Optimal Kicking Angle
For this assignment we are given the following scenario:
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. Note: You will need to find out the width of the uprights and the width of the hash marks . . . make a sketchpad model. Is there any merit to some commentators’ argument to take a penalty in order to have a "better angle" on the field goal kick?
In order to solve the problem we must first understand the proportions of a college football field. The field goal posts are 18.5 feet wide and are centered between the hash marks. We are given the length of the field goal posts in feet, so for this problem we will look at the dimensions in feet rather than yards. The playing field is 300 feet long, with two end zones that are 30 feet each, for a total of 360 yards in length. The width of the football field is 160 feet. The hash marks are 40 feet apart and are each 20 feet from the center of the field. Using Cabri 3d, a program that allows us to explore geometry in 3 dimensions, we can create a college football field in exact proportions to an actual field.
After creating our model football field we can easily test various spots on the hash marks to find the optimal kicking angle. We start by analyzing a kick from between the 30 and 40-yard line. Here we see that the kicker has a 7.54-degree margin of error from the right hash mark.
Let’s see what happens to this angle as we move it closer to the end zone. By moving the location of the kick up and down the right hash we should eventually get a good estimate of the location for the optimal kicking angle.
Here the ball is placed just outside the goal line, while in a game situation this would never actually happen we see that the angle has increased. We continue to move it closer to see if the angle continues to increase.
We now see that inside of the end zone the angle has increased further. This is in fact an estimate of the largest possible angle. We will see in the next image that if we move any closer the angle will continue to move closer and closer to zero.
Here the kicker is just in front of the back of the end zone. We see that the angle has been reduced from 27.7 degrees to 17.72 degrees, so it appears that the maximum kicking angle possible, from the right hash mark, appears to be roughly 27.7 degrees and occurs somewhere around 6 yards from the back of the end zone.
The image that we have looked us here has given us very good estimates and a quick answer to the original question: It never makes sense to take a penalty because you are never going to increase your kicking angle unless you are kicking from inside the end zone(which is impossible).
What if we wanted to find the answer to where the maximum kicking angle occurs without the use of the technology that we’ve used up to this point? How would we go about doing this? For this problem we are looking for what distance from the back of the end zone will create the optimal kicking angle when kicking from the hash. This angle occurs when the circle inscribing the kicking angle is tangent to the hash mark. While this is a sufficient method to solve the problem another interesting method was developed in a discussion I had with one of my students, Preston Stanfield.
To begin, we simplify the image so that it is nothing more than the hash marks and the field goal posts. Here FG represents the distance between the posts, 18.5 feet. HM represents the distance between the hashes, 40 feet. Subtracting 18.5 from 40, and then dividing by two found the distances of HF and GM.
Since we are looking for angle FAG, we will not need anything left of point F. We therefore eliminate by drawing a perpendicular line from F to the opposite side.
Now to solve the problem: we are looking for angle FAG, but in order to find it we will use the two right triangles, triangle FLA and triangle AMG. Since angle LAM is a right angle, we are able to set up the following equation:
While this still gives us an unknown value for both the maximum kicking angle and where the angle occurs, x, if we plug this angle into the program “graphing calculator”, we get some help.
The value of this maximum occurs at:
So, when the kicker is approximately 17.7324 feet away from the end zone, the optimal kicking angle occurs. This confirms are earlier conjecture based on the angles we gathered from the model. At 17.7 feet the kicker would be inside the end zone.
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