# The Physics of Soccer

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Rolling Motion and Friction

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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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