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Physic Behind A Curveball In Soccer

More than 240 million people around the world play soccer regularly according to the Federation Internationale de Football Association. The game has evolved from kicking a rudimentary animal-hide ball around into the World Cup sport it is today. Researches, trace soccer’s discovery to  more than 2,000 years ago in ancient China. Greece, Rome, and parts of Central America also claim to have started the sport; but it was England that transitioned soccer, or what the British and many other people around the world call “football”. Soccer’s or even “football’s” first uniform rule forbids to trip an opponent or to touch the ball with your hand. As the years goes by, the world evolves and more rules are applied to the game. For example, the penalty kick was introduced in 1891. FIFA became a member of the International Football Association Board of Great Britain in 1913. Red and yellow cards were introduced during the 1970 World Cup finals.

The most recent changes include goalkeepers being banned from handling deliberate back passes in 1992 and tackles from behind becoming red-card penalties in 1998. In 1997, in a game between France and Brazil, a young Brazilian player name Roberto Carlos, scored with no direct line to the goal in a free kick and eventually made it to the hall of fame, because of his legendary kick. Carlos decided to curve the ball and when he kicked it, it went flying wide over the players but before it went out of bounce, it moved left and entered the goal. In the line below, we will see the physic behind how and why it occurred.

According to Newton’s first law of motion: an object will move in the same motion and velocity until a force is apply on it. When Carlos hit the ball, he gave it motion and velocity. The trick was in the spin, and to achieve it, he putted his feet on the lower half of the ball sending it high into the right but also rotating around its axis. While the ball was in its direct route, the air was flowing on both sides and slowed it down. On one side, the air moved in the opposite direction to the ball’s spin, causing increased pressure, while on the other side, the air moved in the same direction as the spin, creating an area of lower pressure. That difference made the ball curve towards the lower pressure zone. This phenomenon is called the Magnus effect.

In the figure below, we went a little deeper into it to show the algebra behind it. The way the force is generated is complicated. the magnitude of the force F depends on the radius of the ball b, the spin of the ball s, the velocity V of the kick, the density r of the air, and an experimentally determined lift coefficient Cl.F = Cl * 4 /3 * (4 * pi^2 * r * s * V * b^3) Pi is equal to 3.14159 .. the ratio of the circumference to the diameter of a circle.

When the ball is in the air, the force acts on it and deflected it along his flight path. When the forces act on the ball, which slow it down and change the magnitude and direction of the force, we have a constant force always acting perpendicular to the flow direction. The resulting flight path is a circular arc. On the figure, we see the trajectory of the soccer ball as it moves from right to left. The radius of curvature R of the flight path depends on the velocity V of the kick and the acceleration a produced by the side force.

R = V^2 / a

The acceleration can also be defined by Newton’s second law of motion using the force for a spinning ball and the mass M of the ball. a = F / m.

The radius of curvature depends on the force and all the element that affect the force will also affect the target R = (3 * m * V) / (16 * Cl * r * s * b^3 * pi^2)

This equation can predict the trajectory of a spinning ball. Higher spin S produces a smaller radius of curvature R and a sharper curve. Higher velocity V produces a larger radius of curvature and a straighter curve. A ball with a smaller mass, like a ping-pong ball, has a lower radius of curvature and curves more. At higher altitudes, the density R is lower producing a larger radius of curvature and a straighter path.

With the radius of curvature and the distance of the kick D we can also calculate the distance that the ball is deflected (Yd) along the flight path. There is a right triangle formed by the radius of curvature R, the distance D at the top, and the radius of curvature minus the deflection distance R – Yd on the right. We can then use the Pythagorean Theorem to relate the sides of this triangle:

R^2 = D^2 + (R – Yd)^2R^2 – D^2 = (R -Yd)^2sqrt(R^2 – D^2) = R -YdYd = R – sqrt(R^2 – D^2)

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