I showed the students complex numbers today, in the context of solving the damped harmonic oscillator. I took my time, and treated it as a cultural romp rather than a physics problem, so I enjoyed myself. What I was surprised to learn is that almost every student in the room knew that the square root of negative one is i. What is up with high school math that every student in the room learned what an imaginary number is, while not a single one learned how to program a computer? Which is easier, which is more relevant to them, and which is more natural given their interests and materials? I would say: Programming, programming, and programming!
My midterm exam was today, and I think I made it too hard. I haven't finished grading it, but I don't have any student who is likely to get perfect, and many who got very little completely right. It is my own damned fault, because most of the problems in the exam were multi-faceted and conceptually rich. Now I have the job of reassuring them that I won't kill them at grading time. Evaluation is very tricky, because it creates an adversarial relationship between student and teacher, exactly when cooperation and trust are required for learning.
Here's an energy example problem I discuss briefly every year, because it brings up a serious student misconception. I did it today in class, and it worked as usual, although about one fifth of the class got it right straight off the bat.
A block slides from rest down a long, slanted ramp that ends with a small, up-turnedski jump(I usually draw the end of the ski jump at about 45 degrees elevation above the horizontal). Air resistance and friction are negligible. After sliding down the ramp and leaving the jump, the block will fly on a parabolic trajectory. Will the peak of that parabolic trajectory come up above the vertical height of the starting point, exactly to the height of the starting point, very slightly below the height of the starting point, or well below the height of the starting point?
The students want to go with
slightly below. The correct answer is
well below, because the trajectory in gravity never brings the horizontal component of velocity to zero, and therefore never brings the kinetic energy to zero, or even close to zero. This leads to a nice discussion and an instructive comparison with the typical
roller-coaster problems out there.
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.
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.
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!
Hooke's Law is usually described as F=−kx, and thought only to apply to idealized springs, but really Hooke's Law is that stress is proportional to strain, and it applies not just to idealized springs but to every object in the entire universe (for small distortions), for extremely fundamental reasons. What an observation Hooke made! I tried to impress this upon my class today.
In class yesterday I compared two problems: the block on an inclined plane (sliding without friction) and the car sliding around an icy (frictionless), banked curve. In the latter, the challenge is to enter the banked turn at exactly the right speed that you make it out the other side without either sliding uphill or down. The nice thing about doing both these in one lecture is that they have nearly identical free-body diagrams, both of which have a massive particle acted upon by gravity and a normal force (at the same angle to the vertical if you set it all up correctly), and yet the physical situations are so different and the accelerations point in different directions. The comparison brings out a lot of conceptual material about contact forces and kinematic constraints.
The banked turn example is also hilarious, and has many nice details, such as that in general if you enter a conical banked curve and slide around to the other side without friction, you will come out with the car pointing in a strange (and non-trivial to calculate) direction.