Acceleration, g, and forces
The subjects of this chapter are all concerned with acceleration or deceleration of the ball. In order to give some intuitive feel for the accelerations and forces involved the accelerations will be expressed in terms of the acceleration due to gravity, which is written as g, and forces will be described by the force of an equivalent weight. Because most British people think of speeds in terms of miles per hour and weight in terms of pounds these units will be used. In scientific work the basic units are the metre, kilogram and second and in the final, theoretical, chapter we shall change to these units.
Objects falling freely under gravity have an acceleration of 22 miles per hour per second (9.8 metres per second per second), so in each second the vertical velocity increases by 22 miles per hour. Thus an acceleration of 220 miles per hour per second is 10g.
Forces will be given in pounds. For example a force of 140 pounds is equal to the gravitational force of 140 pounds weight (10 stone). The gravitational force on an object produces an acceleration g and, correspondingly, an accelera¬tion, g, of the object requires a force equal to its weight. Similarly, to accelerate an object by 10g, for example, requires a force equal to 10 times its weight.
1 yard = 0.91 metre
1 mile/hour = 1.47 feet/second
= 0.45 metre/second
1 pound = 0.45 kilogram
Usually the throw-in is used to pass the ball directly to a well placed colleague. The distance thrown is generally not great and the required accuracy is easily achieved by any player. A more difficult challenge arises when the ball is to be thrown well into the penalty area to put pressure on the opponent’s goal. To reach the goal-area calls for a throw approaching 30 yards, and long throws of this type often become a speciality of players with the necessary skill.
A short throw of, say, 10 yards needs a throw speed of around 20 miles per hour. Taking a hand movement of 1 foot the required force is typically 10-15 pounds.
A throw to the centre of the pitch, as illustrated in figure 3.1, requires a throw of almost 40 yards. In the absence of air resistance this challenging throw would require the ball to be thrown with a speed of 40 miles per hour. The effect of air drag increases the required speed to about 45 miles per hour.
To give the ball such a high speed the thrower must apply a large force over as long a path as possible. Although a short run up to the throwing position is helpful, both feet must be in contact with the ground during the throw. This limits the distance the arms can move. The back is initially arched with the ball behind the head, and the muscles of the body and arms are then used to push the ball forward and upward. For a long throw the ball remains in contact with the hands over a distance of about 2 feet. Taking this figure the average acceleration of the ball needed to reach 45 miles per hour is 34g. Since the ball weighs approximately a pound this means that the average force on the ball must be about 34 pounds; the maximum force will of course be some¬what larger.
Figure 3.1. Throw to centre of the pitch.
The record for the longest throw was achieved by the American college player Michael Lochnor, who threw the ball 52.7 yards in 1998. The record was previously held by David Challinor of Tranmere Rovers who reached 50.7 yards, and this throw remains the British record.
Goalkeepers often trust their throw rather than their kick. The ball can be quite accurately rolled or thrown to a nearby colleague. Sometimes the goalkeeper chooses to hurl the ball toward the half-way line rather than kick it, and an impressive range can be obtained in this way. Despite the use of only one arm these throws can carry farther than a throw-in. This is partly because of the longer contact with the ball during the throw, allowing the force to be applied for more time, and partly because of the greater use of the body muscles. The greater ease of obtaining the optimum angle of throw for a long range is probably another factor. For a long throw the hand remains in contact with the ball for about 6 feet, and the contact time for the throw is typically several times as long as for a throw-in.
A well-headed ball is struck with the upper part of the forehead and the ball essentially bounces from the head. The types of header are characterised by the way in which momentum is transferred between the head and the ball.
When a defender heads away a long ball his neck is braced and the bounce of the ball from his head transfers momentum to his body. Another situation in which momen¬tum is taken by the body is in the diving header. In this case the whole body is launched at the ball and it is the speed of the body which determines the resulting motion of the ball.
In more vigorous headers the muscles are used to thrust the head at the ball. This type of header is commonly used by strikers to propel a cross from the side of the pitch toward the goal. When the head strikes the ball, momentum is transferred to the ball and the head is slowed. Because the head weighs several times as much as the ball and because it is anchored at the neck the change in speed of the head through the impact is typically less than 10% of the speed given to the ball. In heading the ball the movement of the head is restricted to a few inches, and the velocity given to the ball is much less than that possible for a kick.
Sometimes the head is struck by an unseen ball, or before the player can prepare himself. It is then possible for all the ball’s loss of momentum to be transferred to the head. In a severe case of a 50 mile per hour ball, the head could be moved an inch in a hundredth of a second, the force on the head corresponding to an acceleration of 50g. Accelerations larger than this can lead to unconsciousness.
Wherever possible, goalkeepers aim to take charge of a ball close to goal by catching it. There are two circumstances where this is not possible. Firstly there is the ball which is flighted into a group of players near the goal and goalkeeper doesn’t have sufficient access to the ball to be confident of catching it. If he can he will then punch the ball as far away from the goal as possible. The punch is less powerful than the kick and the distance of movement of the fist is limited to about a foot. However, the ball bounces off the fist, taking a higher speed than the fist speed. Typically a range of about 20 yards is obtained, corresponding to a fist speed of about 20 miles per hour.
The second situation where a punch is called for is where a shot is too far out of the goalkeeper’s reach for a catch to be safely made and a punch is the best response. When the punch follows a dive by the goalkeeper, considerable accuracy is called for because of the brief time that a punch is possible. For example, a ball moving at 50 miles per hour passes through its own diameter in one hundredth of a second.
While the punch is usually the prerogative of the goal¬keeper, it is also possible to score a goal with a punch.
Goalkeepers make two kinds of catch. The simpler kind is the catch to the body. In this case most of the momentum of the ball is transferred to the body. Because of the comparatively large mass of the body the ball is brought to rest in a short distance. The goalkeeper then has to trap the ball with his hands to prevent it bouncing away.
In the other type ofcatch the ball is taken entirely with the hands. With regard to the mechanics, this catch is the inverse of a throw. The ball is received by the hands with its incoming speed and is then decelerated to rest. During the deceleration the momentum of the ball is transferred to the hands and arms through the force on the hands. The skill in this catch is to move the hands with the ball while it is brought to rest. Too small a hand movement creates a too rapid deceleration.
of the ball and the resulting large force makes the ball difficult to hold. The movement of the hands during the catch is nevertheless usually quite small, typically a few inches.
Taking as an example a shot with the ball moving at 50 miles per hour, and the goalkeeper’s hands moving back 6 inches during the catch, the average deceleration of the ball is 170g, so the transient force on the hands is 170 pounds, which is roughly the weight of the goalkeeper. The catch is completed in just over a hundredth of a second.
When a pass is received by a player the ball must be brought under control, and in tight situations this must be done without giving opponents a chance to seize the ball. The basic problem with receiving arises when the ball comes to the player at speed. If the ball is simply blocked by the foot, it bounces away with a possible loss of possession. The ball is controlled by arranging that the foot is moving in the same direction as the ball at the time of impact. The mechanics are quite straightforward - essentially the same as for a bounce, but with a moving surface. Thus, allowing for the coefficient of restitution, the speed of the foot can be chosen to be such that the ball is stationary after the bounce. It turns out that the rule is that the foot must be moving at a speed equal to the speed of the ball multiplied by e/(1 + e) where is the coefficient of restitution. If, say, the ball is moving at a speed of 25 miles per hour and the coefficient of restitution is 3, then the foot must be moving back at a speed of 10 miles per hour. This ideal case, where the ball is brought to rest, is illustrated in figure 3.3.
To receive a fast ball successfully it is not only necessary to achieve the correct speed of the foot, but also requires good timing. A ball travelling at 30 miles per hour moves a distance equal to its own diameter in about a sixtieth of a second, and this gives an idea of the difficulty involved. The player’s reaction time is more than ten times longer than this, showing that the art lies in the anticipation.
Figure 3.3. Controlling the ball.
Trapping the ball under the foot presents a similar challenge to that of receiving a fast pass in that the time available is very brief. A particular need to trap the ball arises when it reaches the player coming downwards at a high angle. To prevent the ball bouncing away the foot is placed on top of it at the moment of the bounce. Easier said than done.
As the ball approaches, the foot must be clear of it so that the ball can reach the ground. Then, when the ball reaches the ground the foot must be instantly placed over it, trapping the foot withdrawn at impact ball comes to rest ball between the foot and the ground. This is sometimes done with great precision. The ‘window’ of time within which trapping is possible is determined by the requirement that the foot is placed over the ball in the time it takes for the ball to reach the ground and bounce back up to the foot, as illustrated in figure 3.4.
Figure 3.4. Trapping the ball requires a well timed placement of the foot.
We can obtain an estimate of the time available by taking the time for the top of the ball to move downwards from the level of the foot and then to move upwards to that level again. The upward velocity will be reduced by the coefficient of restitution but for an approximate answer this effect is neglected. If the vertical distance between the ball and the foot at the time of bounce is, say, 3 inches then taking a hundredth of a second for the duration of the bounce, a ball travelling at 30 miles per hour will allow about a fiftieth of a second to move the foot into place. As with receiving a fast pass, anticipation is the essential element.
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