Football Physics


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Any action in nature responds to the exact laws of physics. The football game does not escape this fact. Goal shots, penalties, ball passes, sprints, jumps, and headers are regulated by physical equations, above all in sports. Many soccer teams, while training, spend a few hours learning the rules to apply them later during the match. This article describes some interesting experiments, even if the complete observation of all phenomena of the game requires writing thousands of pages of theory. It looks like a video game, but in reality it is perhaps the most complicated software out there, especially one that contains sports content. In them, the creators have programmed various physical laws to bring the behavior of the ball and players closer to reality. The latest releases of the most popular soccer games look like real events. It is not enough to be a good programmer to make a video game. It is essential to have a thorough knowledge of physics and mathematics to write very complex algorithms that coordinate the movement of the ball and every part of a player’s body. The force of gravity is decisive; the evolution of most physical events depends on the force of gravity. He is able to decisively influence almost all the events that happen on the field. On another planet, soccer moves will be very different from those on Earth, with the same forces used. The first effect of Earth’s gravity can be observed with a goalkeeper’s throw (see Figure 1). Using his legs, he sends the ball (which is not a ball) a great distance. Sometimes, goalkeepers score in the opponent’s goal, exactly with their raises. When the ball is kicked, it must oppose two forces: the force of gravity that pushes it down, and the resistance of the air (considered a liquid), which acts in the opposite direction of the throw, which suppresses the run. Figure 1: The force of gravity also has a decisive effect on the goal kick. Assume the following static and dynamic data: Field length: about 70 m Rubber ball: 600 g mass The ball is drawn slightly larger to show in the figure the initial velocity X: 19.9 m / s Initial velocity r: 19.8 m / s Total velocity: 28 m / s Angle: 45 degrees Remember that the acceleration of gravity is not the same for the entire universe but changes according to different parameters. For the planets and satellites used in the test, the values ​​are as follows: Earth: 9.81 m / s 2 Moon: 1.62 m / s 2 Jupiter: 24.79 m / s 2 Sun: 274 m / s 2 After the goalkeeper leaves, the ball will lose access To different distances and heights (see diagram in Figure 2), at different times, and specifically: on the ground the distance traveled by the ball: about 70 meters Maximum altitude: about 19 meters Time spent: about 4 seconds on the moon Distance covered by the ball: About 270 meters Maximum altitude: about 85 meters Time spent: About 20 seconds on Jupiter Distance covered by the ball: About 30 meters Maximum altitude: About 7 meters Time spent: Approx. 1.5 seconds at the sun Distance traveled by the ball: About 2.5 meters Maximum altitude : Approx 1 meter Time spent: approx 0.1 sec. Physical behaviors of the ball vary, also depending on the reference system used. Figure 2: Diagram of the X and Y position of the ball after a goalkeeper’s shot, on the offside planet: a perfect alignment issue Offside reporting is often very complex, and some errors are not attributed to the goalkeeper but to visual and perspective errors. Their errors are large in number, and although they are professionals, their supervision will be inevitable because they result from a physical limitation of human visual perception and not due to their lack of aptitude. The line men tend to always position themselves within a meter of the last cannons and thus do not fully align with the suspected offside position. The errors are precisely due to this inaccuracy: if you examine the action from this visual angle, the images of the action are affected by an error in perspective. As can be seen in Figure 3, if the defender is between the attacker and the assistant referee, he “believes” the situation is erratic even if the players are on the line. On the other hand, if the striker is between the assistant referee and the defender, then it appears that the players are the same and offside is not declared. The only way to avoid omissions is to analyze sequences from a valid control point, even if this solution presents significant technical and operational difficulties. Figure 3: Incorrect positioning of the Referee Server can negatively affect the intrusion assessment. Penalty kick: a matter of too high a speed many physical concepts involve in the penalty kick, but scoring a goal in this way is often a matter of luck (see Figure 4). When the attacker’s shot is too strong and gives the ball an average velocity of about 110 km / h (equivalent to about 30 m / s), the goalkeeper, who is 11 meters from the ball at the time of the kick, does not have time to follow and predict the trajectory of the ball And put himself correctly to block the goal. The penalty kick consists of at least two components: one horizontal and the other vertical. To maximize power, some players like to keep the ball low and angled with great precision. On average, it has the following dynamic data: Initial velocity: 30.6 m / s (110.16 km / h) Final velocity: 30 m / s (108 km / h) Elapsed time from shot to goal: 0.3 seconds. Ball: 2 m / s2 Force: 1.2 N Linear Kinetic Energy: 275 J The goalkeeper should have very fast reflexes, with reaction times of less than 40ms. In the short interval of 0.3 seconds, the goalkeeper should (theoretically) take the following actions: Observe the movements of the player taking the penalty kick. Monitor the direction of the ball. Catching the ball is a set of actions that must be performed in a very short time and, in most cases, the goalkeeper is instinctively drenched. Only devices can do it right. This is also the reason why some goalkeepers have decided to position themselves above the goal line, only to get a few hundredths of a second to respond. A striker has the advantage over a goalkeeper in knowing the strength and direction of the shot in advance. Unable to cope with a lot of movement in a short time, goalkeepers dive into anyway, trying to guess the angle of the shot in hopes of having a greater chance of guessing direction. That is why the penalties were taken about 88% of those taken. Figure 4: The penalty kick is a combination of skill and luck. Goal Probability Relative to Soccer Goal Size The probability of scoring a goal also depends on goal size. The larger the visual area, the greater the possibility of entering the ball. Figure 5 assumes a “useful” imaging area the size of the red surface. The Monte Carlo algorithm simulates shots taken in this red zone and calculates the number of shots taken into the goal. Figure 5: The probability of scoring a goal also depends on the size of the goal. In the first simulation, the dimensions of the useful area and the door are as follows: Useful area (red): 18.32mx 5m Target area: 7.32mx 2.44m With one million rounds, the chance of scoring is 19.48%. The simulation, of course, does not take into account the player’s skills but only assesses the impact of the ball in the goal area. The simulations are random, but for the law of large numbers the probabilities of the results follow exactly the theoretical probabilities, as shown by the simulation graphs in Fig 6. In other words, the distribution of events is constant. Figure 6: Graph showing the number of targets scored (out of a million rounds) in many different simulations. In the second simulation, the door size was reduced as follows: after 1 million virtual shots, the probability of recording decreased to 13.65% (see graph in Fig.7). The following algorithm, very simple, has been implemented with Pari / GP software but is adaptable to any other programming language. Figure 7: The probability of the goal is proportional to the size of the soccer goal. {area_x = floor (18.32 * 100); area_y = floor (5.00 * 100); football_goal_x = floor (7.32 * 100); football_goal_y = floor (2.44 * 100); tot_goals = 0; Limit = 100,000; For (k = 1, limit, rand_x = random (area_x); rand_y = random (area_y); cond1 = rand_x> ((area_x – football_goal_x) / 2); cond2 = rand_x <(football_goal_x + football) ((area_x --_goal_x)) / 2)); cond3 = rand_y


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