Mathematics of kicking a field goal
The mathematics of kicking a field goal in American football involves a blend of probability, statistics, and physics, significantly influencing a coach's decision-making process. A field goal scores three points when the ball is successfully kicked through a goal comprised of a crossbar and two upright posts, each with specific dimensions. The trajectory of the ball typically follows a parabolic path, influenced by factors such as wind, rain, and altitude, which can alter its distance and height. Coaches analyze extensive data from various conditions and venues to assess the success probability of kicks, considering both their team's performance and opponents' defenses. This data-driven approach has gained prominence in modern sports, paralleling the rise of analytical methods like those popularized in the "Moneyball" era. As a result, kickers have improved in both accuracy and distance, with long-range kicks becoming more common and expected. The evolution of field goal kicking exemplifies how mathematical modeling and data analytics play crucial roles in enhancing athletic performance and strategic decisions in football.
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Mathematics of kicking a field goal
SUMMARY: Probability, statistics, and physics govern a football coach’s decision to attempt a field goal.
Football is one of the most popular sports in the United States, and successfully kicking a field goal is one of several ways that a football team scored points. Every kick that made was unique, and a solid knowledge of mathematics was necessary to tailor the kick to be successful. The path of the ball was parabolic in nature and may have differed based on environmental conditions. Geometry played a major role in the strategy used when kicking a field goal. Coaching staffs gathered large amounts of data not only from their own team but from others, as well to understand how these conditions altered kicking success. The data were used to create information about the probability of success to then determine if and when a field goal will be attempted. Mathematicians have also studied and modeled these problems using a variety of methods such as partial derivatives and concepts from algebraic topology. Many models focused on such critical variables as goal post placement with respect to inbounds lines, goal post height and distance between uprights, the kicker’s distance from the goal post base, wind factors, and kicking angles. These models could also be applied to other sports that have goals, like hockey and soccer.
Process
American football has been played for over 100 years and has evolved over the years into the game played today. Scoring can occur in a variety of ways including the field goal, which gives the kicking team three points. A field goal is scored when the ball is kicked from the ground and through a goal. The goal is made up of a horizontal crossbar that is 10 feet off of the ground and two upright side posts that are 18.5 feet apart in the National Football League and 23 feet 4 inches apart for college and high school games. The ball must pass between the two upright posts while going over the crossbar, which is located 10 yards behind the end zone line. The ball is kicked from a position on the field that is either where the last down was markedif the ball is in the center of the fieldor on hash marks that are lined up with the two side posts of the goal.
Thus, the path the ball must travel to score a goal is not a set path, but rather is based on the starting position of the ball. As long as the ball clears the crossbar between the two side posts, it is successful. If the kick is not successful, the other team gets possession of the ball at the location of the ball prior to the kick. Because of this potential exchange of possession of the football, the likelihood that a kick will be successful must be well understood.
Flight Path Factors
The path that the ball travels is very close to a parabola, with differences potentially being created by environmental conditions such as wind, rain, or altitude. Wind can act to either increase or decrease the distance of flight, depending on the wind direction. Rain acts to decrease both the ball’s distance of flight and height. Higher altitudes mean there is less air resistance, so the ball will travel both farther and higher when games are played in stadiums high above sea level. Other factors that influence the ball path include aerodynamic forces related to ball spin. These can lead the ball to move to one side or the other and make a ball that has the appropriate distance and height fail to score. These forces can also shorten a ball distance if the ball is spinning backward.
Coaches gather large sets of data about successful and unsuccessful kicks in different venues to better understand how the environmental conditions might alter the kick path. A team with an outdoor stadium often has an advantage over a team whose home field is in an enclosed dome. Along with venue information, coaches study information about their kickers’ practice kicks and the opponent’s defense to determine the probability of a kick leading to a score. Collectively, this information helps coaches decide whether they want to take the risk of kicking a field goal or whether punting the ball down the field is the wiser decision.
The use of mathematical modeling in twenty-first century sports could be attributed to the widespread use of “data analytics.” This practice was popularly termed “Moneyball” based on the 2003 book "Moneyball" by Michael Lewis. Moneyball was the process which enabled the Oakland A’s baseball team to field very competitive teams in the 1990s on a relatively low budget. The A’s allowed data analytics to inform personnel acquisition efforts whereas most other teams relied on talent scouts.
In the mid-2020s, data analytics became fundamental aspect of most sporting events. Teams that did not employ data analytics for decision-making placed themselves at a serious competitive disadvantage relative to other teams. Even to the casual sports observer, the impact of data analytics was obvious. In both professional and college football, placekickers began to make field goals of increasing distance and accuracy. In 1970 New Orleans Saints kicker Tom Dempsey kicked a game-winning 63-yard field goal to beat the Detroit Lions. Dempsey’s kick broke the previous record for the NFL-longest field goal of 56 yards previously stood for 17 years. Dempsey’s own kick remained the NFL record for 43 years.
In the 2020s, after the advent of mathematical modeling, field goals of beyond fifty yards became routine and expected. In previous decades, field goals of this distance had been considered low-percentage and risky. In 2024, Dempsey’s old record of 63 yards was shared by six other kickers. In the 2024 NFL preseason, Dallas Cowboys placekicker Brandon Aubrey matched the then-current NFL record of 66 yards set by Baltimore Raven kicker Justin Tucker in 2021. Aubrey’s kick did not count toward official record-keeping as it was made during the preseason. Later in the 2024 season, Aubrey successfully made a 66-yard field goal in a regular season game that was negated by a penalty. Also in 2024, Aubrey completed a 65-yard field goal that was good for second place on the all-time record. In 2024, it had become realistic to expect that a field goal of 70 yards would become possible in the future.
Bibliography
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