numerical homework

Many students found it nearly impossible to complete the problem-set I gave with the golf shot with air resistance (PDF). I will have to analyze the problem sets to find out why. This was the third problem I have given on a problem set that involved making a numerical integration spreadsheet, so it wasn't integration per se that was hard for them. On the other hand, this was definitely the most physically challenging of the numerical integration problems I have given. Once again, I learned that the fact that I could do the problem in 20 minutes in Microsoft (tm) Excel (tm) does not mean that the students will find it easy! And I don't want to denigrate the students, many of whom clearly put in long hours on that problem. This is a level of enthusiasm I want to harness in this class!

After the mid-term, I will back off to a problem involving the numerical integration of sine and cosine, which is a straight-up math problem, but nonetheless very instructive.

golf and math

Andrei Gruzinov (NYU) proved analytically my conjecture, made yesterday, that the distance a golf ball flies is dependent on the initial muzzle velocity only logarithmically, in the air-resistance-dominated limit. Let's hear it for uninformed intuition! Actually, I have to admit that my intuition was highly informed by messing around with numerical integration spreadsheets. Gruzinov's analysis also confirms my conclusion that good golfers hit the ball about as far as it is physically possible to hit, given annoying limits like the speed of sound.

golf and air resistance

I just posted a problem set with a problem that involves numerical integration of a golf shot with air resistance, and comparison to the no-air case. With air, the golf ball must be hit far, far, far harder! In fact, if one assumes standard ram-pressure air resistance and a 45-degree elevation shot, it is essentially impossible to hit a ball 250 yards (as good golfers have no trouble doing). Of course, when air resistance comes in, it is better to reduce the elevation angle (as good golfers do!), which makes the shot possible again.

The amazing fact is that golf shots are enormously affected by air resistance, and no air-free calculation is in the least bit relevant. For shots of hundreds of yards, the with-air shot requires factors of ten more muzzle velocity, or factors of hundred more initial kinetic energy, if the elevation angle is held constant. These factors reduce a bit if you have the freedom to drop the elevation angle. Sweet!