Discussions with Yann LeCun (NYU) and Jon Barron (NYU) got me excited again about my project to produce planetarium software for the One Laptop Per Child project. My planetarium is a totally human-readable, child-hackable, ascii python executable. And it is fast to compensate for the slow hardware. That said, the OLPC is the best possible platform for field support for amateur and student observing.
A very nice paper appeared on the arXiv today on sheep–sheep collisions. It makes the point, which I stressed here, that collisions involve immense forces. It also makes some realistic estimates of the physical properties of the horns and skulls of bighorn sheep. But perhaps my favorite thing about the paper is that it begins by deconstructing a laughably wrong analysis in one of the many bad textbooks.
In class today I did
the rocket, the calculation of the trajectory of a device that is propelled by a continuous stream of momentum-carrying propellant. It accelerates as it sheds mass, and the final velocity is related to the initial velocity by a logarithm of the mass ratio (initial to final). I then worked out how big the Space Shuttle's fuel tanks need to be—relative to the orbiter—to get the orbiter to orbit, assuming that the propellant is expelled at around the speed of sound. Insane! It turns out that the space shuttle boosters expel propellant at around 4000 m/s, far, far higher than the speed of sound. If I know anything about hydrodynamics, this is non-trivial. Kudos to those NASA engineers.
There is a nice old problem of the force exerted by an engine pulling a set of train cars, all of which are initially at rest on a frictionless track, and all of which end up moving at speed v. You can think about it in terms of momentum (the change in momentum is due to a force acting over a time) or in terms of kinetic energy (the kinetic energy is produced by a force acting over a distance). The two ways of thinking about it get different answers by a factor of two!
The resolution is simple: The momentum of the train must be created by a force, and since it is a vector law in a one-dimensional problem, the total force times the total time must equal the total momentum. Any other solution fails to conserve momentum. The work done by the engine can go into kinetic energy, but it can also go into other forms of energy (like oscillation or dissipation in the train linkages). Energy is conserved as long as the kinetic energy is less than the work done. This resolves the discrepancy, and creates the nice result that when a train is accelerated, there must be dissipation, or else the train will be left vibrating or oscillating as it goes. I think this is an example of impedance matching.
Sanjoy Mahajan (MIT) and I have spent a lot of time talking about balls rolling down planes, in part because it is a very rich physics problem, and in part because it was the experiment that allowed Galileo to infer the constant acceleration behavior and galilean relativity. I started on this problem in class yesterday, but considering only the three energies: potential, linear kinetic, and rotational kinetic. When I asked the class to predict the outcome, I was surprised that I could get all three answers to the question "will the tube roll down the plane faster, slower, or at the same speed as a block sliding with little friction?" Even when we were done, not everyone got the inference, but I have to admit, the issue is subtle. Next time I will start to look at the problem from the point of view of forces; unfortunately, the class isn't quite ready for torques yet.