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The ball and the bounce

ball-manThe 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.

1.1

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.

1.2

Figure 1.2. Pressure forces on opposing surfaces cancel.

In the first, and more intuitive, we say that it is the upward force from the ground which first lows 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.

1.3

Figure 1.3. Two descriptions of the force balance during the bounce.

Because the force on the ball is proportional to the area of contact with the ground and the area of contact is itself determined by the distance of the centre of the ball from the ground, it is possible to calculate the motion of the ball. The result is illustrated in the graph of figure 1.4 which plots the height of the centre of the ball against time. As we would expect, the calculation involves the mass and radius of the ball and the excess pressure inside it. These are precisely the quantities specified by the rules governing the ball. It is perhaps surprising that these are the only quantities involved, and that the rules determine the duration of the bounce. This turns out to be just under a hundredth of a second. The bounce time is somewhat shorter than the raming time of television pictures and in television transmissions the brief contact with the ground is often missed. Fortunately our brain fills in the gap for us.

1.4

Figure 1.4. Motion of ball during bounce.

Apart from small corrections the duration of the bounce is independent of the speed of the ball. A faster ball is more deformed but the resulting larger force means that the acceleration is higher and the two effects cancel. During the bounce the force on the ball is quite large. For a ball falling to the ground at 35 miles per hour the force rises to a quarter a ton – about 500 times the weight of the ball. The area of casing in contact with the ground increases during the first half of the bounce. The upward force increases with the area of contact, and so the force also increases during the first half of the bounce. At the time of maximum deformation, and therefore maximum force, the ball’s vertical velocity is instantaneously zero. From then on the process is reversed, the contact area decreasing and the force falling to zero as the ball loses contact with the ground.

If the ball were perfectly elastic and the ground completely rigid, the speed after a vertical  bounce would be equal to that before the bounce. In reality the speed immediately after the  bounce is somewhat less than that immediately before the bounce, some of the ball’s energy being lost in the deformation.

The lost energy appears in a very slight heating of the ball. The change in speed of the ball in the bounce is conveniently represented by a quantity called the ‘coefficient of restitution’. This is the ratio, usually written e, of the speed after a vertical bounce to that before it,

formula

A perfectly elastic ball bouncing on a hard surface would have e ¼ 1 whereas a completely limp ball which did not bounce at all would have e ¼ 0. For a football on hard ground e is typically 0.8, the speed being reduced by 20%.

1.5

Figure 1.5. Showing how the bouncing changes with the coefficient of restitution.

Grass reduces the coefficient of restitution, the bending of the blades causing further energy loss. For long grass the resulting coefficient depends on the speed of the ball as well as the length of the grass.

Figure 1.5(a) shows a sequence of bounces for a hard surface (e ¼ 0:8). This illustrates the unsatisfactory nature of too bouncy a surface. Figure 1.5(b) shows the much more rapid decay of successive bounces for a ball bouncing on short grass (e ¼ 0:6).

The bounce in play

The bounce described above is the simple one in which the ball falls vertically to the ground. In a game, the ball also has a horizontal motion and this introduces further aspects of the bounce. In the ideal case of a perfectly elastic ball bouncing on a perfectly smooth surface the horizontal velocity of the ball is unchanged during the bounce and the vertical velocity takes a value equal and opposite to that before the bounce, as shown in figure 1.4. The symmetry means that the angle to the ground is the same before and after. In reality the bounce is affected by the imperfect elasticity of the ball, by the friction between the ball and the ground, and by spin. Even if the ball is not spinning before the bounce, it will be spinning when it leaves the ground. We will now analyse in a simplified way the effect of these complications on the bounce.

In the case where the bounce surface is very slippy, as it would be on ice for example, the ball slides throughout the bounce and is still sliding as it leaves the ground. The motion is as shown in figure 1.6. The coefficient of restitution has been taken to be 0.8 and the resulting reduction in vertical velocity after the bounce has lowered the angle of the trajectory slightly.

1.6

Figure 1.6. Bounce on a slippy surface.

In the more general case the ball slides at the start of the bounce, and the sliding produces friction between the ball and the ground. There are then two effects. Firstly the friction causes the ball to slow, and secondly the ball starts to rotate, as illustrated in figure 1.7. The friction slows the bottom surface of the ball, and the larger forward velocity of the upper surface then gives the ball a rotation.

If the surface is sufficiently rough, friction brings the bottom surface of the ball to rest. This slows the forward motion of the ball but, of course, does not stop it. The ball then rolls about the contact with the ground as shown in figure 1.8. Since the rotation requires energy, this energy must come from the forward motion of the ball. Finally, the now rotating ball leaves the ground.

1.7

Figure 1.7. Friction slows bottom surface causing the ball to rotate.

For the case we have considered it is possible to calculate the change in the horizontal velocity resulting from the bounce. It turns out that the horizontal velocity after the bounce is three fifths of the initial horizontal velocity, the lost energy having gone into rotation and frictional heating.

1.8

Figure 1.8. Sequence of events when the ball bounces on a surface sufficiently rough that initial sliding is replaced by rolling.

Television commentators sometimes say of a ball bouncing on a slippy wet surface that it has ‘speeded up’ or ‘picked up pace’. This is improbable. It seems likely that we have become familiar with the slowing of the ball at a bounce, as described above, and we are surprised when on a slippy surface it doesn’t occur, leaving the impression of speeding up.

1.9

Figure 1.9. At low angles the ball slides throughout the bounce, at higher angles it rolls before it leaves the ground.

Whether a ball slides throughout the bounce, or starts to roll, depends partly on the state of the ground. For a given surface the most important factor is the angle of impact of the ball. For a ball to roll there must be a sufficient force on the ground and this force increases with the vertical component of the velocity. In addition, it is easier to slow the bottom surface of the ball to produce rolling if the horizontal velocity is low. Combining these two requirements, high vertical velocity and low horizontal velocity, it is seen that rolling requires a sufficiently large angle of impact. At low angles the ball slides and, depending on the nature of the ground, there is a critical angle above which the ball rolls as illustrated in figure 1.9. With a ball that is rotating before the bounce the behaviour is more complicated, depending on the direction and magnitude of the rotation. Indeed, it is possible for a ball to actually speed up at a bounce, but this requires a rotation which is sufficiently rapid that the bottom surface of the ball is moving in the opposite direction to the motion of the ball itself as shown in figure 1.10. This is an unusual circumstance which occasionally arises with a slowly moving ball, or when the ball has been spun by hitting the underside of the crossbar.

1.10

Figure 1.10. A fast spinning ball can ‘speed up’ during the bounce.

Players can use the opposite effect of backspin on the ball to slow a flighted pass at the first bounce. The backspin slows the run of the ball and can make it easier for the receiving player to keep possession.

Bounce off the crossbar

When the ball bounces off the crossbar, the bounce is very sensitive to the location of the point of impact. The rules specify that the depth of the bar must not exceed 5 inches, and an inch difference in the point of impact has a large effect.

Figure 1.11(a) shows four different bounce positions on the underside of a circular crossbar. For the highest the top of the ball is 1 inch above the centre of the crossbar and the other positions of the ball are successively 1 inch lower.

Figure 1.11(b) gives the corresponding bounce directions, taking the initial direction of the ball to be horizontal and the coefficient of restitution to be 0.7. It is seen that over the 3 inch range in heights the direction of the ball after the bounce changes by almost a right angle.

1.11

Figure 1.11. Bounce from the crossbar. (a) Positions of bounce. (b) Angles of bounce.

As with a bounce on the ground, the bounce from the crossbar induces a spin. Calculation shows that a ball striking the crossbar at 30 miles an hour can be given a spin frequency of around 10 revolutions per second. This corresponds to the lowest of the trajectories in figure 1.11. For even lower trajectories the possibility of slip between the ball and bar arises.

When the ball reaches the ground the spin leads to a change in horizontal velocity during the bounce. For example, the 30 miles per hour ball which is deflected vertically downward is calculated to hit the ground with a velocity of about 26 miles per hour and a spin of 9 rotations per second. After the bounce on the ground the ball moves away from the goal, the spin having given it a forward velocity of about 6 miles per hour.

This, of course, is reminiscent of the famous ‘goal’ scored by England against Germany in the 1966 World Cup Final. In that case the ball must have struck quite low on the bar, close to the third case of figure 1.11. The ball fell from the bar to the goal-line and then bounced forward, to be headed back over the bar by a German defender. Had the ball struck the bar a quarter of an inch lower it would have reached the ground fully over the line.

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