In this assignment the best (Maximum) angle for a football kicker--kicking from the hash mark--is determined by calculating the distance from the goal post that gives the best angle.
A diagram is provided below for clarity. The diagram is not drawn to scale.
The dimensions of the football field are as follows (NCAA):
Length of playing field 300ft.
Length of end zone 30ft.
Width of field 160ft.
Distance between uprights (goal posts) 18.5ft.
Distance from sideline to hash marks 60ft.
Distance between left and right hash marks 40ft.
parallel distance from hash marks to goal post 10.75ft.
This diagram is a blow-up of the problem at hand. The task is to determine angle LKR of the shaded triangle.
The distance from the hash mark (D) to the right goal post (R) is 10.75ft.
The distance from (D) to the left goal post is 29.25ft. (10.75ft. +18.5ft.).
The question here is whether to take a penalty before the field goal try. The answer would depend on what yard line the ball is on. So by finding the best angle for the kick, the question can be answered.
* Notice that tan(F)=DO/X, tan(F+E)=(DO+AL)/X, where DR=10.75ft. and AL=18.5ft.
So, angle-F = arctan(DO/X), and angles E+F =arctan([DO+AL])/X.
This leads to the maximum angle=M=arctan([DO+AL])/X - arctan(DO/X).
Now, by substituting the numerical values into the equation, we solve for M, the maximum angle for the kick.
Solution: M=arctan(10.75+18.5)/X - arctan(10.75/X)
Next we find the derivative of this equation and set it equal to 0 to find the maximum angle for the kick.
The derivative, y'=10.75/(x^2+115.563)-29.25/(x^2+855.5625)
29.25/(x^2+855.56) = 10.75/(x^2+115.56)
10.75(x^2+855.56) = 29.25(x^2+115.56)
(10.75/29.25)(x^2+855.56) = x^2+115.56
0.3675x^2+314.438 = x^2+115.56
314.438 -115.56 = x^2-0.3675x^2
198.878 = 0.6325x^2
314.432 = x^2
17.73 = x
These calculations give the maximum angle at a distance of 17.73 ft. from the back of the end line. This implies the best angle would occur when the ball is placed inside the end zone. So, the best angle possible to kick from is when the ball is spotted as close to the end zone as possible. Hence, the team should not take a penalty hoping for a better kicking angle.
Note M = Maximum angle = 27.5 degrees. With the ball on the goal line (as close to the end zone as possible) and the kick 7 yards back, on or about the 7 yard line, the angle is 19.9 degrees. This is the best practical angle.
Lets suppose the team takes a 5 yard delay of game penalty. In this case the kick would be from the 12 yard line. This angle is 18.3 degrees, not the best kicking angle.
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