We may study the scientific aspect of soccer and thereby increase our understandings on this sport. Here, let us study the basic facts and principle of soccer by scientific analysis.
The Soccer Ball
"There can be no game of football without a ball and the better the ball, the better the game." [i]
The birth of contemporary football can be traced back to the 19th century. It began in 1863 in England when the world's first football association --- The Football Association in England --- was founded. At that time, most of the soccer balls were made of pig or ox bladder encased with leather. The major deficiencies of these balls were the difficulty of making a regular shape ball and maintaining the shape of the ball when use.
By 1900s, most balls were made of rubber bladder instead of the animals bladder while leather was still used as the covering material of balls. The use of the rubber bladders is to ensure that the ball can remain hard and avoid losing its shape. Thus these balls would bounce easier and yet could be kicked. However, leather will absorb water if it is wet. Therefore, the leather casing ball would absorb water during rain and become very heavy, which caused many head injuries at that time.
1910 soccer ball [ii]
1950 soccer ball [ii]
2004 Euro Cup ball [ii]
In the late 1980s, the leather casing ball was replaced by totally synthetic ball in soccer competitions. The covering material of the totally synthetic ball is synthetic leather made from polymer. For high quality ball, the casing is made of the synthetic leather panels stitched together through pre-punched holes by waxed threads. The bladder of a totally synthetic ball is usually latex or butyl bladder. The ball is then inflated by pumping air into its bladder through a tiny hole on the casing. The totally synthetic ball could resist water absorption and reliably maintain its shape.
Nowadays, the official soccer rules called the "Laws of the game", which are maintained by the International Football Association Board (IFAB), specify the qualities of the ball used in soccer matches. According to the laws, the soccer ball should satisfy the following descriptions:
|How Things Work|
Flight of Ball in Air
Suppose a soccer ball is kicked and then flies into the sky with no spin. It then travels through the air before touching the ground. However, the air resists the ball to move through it. Thus the horizontal displacement of the ball in air would be much shorter than that in case of vacuum. For example, a soccer ball kicked without spin at a speed of 35 m/s and at an angle of 45 degrees to the ground would transverse a horizontal distance of 66m in air; while the distance would be 125m in vacuum. This example clearly illustrates the importance of air resistance in the motion of the ball. So let us investigate how air affects the flight of a ball.
When a ball is moving through the air, it experiences a drag force exerted by the
passing air. For simplicity, we study this problem by taking the ball being
stationary with air flowing over it. Indeed, there are two types of airflow, known
as laminar airflow and turbulent
airflow, whose behavior are remarkably different. In laminar airflow,
viscosity dominates the flow, keeping it smooth and orderly;
while in turbulent airflow, inertia dominates the airflow, ripping it apart into
swirling eddies. Whether the airflow is laminar or turbulent depends on the size
of the obstacle that the air encounters and the speed of the air passing the
|How Things Work|
2. Laminar airflow for a slow-moving ball
Let us first consider a ball moving slowly through the air. If the ball moves slowly enough, the airflow is laminar. The below figure shows the laminar airflow around a slow-moving ball. In the figure, the lines indicating the airflow, along which each small piece of air follows, are called the streamlines. For laminar flow, the air between two streamlines remains between those streamlines throughout its motion.
The airflow pattern around a soccer ball in case of laminar airflow.
Since the flow is laminar, the air separates neatly in front of the ball and then recombines again after passing the ball, leaving a smooth and free-of-turbulence wake --- an air trail left behind the ball. So the streamlines bend outward on approaching the ball (point A) and bend inward again on leaving the ball (point C). There must be change in air's speed and pressure so that the air can change its flowing direction. Specifically, air experiences a drop in speed and a rise in pressure in the region where it bends away from the ball's surface (near point A and C); while air experiences a rise in speed and a drop in pressure in the region where it bends toward the ball's surface (near points B and D). Thus, as indicated by the widely spaced streamlines, the air moves slower at the high-pressure region near the front and back of the ball (points A and C). Moreover, at the low-pressure region near the sides of the ball (points B and D), the air moves faster as indicated by the narrowly spaced streamlines. It is consistent with the Bernoulli's principle that the speed of a stream of moving fluid will increase if it flows into a narrower channel where the pressure is decreased.
The pressure and speed of the air flowing at different region around a soccer ball.
It may seem strange that why the air can flow from the low pressure region near the sides of the ball to the high pressure region at the back of the ball. It is because the low-pressure air sweeping the sides of the ball have enough forward momentum (= mass x velocity) to move to the back of the ball. Thus the airflow slows down when it move from the low pressure region to the high pressure region.
Although there is change in the speed and pressure of the air on moving past the ball, the airflow is symmetric and the forces exerted by the air pressure are also symmetric. The high pressure in front of the ball is balanced by the high pressure at the back of the ball. Thus the pressure forces cancel out completely and so the ball experiences no net pressure force. As a result, the only force that the passing air acting on the ball is the viscous drag - the friction-like downstream force due to the sliding of the viscous air across the ball's surface.
3. Turbulent airflow for a fast-moving ball
When the ball moves faster, the airflow become turbulent. Unlike in laminar flow, the air pressure is not distributed symmetrically around the ball in the turbulent flow. Thus the pressure forces cannot balance with each other and the ball experiences pressure drag - downstream force exerted by unbalanced pressures in the moving air. In other words, it is the imbalance pressure which slows down the motion of the ball in turbulent airflow.
The airflow would become turbulent so that the ball would experience pressure drag when the Reynolds number exceeds about 2000. Thus the Reynolds number gives an indication whether the airflow is turbulent or laminar. Researchers showed that the Reynolds number depends on the ball's size and speed as well as the viscosity and density of air. At low Reynolds number, the air's viscosity supports laminar flow over the ball's surface. At high Reynolds number, the air's inertia prevents it curving around the ball's surface and air's viscosity triggers the air to swirl about.
How the air's viscosity triggers the swirling air in turbulent flow? We have to look at the air near the ball's surface. Viscous force within the air keeps the air touching the surface to remain at rest and also slows down the nearby air to form the boundary layer. In the boundary layer, the air moves more slowly and have less total energy that the freely flowing air farther away from the ball. Outside the boundary layer, the viscosity of the air can be neglected.
The boundary layer around a fast-moving soccer ball.
When the air flows toward the back of the ball, it travels from a low pressure region to a high pressure region. Although the air outside the ball have enough total energy to reach the back of the ball under the unfavorable pressure change, the air in the boundary layer does not. At high Reynolds number, the viscous forces between the freely moving air and boundary layer is not strong enough to push the boundary layer moving to the back of the ball. Eventually, the boundary layer stops and reverses directions. The airflow would be messed up by this boundary layer reversal that the freely flowing air separates from the surface and the separated airflow produces eddies behind the ball. These eddies are then confined together to form a turbulent wake similar to that left behind a ship moving through the water.
Eddying wakes are formed after a fast-moving soccer ball.
Because of this turbulent wake, the air behind the ball no longer slows down and its pressure no longer rises. Thus the air pressure in front of the ball is higher than that behind the ball. Hence, the ball experiences pressure drag pushing it backward for turbulent airflow. The pressure drag is the main source of the drag experienced by a ball moving through the air for turbulent airflow. Alternatively, we can say that the slowing of the ball is due to the loss in the kinetic energy of the ball for the extra kinetic energy of the eddying air in the turbulent wake.
5. What will happen for a very-fast-moving ball?
If the speed of the ball is further increased, the freely moving air will separate from the ball's surface earlier since the Reynolds number increases further. That is to say, the separation points move to a more upstream position. As a result, the turbulent wake become wider resulting in a greater pressure drag on the ball. In brief, the air drag will increase if the speed of the ball increases for a fast-moving ball. However, if the speed of the ball is increased so much that exceeding the critical speed, the drag forces would behave in a very different manner as a function of the air's speed.
When a ball travels fast enough that its Reynolds number exceeds about 100000, the boundary layer itself also becomes turbulent. Since the air in the turbulent boundary layer have more kinetic energy than that in the laminar boundary layer, the turbulent boundary layer can move to a farther position at the back of the ball before stopped by the rising air pressure. Thus the freely flowing air outside the turbulent boundary layer would separate at a more downstream position leading to a smaller wake and reduced drag. So there is a sudden drop in air resistance when the boundary layer changes from being laminar to turbulent.
Eddying wakes formed after a soccer ball would be larger if the ball moves with a faster speed.
Note that both v1 and v2 are smaller than the critical speed.
The turbulent boundary layer gives rise to a reduced drag if the ball's speed exceeds the critical speed.
The critical speed is defined to be the speed of the ball at which the boundary layer become turbulent and thus the air drag is reduced. Above the critical speed, the drag force falls with increasing speed until it exceeds another critical value. From the below figure, we can see that the critical speed of the soccer ball is around 20 m/s.
The air drag on a soccer ball as a function of its speed [v].
Any seams or irregularities on the sphere could "encourage" the formation of a turbulent boundary layer around the ball. For such case, the critical speed would be reduced and thus the air drag would be less for a fast moving ball with rough surface. That's why a tennis ball has fuzz and a golf ball has dimples.
In summary, we have found that the air flowing around a non-spinning ball moving through the air exhibits different properties in case of different Reynolds number as summarized in the table below. As a result, the air drag experienced by a ball has different characteristics if the ball moves with different speed.
Curving of Ball in Air
|How Things Work|
|In the previous section, we have investigated how the air drag affects the motion
of a soccer ball traveling through the air. Indeed, air drag is not the only
possible force exerted by air sweeping past a moving ball. It is also possible for
the passing air to exert lift forces acting perpendicular to the airstream on a
moving ball. Unlike drag force, lift force pushes the ball sideway to either
upward or downward directions which causes the ball to curve in flight.
The "banana shot", a ranged shot curling on its way to the goal, is a very useful trick for deceiving the defensive walls and fooling the goalkeepers so that the ball can score. In soccer matches, it's not rare for player to kick the "banana shot". For example, David Beckham is famous for being able to kick the "banana shot" to get score for his team. If you are careful enough, you probably notice that the ball is spinning in all the "banana shots". In fact, the ball must be spinning rapidly in order to produce a curved flight. The phenomenon that a rapidly spinning ball moving in air would be deflected by the resultant lift force to follow a curved path is called the Magnus effect. Let us take a look on how a spinning ball interacts with the surrounding air to give rise for the Magnus effect.
Suppose a ball is kicked to rotate clockwise rapidly about an axis perpendicular to its moving direction during its motion to the left (see below figure). Then the ball would experience two lift forces, the so-called Magnus force and the wake deflection force, pushing it upward when it travels through the air. Thus the ball moves with a curved path to reach the goalkeeper's right.
A soccer ball is shot to spin clockwise during its motion to the goalkeeper.
The Magnus force, discovered by the German physicist H. G. Magnus (1802 - 1870), is due to the interaction between the spinning ball and the viscous air. For a spinning ball, the passing air is moving in the same direction as the surface of contact on one side of the ball while it is moving in an opposite direction as the surface of contact on the other side. Therefore, the relative speed of the air is smaller at the side where it flows along with the rotating surface of the ball. Thus the Reynolds number is also smaller at this side. Hence, as explained above, the viscous air will separate from the ball's surface at a more downstream position on the side that the air moves in the same direction as the rotating surface. In contrast, on the other side of the ball, the point of separation is much earlier because the Reynolds number is larger over there. (Of course, we have assumed that the relative speed of the air does not exceed the critical value on both sides of the ball.) As a result, the airflow pattern around the spinning ball is not symmetric. The moving direction of the airstream is twisted to the side that the air moves opposite to the rotating surface. Moreover, the turbulent wake is also deflected toward this side. According to the Newton's 3rd law of motion, there is a reaction force that the ball pushes the airstream as well as the turbulent wake to the opposite side. The Magnus force is the resultant sideway reaction force due to the deflection of the air flow while the wake deflection force is the resultant sideway reaction force due to the deflection of the turbulent wake. The below figure illustrates how these two force arisen for a spinning soccer ball.
The Magnus force and wake deflection force on a soccer ball which is spinning clockwise.
We should remind that both the Magnus force and the wake deflection would deflect the spinning ball to the same direction. Besides, it can be shown that these lift forces would be maximum if the axis of rotation of the ball is perpendicular to the line of motion of the ball while these forces would be zero if the axis of rotation is parallel to the line of motion.
Physics of injuries caused by collision
Almost everybody would agree that the aim of every sport player is winning the
game through competitions. Thus, collision occurs frequently in many sport
competitions including soccer matches. For example, a soccer player's feet hit the
ground when he/she runs on the field. It is also not rare that a soccer player
carrying the ball to approach the opponent's goal smashes with the defending
players of the opponent's team. In addition, player sometimes gets score by using
his/her head to hit the ball into the goal. However, if the collision is too
strong, players might hurt by minor injuries like cuts, bruises, abrasions,
sprains and twists. For more violent collision, players might have more severe
injuries like broken clavicles, cracked ribs, torn knee alignments, smashed noses,
fractured jaws, pulled muscles and so on. In the most severe cases, players might
suffer from paralysis or death due to either broken spines, snapped necks or
A player may hurt if he strikes the soccer ball with his head [viii]
In the last two decades, researchers have paid much efforts in developing sports medicine and protective equipment that prevent injuries and speed up athlete's recovery. However, even the most sophisticated medical techniques and equipment can't guarantee that an injured athlete can have total recovery. Therefore, it is necessary for us to learn how collisions are governed by physics and human physiology in order to avoid injuries in sport competitions such as soccer matches.