Problem of the Week

What is the best place from which to kick a field goal in football?

Here are the particulars.  An American football field has a goal post positioned at each end; see the picture of a football field.  In order to score a field goal, a kicker must kick the ball through the upright bars of the goal post while standing 50 feet from an imaginary line running down the center of the field, see below.  From which distance to the goal post, x, is the visual angle q maximal?  (Notice that the larger the angle, the less "accurate" a kicker has to be.)

(Notes for football fans only: These were the rules in 1974.  In that year the rules were changed and the distance of the kicker to the center line was decreased from 50 feet to just 9.25 feet, that is, to just in front of the upright portion of the goal post. How did this rule change alter this optimal angle, and is it ever to a team's advantage, under the present rules, to take an intentional penalty of five yards in order to increase the angle, q?  Was it ever a good idea to do so?)

You are visitor number 9314 to this page.