically downwards, this time the direction of the air resistance force also changes at every point on the flight curve of a football. That doesn't exactly make the mathematical calculation of a flight path any easier.

Air resistance for a (perfectly smooth) sphere

In principle, however, the air resistance of a soccer ball can be calculated very precisely. The figure shows such a curve for a completely smooth football. The air resistance is given here in multiples of the ball's weight as a function of the ball's speed.

It can be seen that the air resistance increases quadratically with the speed, exactly as we have just discussed. However, for speeds in the range of around 75–90 km / h, above the so-called “critical speed”, something very strange happens: The air resistance now decreases sharply with increasing speed! How can this happen? The explanation for this is certainly not easy. We had seen that the shedding of air eddies in the boundary layer around the soccer ball, which is only a few millimeters thick, is responsible for the air resistance and its increase with increasing speed. At the critical speed something strange happens: The air in the boundary layer itself becomes turbulent, that is, the boundary layer itself begins to swirl microscopically. Paradoxically, this has the result that the larger eddies can now detach from the ball surface much later than before, since the microscopic eddies dominate the boundary layer and this determines the air resistance. As a result, it then follows that the wake vortex becomes smaller and the overall air resistance is drastically reduced. This should also be illustrated schematically with a figure (right).

Operations at "critical speed"

But if the air resistance of a football were to depend on the speed as shown in the penultimate figure, then this would have immense consequences for the game. With every somewhat stronger shot, the speed of which is above the "critical speed", the air resistance would initially increase with decreasing speed, run through a maximum and then decrease again. The trajectory of the ball would look strange - the ball would "flutter" in the air.

The previous considerations were only valid for a perfectly smooth ball the size of a soccer ball. Every football, even the smooth “team spirit” ball from the last World Cup in 2006, has seams or other roughening on the surface. This rough surface now ensures that microscopic eddies occur in the boundary layer even at very low speeds. These eddies are effectively detached by the unevenness on the surface and prevent the large eddies from forming, which otherwise determine the air resistance. Overall, there is a net gain, because the swirled boundary layer causes a reduction in the overall air resistance, as the figure above shows. Furthermore, it is also prevented that there is such a thing as a “critical speed” above which the air resistance decreases. The air resistance is then a monotonically increasing function of the speed, as can be seen in the figure.

Air resistance for a real soccer ball

The red curve is the measurement of air resistance for a real soccer ball, as the Englishman John Wesson did for his book The Science of Soccer carried out at an airfield with a self-made apparatus. You can now see that there is an almost even increase, which is indicated by the blue straight line with the formula F.Airw. = βv can be approximated. The air resistance of a soccer ball is therefore an evenly increasing function. The reason for this is its irregularly shaped surface due to the seams.

Why didn't I just show you the air resistance curves of all balls, as indicated by the manufacturers for balls such as the "Teamgeist" ball, but the measurement above by "Amateurs" John Wesson, which he made for a conventional soccer ball in 2001 ? Quite simply: the manufacturers do not specify these curves anywhere! The curve above is like the “Coca-Cola formula” of a soccer ball - the secret recipe from which all flight characteristics can be determined! The exact air resistance curve must therefore be measured precisely in the wind tunnel for every new model of a football with a new surface structure, so that one can be absolutely certain that the phenomenon of critical speed does not occur and the ball does not start to "flutter" at high speeds. However, measurements in the wind tunnel are expensive and not affordable for ordinary people. You don't want to leave them to the competition "for free" - that's why there are no air resistance curves from soccer ball manufacturers.

In conclusion, we can say that a football only flies so evenly through the air because it has a rough surface. These irregularities of the surface reduce its air resistance and it usually does not “flutter”. Instead, professionals such as Thorsten Frings allow it to be passed over many meters to the teammate in order to score decisive goals. The air resistance curve of a good football, i.e. the air resistance as a function of the speed of the football, should therefore be a steadily increasing, almost straight line. If you know this curve, you also know the exact flight characteristics of the ball. This air resistance curve is just as important for the manufacturer as the "Coca-Cola formula" for the drink of the same name.

As you have seen, football can be quite complicated at times. The mere flight of a soccer ball through the air is a highly complex process from a physical point of view, and I haven't told you everything yet.

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### Book reference

More detailed descriptions of the flight of a football can be found in the book by John Wesson, The Science of Soccerthat was released in 2002. This very readable book was translated into German in 2005 and is published by the Spektrum-Verlag (Elsevier) under the titleFootball - science with a kick for 15 euros.