The Physics of Soccer

The Physics of Soccer

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Rolling Motion and Friction

Suppose you kick a soccer ball without giving it any spin. Your foot, therefore, gives the ball an initial speed (v) and an initial angular speed of 0. Since grass is not frictionless, the ball initially slides across the field, then starts to rotate and, eventually, starts rolling without slipping. A soccer ball rolls without slipping when its center-of-mass speed equals its angular speed (around its center of mass). OK, now suppose you want to kick the ball so that it immediately starts rolling without slipping. How? You would give the ball "topspin" by striking the ball a distance (s) above an imaginary horizontal line that passes through the ball's center. But where? ANSWER: s=0.4R. You would strike the ball a little less than half the radius of the ball above its center line.

When two objects slide across one another, they exert a frictional force against each other. These forces are always tangent to the surfaces. A soccer ball and its interaction with the field is an example of this. The frictional force is opposite the direction that the ball is traveling. Physics gives us the following equation: f=mN for objects that slide against one another; where the frictional force (f) is equal to the upward "normal force" that the surface exerts on the ball (N) multiplied by the coefficient of friction (m). The coefficient of friction is not a constant, but will vary with the ball and surface type. The more friction there is between the ball and the field, the slower the ball will move after a bounce. Balls that skid, on the other hand, do not generate as much friction and subsequently do not slow down as much. So, the coefficient of friction tells us how fast (or slow) a ball will travel: The higher the coefficient, the slower the ball. A device similar to the Stimpmeter®, which is used to measure the "speed" of a golf green, could measure a soccer field's coefficient of friction by rolling a small ball on grass and measuring the distance it travels before stopping.

Projectile Motion

When projectile motion is treated in basic physics courses, the influence of air resistance is often neglected in the calculations and the trajectory of a projectile becomes a parabola where the horizontal velocity component is contant and the vertical component is subject to gravity. However, for someone watching a game of soccer, it is clear that the motion of a soccer ball is governed not only by gravity, but also by air resistance.

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The result from the simplified calculations are obviously not relevant, the range in reality being much shorter than predicted by simple theory and the real path often being quite asymmetric.  In general, the greater the air pressure, the greater the air resistance. As the soccer ball moves through the air, the air in front of it experiences a rise in air pressure and pushes the ball in the direction opposite its motion. While there are various other changes in air pressure around the ball's surface, this rising pressure in front of the ball remains largely unbalanced and it slows the ball down. The higher the air pressure was to start with, the greater its rise in front of the ball and the stronger the backward push of air resistance. Thus if you were to play soccer in the Rocky Mountains, where the air pressure is much less, you'd be able to kick the ball significantly farther.

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When trying to aim a header it may not go where you want or expect it to go. Why? Because it's all about vectors. Since the "angle of incidence equals the angle of reflection", try heading the ball towards where it was served. The combinations of vectors (the motion of your head , the motion of the ball, and the bounce -reflection- off of your head) will give the ball its final speed and direction.

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