vectors and their derivatives

The time derivative of velocity is acceleration, both vectors of course. But I was reminded in office hours today of just how hard it is to get across the idea that the velocity vector and the acceleration vector can point in totally different directions. And some students have trouble seeing this when a ballistic stone is going upwards along some (parabolic) arc, some have trouble seeing it when it is going down, and some have trouble seeing it at the top. That is, different students have very different problems visualizing the differences of the vectors over time.

I said in lecture that this issue was deep but I didn't emphasize it enough. I feel like it is so big it almost needs its own week!

Stokes vs ram pressure

Macroscopically, air resistance is ram pressure (proportional to cross-sectional area times velocity squared). Microscopically, drag is Stokes-like (proportional to radius times velocity). Where does the cross-over happen? I didn't have the guts to put that on problem set one of my course for pre-health students, but it will be in around problem set eight. It could be on problem set one, because the transition can be obtained purely by dimensional analysis.

In transport processes, there are often qualitatively different effects working at small scales than at large. Another good example is diffusion vs convection.

what does a future doctor not need to know?

My big challenge in preparing my General Physics I syllabus is to figure out what to cut, when the majority of the students are pre-health. I cut thermodynamics, because we have learned that it is taught also in chemistry (and other places). I then wanted to add more material about fluids and elastic solids (pretty relevant to medicine, it seems), so what to cut? I ended up cutting most of rotation, spinning, and angular momentum. Why? To understand the body, you do need to know about torques (how does your arm work, static structures, and so on) but you don't really need to conserve angular momentum. Or do you? The centrifuge spins, but it doesn't have angular dynamics.

(I will be doing the centrifuge.)

parallax distance

The parallax distance to an object is the distance you get by moving your (single) eye's position by a transverse distance x, measuring the angular displacement θ of the object given that move, and dividing the transverse distance by the angle.

Problem-set problem: You are outside after the rain late in the day and you see a rainbow. What is the parallax distance to the rainbow?

(Thanks to Andrei Gruzinov at NYU.)

memorizing capitals

Cheryn (9 years old): What's the capital of Kansas?

Hogg (40 years old): I don't know; it is a useless piece of knowledge, because I can look it up in a few seconds on the internet.

Dustin (30 years old): In other words "I don't care, because I can just ask my iPhone." but then your teacher says "Ah, but what will you do if your iPhone isn't working?" answer: "I am in New York City; if the iPhones aren't working, I don't need to know the capital of Kansas, I need to know how to shoot, skin, and dress a squirrel!"

don't make students hate math!

In my previous post I implied, inadvertently, that we should only teach useful math. That was not my point. My point was that we should not teach math if the effect of that teaching is to cause most students to hate it. Of course if we could teach it such that the effect was to cause most students to love it, I would be all for teaching it!

don't teach math!

In a nice conversation about writing for education, Adam Gidwitz (the author of A Tale Dark and Grimm) pointed me to The Mathematician's Lament. The book makes (much more clearly than I) a point I have been making informally for years: If you want students to know and love math, you definitely should not teach it in school! (Same for literature.) The mathematics requirements in school empty the subject of its meaning and point, and are useless to boot. How many non-scientists use the quadratic formula, ever? Discovering the formula would be fun, using it is a drag (and exceedingly rare).