Maximization and an Awesome Connection

Last week, AP Physics took kinematics to the max (or min).  The prompt for the day:

     Pick a situation and maximize (or minimize) something related to a motion.

In a few small groups, everyone ended up landing on trying to find the angle that maximizes projectile range down (or up) a hill.  This is a fun problem, and really tests the algebra/bookkeeping skills.

This is also a good chance to test out not one but two cool applications of the quadratic formula.  You have a couple of equations that are quadratic not in theta but in the tangent of theta!

Getting all of the way down to the end requires some slick tricks like that, but also a great deal of discipline and the ability to work quickly but very accurately.  Thinking back to my undergrad physics courses oh-so-long ago, I remember that being an under-advertised but very important skill.  The whole idea of junior-level mechanics and E&M and certainly of undergraduate quantum mechanics seems to be to exhaust all of the problems that can be done analytically, which means that they get... ahem... robust, in terms of the algebra.

One group finished quickly enough to test their prediction with a ball launcher and looooooong ramp:

The prediction worked like a charm, and the ball's maximum range occurred at the predicted launch angle.  They made some pencil marks (did you guys erase those?!) on the track, and here was where it got real.

Even though the maximum range did occur at the predicted angle, they noticed that the range didn't change much even with what seemed to be a relatively large angle change.  It decreased, just not as much as they had expected.  On one side of the optimum angle, the range changed very little for up to 10 or 15 degrees of angle change; on the other side, it was significantly more sensitive.

This observation led to a couple of great revelations (without any input from me):

• This tells us a good bit about the shape of the function.  On one side, the function slopes away from the maximum relatively slowly, but falls off much more quickly on the other side
• This is, 100%, the "algebraic test" for extrema!  OK, we may have made that name up, but it's the process where, instead of taking a second derivative to test the type of extremum you've found, you test a value to the left and one to the right.  This test works well if the derivative's complex.  Visually, this is it, in the flesh!

That connection - the idea that we can just see what some arcane mathematical procedure is about by doing an experiment - is fantastic.  I'd love to take credit for having designed the whole situation to force that epiphany, but it just happened.  That makes it even better!