infinite jerk

Last week I worked through some kinematics, and it led to a discussion in lecture about the jerk, that is, the time derivative of the acceleration. I was emphasizing that the velocity must be a continuous function of time (lest we have infinite accelerations and hence infinite forces), but then I realized that most people in the class (physics majors) felt that the acceleration must also be a continuous function of time.

After a bit of thought I agreed with them, but not for the reasons they wanted; indeed several were unsatisfied with my discussion: As far as I can tell, the only thing that limits the jerk is the propagation of information. When you slam on the brakes on your car, it takes a finite time for the brake shoe to come in contact with the wheel. This is not really a fundamental problem with infinite jerk (the jerk appears in no physical law), but in any real situation because information propagation (and other kinds of changes) happen at finite speeds, the jerk can never really be infinite.

vector subleties

It is vectors all week this week, in my class, and in two classes I have taught for others. It is understandable that they confuse students, even physics majors with good backgrounds. Here are some subleties that I like to point out:

• Vectors have a magnitude and a direction, but that is not sufficient. They also have a coordinate-free existence or description, and they form a linear space (with the usual linear operators). In this sense, despite what every textbook says, the unit vectors that define the coordinate system are not vectors!
• Although vectors carry around all this geometric baggage, they have a magnitude and a direction and nothing else. I can still confuse the physics majors by sliding around vectors on the board. There is no position associated with a velocity vector, and we confuse the students by always drawing the velocity as coming from the object that is moving.
• Multiplication of a vector by a scalar is usually conceived as changing the magnitude of the vector, which it does, but it also changes the units, in many cases of interest (for example when a displacement is multiplied by an inverse time to make a velocity). So it often produces a new vector that is not longer than the original vector, nor shorter, but really incomparable.
• There is a perfect symmetry between the relationship between velocity and position and the relationship between acceleration and velocity. However, it is far harder for students to understand that the acceleration vector can point perpendicular to the velocity vector than it is to understand that the velocity vector can point perpendicular to the position vector. No amount of class time spent on this point is wasted, in my experience.

everything is an approximation

One of the main things I emphasized in today's class (computing a trajectory in gravity near the surface of the Earth with no air forces) is that every calculation in physics is an approximation. The parabolic trajectory near the surface of the Earth is an approximation to the tip of a very eccentric ellipse, and the eccentric ellipse comes out only in the Newtonian approximation to GR, and even that only holds if there are no other forces acting (and there always are). There are also small adjustments for reduced mass, and if the object has non-trivial extension. Crazy! And in high school this is all taught like it is exact: Just plug numbers into the equations!

dimensions and symmetry

In class, I spend a lot of time on the dimensions of physical quantities. If the left-hand side of an equation is an energy, then the right-hand side must be also. Or if the left-hand side is a quantity measured in J, then the right-hand side must be also. Is it right to call this property a symmetry of physical law? It acts very much like a symmetry, because it selects out of all the things you might write down a very small fraction that could conceivably be physical laws or physical results. On the other hand, these issues are so fundamental, they almost transcend that.

is our children learning?

Teaching style / technique came up at the MPIA mess hall lunch table last week. I said that I use the highly interactive style. Some at the table opined that it probably only helps the best students and is bad for the worst. I differed, but realized that I have essentially no objective evidence. There are certainly studies on the subject, many of which do limited forms of controlled experiments, and they tend to support my view. But it is truly impossible to do a proper differential experiment.

You can't teach two classes at identical quality in two different styles to two identical sets of incoming students. It gets worse the more you think about it. For just one example of the biases: Even if you get two teachers to agree, the one that agrees to do the newer style will tend to be younger; that is, very different when it comes to interacting with the students. And for another: There is no agreed-upon evaluation of student knowledge or aptitude, before or after, and the ones that exist (FCI, for example) favor certain kinds of knowledge.

This relates to a bigger issue I sometimes expound upon from my soap box: When it comes to human or important things, like teaching, hiring, promotion, and supervision, scientists tend to become non-scientists, and throw repeatable empiricism out the window. They are convinced that what they are doing is right despite being able to muster no piece of objective evidence (indeed, evidence is often concealed behind confidentiality or human subjects rules).

For all these reasons, I am a big fan of those who are trying to actually make measurements of the effectiveness of various educational strategies. More power to them, and I will continue to take their advice.

OLPC: the leading-order term

There was a meltdown on the One Laptop Per Child developer list because Negroponte (OLPC chief) said that they would work on a Windows version. In the ensuing discussion (which was, in fact, very enlightening and constructive), many issues came up, about code development and open-source and the educational value of having computers in the hands of children.

One point made by one of the developers, with which I strongly disagree, is that the leading-order term for the educational impact of the OLPC is that it gives children access to the web. Although I love the web (as my non-existent readers know), this is not the leading educational impact of OLPC, if OLPC is successful. If the main point is the web access, then give the students all ASUS or Nokia or Classmate low-cost computers and be done with it!

The leading-order term in the OLPC project is that the computer is a device that can be modified, programmed, altered, and made to do new things. The project de-mystifies computers and electronics and technology and software and the web. It is not access to the worlds information, but an introduction to the world's modifiability and opportunity for innovation. Unfortunately, I don't think everyone on the project agrees, and I don't think that the countries that are investing in OLPC understand. This may bode ill for what might be right now a marriage of convenience between constructivist educators and countries hungry for development (of the economic kind, not the code kind).

teaching physics teachers

I took a break from my no-teaching, all-research sabbatical to make a guest appearance this week in Jhumki Basu's course Recent Advances in Physics in NYU's education program. Her students are building new science units with help and ideas from current researchers. I presented not really my research, but some of my research techniques: estimation and approximation. No surprise there!

I showed on dimensional grounds that cars like the ones we currently drive will never do far better than 30 miles per gallon. 100 maybe. But never 1000. A nice result, with important implications, using only techniques that a high schooler could easily muster.

After I spoke, we discussed, and it was noted by one and all that despite the simplicity of the techniques, in fact estimation and approximation techniques are non-trivial and sophisticated. It is hard to incorporate them incrementally into the existing New York State middle- and high-school curricula. On the other hand, it is my (perhaps optimistic and/or utopian) view that if these things were the focus of quantitative education from day one, they would be easy to have mastered by the end of high school. Of course the teachers I was talking to are going into the system that exists; they can't start from scratch!

Many other interesting things came up, which I hope to blog about at some pont in the future, including students' lack of contact with machinery and hardware and electronics, and the idea (that I hold, but others don't) that education ought to give students skills and tools, rather than knowledge.

mean average rainfall

I dropped in on Sanjoy Mahajan's course 6.055/2.038 Art of approximation in science and engineering at MIT yesterday. We learned about mean average rainfall; you can estimate it pretty well by considering the mean Solar flux, the specific heat of vaporization of water, and the density of water. If you assume all of the Solar flux goes into evaporating the oceans you get 5 m/yr of rainfall, but the true average on the earth is about 1 m/yr; the factor of 5 comes from things like the fact that much of the earth is land, much is covered by clouds, light is reflected, light is absorbed by other processes, and other messy details of the energy budget.

After class, Mahajan and I discussed the size of raindrops, which has a similarly simple calculation: They break up when the stresses exceed the surface tension stress; the main stress is air resistance, which, at terminal velocity, is balancing gravity. I haven't checked the calculation, but Mahajan says this gives you a few mm.