Madden 14 and Calculus

Recently I was playing Madden 14 with a friend on my XBOX 360 when he asked me

Dave, does the wind make that much of a difference when I punt the ball?  If so, should I change my angle a lot, or a little?

I replied with "It would change your angle, but by how much I don't know".  This question also sparked my most recent calculus optimization problem.

I asked the students in my class

If a football punter is kicking into the wind, should he/she worry about the angle they kick at?

Also showed this video

What came of this was a great 10 minute  discussion around questions such as

• How windy is it?
• How much should the angle change by before we worry about it?
• How fast does the ball move?

After doing some "googling" we decided to work on the following problem

If a punter can kick a ball at  40 kph, and there is a head wind of 20kph, what angle should the punter kick it at so the ball travels the farthest distance?

Here is a possible solution: (Yes I do realize we ignored air resistance, as I said I assumed Madden 14 did the same)

When you kick a ball there are vertical and horizontal components and we determined the following, starting at acceleration and integrating, with respect to time.(Using h=horizontal, v=vertical)

We then discussed physics, that if I kick a ball with velocity 40 at an angle of theta, then the vertical velocity is 40sin(theta), and horizontal velocity is 40cos(theta), as well as initial distances were 0, and subtracting 20 off the horizontal velocity due to the wind, and so the formulas become

Now, we realized we have two variables in the equation we are optimizing (d_h(t)) and so we needed a way to relate the angle and time.  Knowing at the top of the kick the vertical velocity would be 0, and that the total time in the air would simply be double this value we get the total time to be: (setting v_h(t)=0, then multiplying by 2)

Now we substituted this into the horizontal distance equation, and took derivative (knowing that the derivative equals 0 at a max), and solved for theta

I allowed my students to solve this last equation graphically to determine when it equaled 0.  

We got an angle of 32 degrees.

We then discussed how easy it would be to simply make the wind speed a letter, the kicker speed another letter and ultimately create a program which could do this for any experience.

I am in the process of making an app and selling it to all football teams and retiring in the next year..jokes!!

Feel free to use, tweak, comment, etc.