The ball is kicked in a variety of ways according to the circum-stances. For a slow accurate pass the ball is pushed with the flat inside face of the foot. For a hard shot the toes are dipped and the ball is struck with the hard upper part of the foot. The kick is usually aimed through the centre of the ball, but in some situations it is an advantage to impart spin to the ball. Backspin is achieved by hitting under the centre of the ball, and sidespin by moving the foot across the ball during the kick.
For a hard kick, such as a penalty or goal kick, there are two basic elements to the mechanics. The first is the swinging of the leg to accelerate the foot, and the second is the brief interaction of the foot with the ball. Roughly, the motion of the foot takes a tenth of a second and the impact lasts for a hundredth of a second.
For the fastest kicks the foot has to be given the maximum speed in order to transfer a high momentum to the ball. To achieve this the knee is bent as the foot is taken back. This allows the foot to be accelerated through a long trajectory, producing a high final speed. The muscles accelerate the thigh, pivoting it about the hip, and accelerate even faster the calf and the foot. As the foot approaches impact with the ball the leg straightens, and at impact the foot is locked firmly with the leg. This sequence is illustrated in figure 2.1.
If the interaction of the foot with the ball were perfectly elastic, with no frictional energy losses, the speed given to the ball would follow simply from two conservation laws. The first is the conservation of energy and the second is the conser-vation of angular momentum. These laws determine the fall in speed of the foot during the impact, and the resulting speed of the ball. If, further, the mass of the ball is taken to be negli¬gible compared with the effective mass of the leg, the speed of the foot would be unchanged on impact. In this idealised case, the ball would then ‘bounce’ off the foot and take a speed equal to twice that of the foot.
Figure 2.1. In a fast kick the upper leg is driven forward and the lower leg whips through for the foot to transfer maximum momentum to the ball.
In reality the leg and the foot are slowed on impact and this reduces the speed of the ball. Frictional losses due to the deformation of the ball cause a further reduction in speed. This reduction can be allowed for by a coefficient of restitution in a similar way to that for a bounce. When these effects are taken into account it turns out that at the start of the impact the foot is moving at a speed about three-quarters of the velocity imparted to the ball. This means that for a hard kick the foot would be travelling at more than 50 miles per hour.
It was seen in figure 2.1 that in a hard kick the thigh is forced forward and the calf and the foot are first pulled forward and then swing through to strike the ball. The mechanics of the process can be illustrated by a simple model in which the upper and lower parts of the leg are represented by rods and the hip and knee are represented by pivots, as illustrated in figure 2.2. Let us take the upper rod to be pulled through with a constant speed and ask how the lower rod, representing the lower leg, moves.
Figure 2.2. Model in which the upper and lower parts of the leg are represented by two pivoting rods, the upper of which is driven around the (hip) pivot.
Figure 2.3 shows what happens. Initially the lower rod is pulled by the lower pivot and moves around with almost the same speed as the pivot. However, the centri¬fugal force on the lower rod ‘throws’ it outward, making it rotate about the lower pivot and increasing its speed as it does so. As the upper rod moves round, the lower rod ‘whips’ around at an increasing rate and in the final stage illustrated the two rods form a straight line. The whipping action gives the foot of the lower rod a speed about three times that of the lower (knee) pivot.
Figure 2.3. The lower rod is pulled around by the upper rod and is thrown outward by the centrifugal force, accelerating the foot of the rod.
This model represents quite well the mechanics of the kick. The motion illustrated by the model is familiar as that of the flail used in the primitive threshing of grain, and is also similar to that of the golf swing. When applied to golf the upper rod represents the arms and the lower rod represents the club.
Since students of elementary physics are sometimes confused by the term centrifugal force used above, perhaps some comment is in order. When a stone is whirled around at the end of a string it is perfectly proper to say that the force from the string prevents the stone from moving in a straight line by providing an inward acceleration. But it is equally correct to say that from the point of view of the stone the inward force from the string balances the outward centrifugal force. This description is more intuitive because we have experienced the centifugal force ourselves, for example when in a car which makes a sharp turn.
Forces on the foot
During the kick there are three forces on the foot, as illustrated in figure 2.4. Firstly, there is the force transmitted from the leg to accelerate the foot towards the ball. Secondly, and particu¬larly for a hard kick, there is the centrifugal force as the foot swings through an arc. The third force is the reaction from the ball which decelerates the foot during impact.
Figure 2.4. The three forces on the foot during a kick.
To see the magnitude of these forces we take an example where the foot is accelerated to 50 miles per hour over a distance of 3 feet. In this case the force on the foot due to acceleration is 30 times its weight and the centrifugal force reaches a somewhat greater value. On impact with the ball the foot’s speed is only reduced by a fraction, but this occurs on a shorter timescale than that for its acceleration and the resulting deceleration force on the foot during impact is about twice the force it experiences during its acceleration.
The scientific unit of power is the watt, familiar from its use with electrical equipment. It is, however, common in English speaking countries to measure mechanical power in terms of horse-power, the relationship being 1 horse-power =750 watts. The name arose when steam engines replaced horses. It was clearly useful to know the power of an engine in terms of the more familiar power of horses. As would be expected, human beings are capable of sustaining only a fraction of a horse¬power. A top athlete can produce a steady power approaching half a horse-power.
The muscles derive their power from burning glucose stored in the muscle, using oxygen carried from the lungs in the bloodstream. The sustainable power is limited by the rate of oxygen intake to the lungs, but short bursts of power can use a limited supply of oxygen which is immediately avail¬able in the muscle. This allows substantial transient powers to be achieved. What is the power developed in a kick?
Both the foot and the leg are accelerated, and the power generated by the muscles is used to produce their combined kinetic energy. For a fast kick the required energy is developed in about a tenth of a second, and the power is calculated by dividing the kinetic energy by this time. It turns out that about 10 horse-power is typically developed in such a kick.
The curled kick
To produce a curved flight of the ball, as illustrated in figure 2.5, it is necessary to impart spin to the ball during the kick. The spin alters the airflow over the ball and the resulting asymmetry produces a sideways force which gives the ball its curved trajectory. We shall look at the reason for this in chapter 4. Viewed from above, a clockwise spin curls the ball to the right, and an anticlockwise spin to the left.
Figure 2.6(a) shows how the foot applies the necessary force by an oblique impact. This sends the ball away spinning and moving at an angle to the direction of the target. The ball then curls around to the target as shown in figure 2.6(b). The amount of bend depends upon the spin rate given to the ball, and the skill lies in achieving the required rotation together with accuracy of direction.
Figure 2.5. Curved flight of spun ball.
Only a small part of the energy transferred to the ball is required to produce a significant spin. If the energy put into the spin in a 50 mile per hour kick is 1% of the directed energy, the ball would spin at 4 revolutions per second.
Figure 2.6. To produce a curved flight the ball is struck at an angle to provide thenecessary spin.
The directional accuracy of a kick is simply measured by the angle between the direction of the kick and the desired direction. However, it is easier to picture the effect of any error by thinking of a ball kicked at a target 12 yards away. This is essentially the distance faced by a penalty taker. Figure 2.7 gives a graph of the distance by which the target would be missed for a range of errors in the angle of the kick.
Figure 2.7. Error at a distance of 12 yards resulting from a given error in the direction of the kick.
There are two sources of inaccuracy in the kick, both arising from the error in the force applied by the foot. The first contribution comes from the error in the direction of the applied force and the second from misplacement of the force. These two components are illustrated separately in figure 2.8.
Figure 2.8. The kick can have errors in both direction and placement on the ball. In (a) and (b) these are shown separately.
It is seen from figure 2.7 that placing the ball within one yard at a distance of 12 yards requires an accuracy of angle of direction of the ball of about 5°. The required accuracy of direction for the foot itself is less for two reasons. Firstly, the ball bounces off the foot with a forward velocity higher than that of the foot by a factor depending on the coefficient of restitution and, secondly, part of the energy supplied by the sideways error force goes into rotation of the ball rather than sideways velocity. For a 5% accuracy of the ball’s direc¬tion these two effects combine to give a requirement on the accuracy of the foot’s direction more like 15°. The geometry of this example is illustrated in figure 2.9.
Figure 2.9. When there is an error in the direction of the applied kick the error in the direction of the ball is much less.
The accuracy of the slower side-foot kick is much better than that of the fast kick struck with the top of the foot. Because of the flatness of the side of the foot the error from placement of the foot on the ball is virtually eliminated, leaving only the error arising from the direction of the foot. This makes the side-foot kick the preferred choice when accuracy is more important than speed.
The fastest kicks are normally unhindered drives at goal, the obvious case being that of a penalty-kick struck with maxi¬mum force. To take an actual case we can look at the penalty shoot-out between England and Germany in the 1996 European Championships. Twelve penalty-kicks were taken and the average speed of the shots was about 70 miles per hour. The fastest kick was the last one, by Moller, with a speed of about 80 miles per hour. Goal-kicks usually produce a somewhat lower speed, probably because of the need to achieve range as well as speed.
It is possible to obtain a higher speed if the ball is moving towards the foot at the time of impact. The speed of the foot relative to the ball is increased by the speed of the incoming ball and consequently the ball ‘bounces’ off the foot with a higher speed. When allowance is made for the unavoidable frictional losses and the loss of momentum of the foot, the increment in the speed of the ball leaving the kick is about half the incoming speed of the ball. Taking a kick which would give a stationary ball a speed of 80 miles per hour we see that a well-struck kick with the ball moving toward the player at 40 miles per hour, which returns the ball in the direction from which it came, could reach 80 +140 = 100 miles per hour, as illustrated in figure 2.10.
Figure 2.10. A kick produces a higher ball speed when the ball is initially moving toward the foot. In this example the kick is such that it would give a stationary ball a speed of 80 miles per hour.
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