Certainly if you are Coach Richt, you want to know the optimal conditions for scoring a field goal when the situation presents itself. It has been argued by sports commentators that the narrowing of the goal posts in college football presents situations when taking a penalty to kick from further away will result in a "better" angle for the kicker. This presents some questions that interested students can use GSP to answer. Do these sports announcers have a point? What is the maximum angle for the field goal kicker? Why has the apparent situation changed with the narrower goal posts?
Before proceding in this investigation, I will first define two terms for the purpose of the discussion. Consider a football field with the width of the goal posts marked at the back of the end zone in pink and the location of the hash marks identified by the blue lines.
Refering to this diagram of the field, the angle shown in pink will be called the angle of sight. This is the angle made from the intersection of the hash marks with the back of the end zone, the point where the kicker kicks from along the hash marks - known as the point of kick origin, and the point where the closest goal post to the kick origin is located. The angle shown in green will be called the angle of the kick. This is the angle measure between the goal posts at a designatd point of kick origin. If we consider that the kicker must always kick the ball at some initial angle aiming for the center of the goal posts, this angle which results in a kick perfectly centered between the uprights will be called the initial angle. We can now consider that the angle of the kick is a value representing the range of error that a kicker has for the initial angle that will still result in a successful field goal attempt. Based on these definitions, we understand that the kicker's best opportunity for a field goal exists when he has the maximum range for error. Therefore we investigate to find the point of kick origin that corresponds to the maximum angle of the kick.
To investigate the maximum angle of the kick, we first have to construct the football field in GSP. To construct exact lengths corresponding to the lengths of the NCAA regulation field, specific points must be plotted on the grid. Considering the dimensions of the field in feet, 160 X 360 including the end zone, some logical choices for points are (0, 0), (0, 1.6), (3.6, 0), and (3.6, 1.6). These points represent the corners of the field, and segments constructed from the points create the boundaries of the field.
Next the points bounding the end zone and the hash marks are constructed. These points are (.3, 0), (.3, 1.6) for the end zone, and (0, .6), (0, 1.0), (3.6, .6), (3.6, 1.0) for the hash marks. The hash marks are located on the field at every yard mark and they are 60 feet from the sideline on both sides of the field.
The last fixed points that should be constructed are those representing the goal posts and the minimum point on the hash marks from which a field goal kick can originate. These points are (0, .7075) and (0, .8925) for the goal posts, and (.51, .6), (.51, 1.0) for the minimum kicking distance. The position for the minimum kicking distance comes from the rules of football that the ball is spotted 7 yards behind the line of scrimmage for a field goal kick. Therefore the minimum spot would be at the 7 yard line.
To investigate the claim regarding the angle, students will need to construct a point on the hash mark line. This point is the vertex of the angle of the kick. The students can see how the measure of the angle changes as the point is moved down the field along the hash mark to different distances from the goal post. As the students vary the location of the point, they can see that the angle becomes smaller as the distance of the kick origin from the goal post increases. The maximum angle occurs when the ball is kicked from the 7 yard line, and is approximately 17.93o
Given the opinions of the sportscasters, curious students can further investigate the effect of the change to the goal posts on the maximum angle and the angle of sight. The width between the goal posts was changed to 23 feet 4 inches several years after the goal posts were moved from the front of the end zone to the back of the end zone, which increased the kick distance by 10 yards. The change in width was made to improve the percentage of successful field goals with the longer distance from the goal posts. (We have already seen the effects of distance on the range of error allowable for a field goal kick in the investigation of the maximum angel of the kick.) The width change was too successful, and the rules were subsequently changed a few years back to return the width of the goal posts to 18 feet 6 inches. Using a similar GSP construction, a football field with the wider goal posts can be used to find the maximum angle.
This maximum angle turns out to be approximately 22.56o, which is more than 25% wider than the maximum angle with the goal post width of 18 feet 6 inches. Obviously the kickers had more room for error with the wider goal posts, but the fact remains that the further the kick is from the end zone, the narrower the margin of error for the kicker. So why would the sports announcers think that kicking from farther down the field improves the kicker's chances?
The angle that probably confuses the announcers is the angle of sight. For reference, the initial angle that results in the ball going through the exact center of the goal posts is the angle of sight added to one-half of the angle of the kick. Let's look at the angle of sight for the current goal post width and the wider goal post width.
Similarly, the maximum value for the angle of sight also occurs at the 7 yard line. For the current goal post width, the maximum angle of sight is about 11.9o. For the former goal post width, the maximum angle of sight is approximately 9.3o. So the angle that "improves" with distance from the goal post is the angle of sight. Because this angle is larger for the narrower goal posts, it leads to the misinterpretation that the angle of the kick is better at a greater distance, because the initial angle value is smaller at a greater distance. It is interesting that the ratio of the angle of the kick to the angle of sight is over 50% larger for the wider goal posts. Could this ratio be the origin of the illusion of a better angle?
Another source for investigation would be to question if there is a way to calculate the angle of the kick and the range of the angles resulting in a successful kick attempt knowing the dimensions of the field and the length of the field goal being attempted. To answer this question we consider the triangles formed by the origin of the kick, the hash marks, and the goal posts.
The tangent of the angle of sight is 10.75 feet divided by the length of the kick in feet. The tangent of the angle of sight + the angle of the kick is 29.25 divided by the length of the kick in feet. Knowing the length of the kick, the inverse tangent function can be used to find the angle of sight. Then the sum of the angle of sight and the angle of the kick can be determined and the difference of this value and the angle of sight is the angle of the kick. The range of the angles that will result in a successful field goal attempt begins at the angle of sight and ends at the sum of the angle of sight and the angle of the kick.
To demonstrate the use of these functions we can try an example. The Bulldogs are stopped on a third down at the 27 yard line of their opponent. Adding 10 yard for the end zone and 7 more yards for the line of scrimmage, this would result in a 44 yard field goal attempt for Billy Bennett. To calculate the range of angles, the length of the kick is converted to 132 feet. This leads to the following equations:
Angle of sight = tan-1(10.75/132) = tan-1(.08144) = 4.7o
Angle of sight + Angle of the kick = tan-1(29.25/132) = tan-1(.2216) = 12.5o
So the range of angles is 4.7o to 12.5o, or in other words the angle of the kick is 7.8o.
As long as Billy stays within a margin of error of 7.8 degrees, the Kick is GOOD!