Field Goal
Janet M. Shiver
EMAT 6680
Investigation
The rules in college football were changed a few years ago, making the uprights five feet narrower than in previous years. 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 commentator’s argument
to take a penalty in order to have a “better angle” on the field goal kick?
To determine which yard line is best for kicking
a field goal from the hash click here. Move the football
up and down the hash marks while watching the angle measure.
Scroll down for the solution.
It appears that the ball should be place approximately
2 yards off the goal line into the end zone.
Next, lets take a look at the
problem from a trigonometric standpoint and try to develop a function to
represent this angle.
First notice that two right triangles have been
formed on the field. Using the tangent function and the diagram above,
we can see that
Solving each equation for
,
we get
.
Since
,
A graph of this
function appears below.
We can see from
the graph that the point at which the angle reaches its maximum occurs when
x equals approximately 8. That is the angle reaches its maximum when the
football is two yards past the goal line in the end zone.
Finally, lets take
a look at this problem geometrically.
We are going to
construct a circle through points A, B and C. To do this we must first determine
the midpoints of the sides of triangle ABC and then construct perpendicular
lines through this point.
The intersection
of the three perpendicular lines determines the center point for our circle.
The angle we want
to maximize is angle ABC and since it is an inscribed angle we know that
it is ˝ the measure of its intercepted arc AC. It follows that if we maximize
arc AC then we will also maximize angle ABC.
Click Here and move point B to determine when the arc
is maximized.
It appears that
the arc is maximized when the area of the circle is minimized. This occurs
when the circle is tangent to the hash marks.
This point of tangency
appears to take place approximately 8 yards from the uprights.








From our investigation,
it appears that the best place to kick a field goal from the hash marks
is two yards into the end zone. Since this is not a possibility in the
game of football, we can conclude that the closer a kicker is to the goal
line the better the angle will be for his kick.








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