Amberly Roberts
Assignment 6: The Mathematics of Field Goal Kicking
The picture above is of Christian Kaufmann, a rising senior at Memphis University School in Tennessee. Christian is my 3rd cousin. He is a top prospect kicker and hopes to play for Vanderbilt University. His resume is impressive both athletically and academically; therefore, I thought of him as I began this investigation. He will be encouraged to read through my findings.
Here's the problem:
The football rules in college football were changed a few years ago have made the uprights 5 feet narrower than previously. Many game commentators have harped about how much harder it is to kick field goals from the hash marks. Assume the field goal is attempted from the hash marks. At what yard marker does the kicker have maximum angle to the two uprights? Is there any merit to some commentators argument to take a penalty in order to have a "better angle" on the field goal kick?
The diagram below shows a model of a field. Click here to open a GSP file that allows you to measure angle DGF while animating point G.
In case you do not have GSP, I will share some of the angle measures with you in the photos below.
Note that field goals are measured using the distance from the back of the end zone to the parallel line determined by the point from which the ball was kicked. The end zone is 10 yards deep.
Now, it appears from the pictures that the angle gets smaller as you move further back. The only thing that you can't see from the pictures is the behavior of the angle when the vertex is actually in the end zone. Since this is not relevant in the game of football, it makes sense to not consider this case. However, for the purposes of this mathematical investigation, we will consider it. Notice that if the vertex of the angle were placed along the back of the end zone, the angle would be a 0 degree angle. Hmm.. If we have observed that the angle constantly decreases in the relevant range of possible kicks, this must mean that the maximum angle is achieved somewhere in the end zone! The spread sheet below shows the relationship between location of kick and angle measure.
Have you ever thought to use a spreadsheet for math investigations? They are a pretty powerful tool! However, you control them. In order for the spreadsheet to produce these results, I had to describe the relationship between distance of kick and m<DGF. How did I do this? Using an equation!

The spreadsheet suggests that the angle dereases as you move further back in the relevant (the bold distances) field goal kicking range. The spreadsheet also suggests that the maximum angle is achieved approximately 6 yards from the back of the endzone. (So, this kick would take place in the endzone) Here is a grpahical representation of the relationship.
The xaxis represents the distance from the back of the endzone. The yaxis represents the measure of the angle. If we want to find the exact place at which the maximum angle is achieved, we could use some calculus! First, take the derivative of the equation This gives us a representation of the slope of the tangent line. We are interested in the place where the tangnet line has a slope of 0. So, equate the derivative to 0 and solve for x. Try this for yourself. 
This conclusion seems a little uninteresting to me. So, I want to explore more. After discussing this problem with a former high school kicker, the following was suggested. Perhaps the angle that the announcers are referering to is one different that the one explored above. When kicking, it is easiest to kick at a 90 degree angle. For example, when kicking from the hash mark, a 90 degree angle is formed by the back of the endzone and a perpendicular line drawn from the kicker to the back of the endzone. This presents a problem because a ball kicked along this path would be outside of the uprights. So, in order for the kick to go through the uprights, the kicker must kick at an angle. The illustation below will hopefully clarify my points.
As the distance increases, the angle decreases (the kick becomes easier) But, don't stop here. Notice that while the amount of increase in distance is a constant 1 yard, the amount of decrease in angle measure varies. The closer you are to the goalposts, the more variance you get by moving back. For example, when choosing between... a 20 and 25 yard field goal, the angle changes by 4.99 degrees. a 25 and 30 yard field goal, the angle changes by 2.93 degrees. a 30 and 35 yard field goal, the angle changes by 2.64 degrees. Notice that the change in the angle is more significant closer to the goal line. This explains why some teams take the penalty. Since the range of the kicker is probably not a factor when making short distance kicks, it may be beneficial to move back to improve the angle of the kick. (Of course, this is based on the assumption that the 90 degree kick is the easiest) Pretty cool.

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Note: I was motivated to complete this particular assignment after reading the work of a classmate, David Park.