walk or take the elevator?

I'm just generally excited about getting back into the classroom after a long sabbatical. I'm thinking about problem-set problems for the Physics Majors. Here's what's in my head right now:

NYC has had a hot summer, with most buildings running air conditioning on a thermostat continuously. To save energy, NYU (and other large entities in NYC) asked their employees to conserve energy in various ways, some of which we might take issue with. Here's an uncontroversial one: You should take the stairs, not the elevator.

But is that uncontroversial? What considerations are required to figure out whether this policy would reduce or increase energy consumption? Obviously—if you take the stairs—you use less elevator energy, but then you drop a metabolic load on the building air-conditioning. Which uses more power in the end? Use a combination of web research and simple physical arguments to make cases, and identify weaknesses in your argument as you change assumptions. Things that matter include: Neither humans nor elevators are 100-percent efficient delivery vehicles for potential energy (in fact, can you see a fundamental argument that elevators must spend more than 50 percent of their energy generating heat?). Elevators are heavy but counter-weighted. Some buildings have very busy elevators, so your contribution to the elevator load is only the marginal contribution; in other buildings you are typically the only person in the elevator. Air conditioning systems have efficiencies limited by fundamental ideas in thermodynamics, but are probably much less efficient than the limits. And so on!

Thanks to Andrei Gruzinov (NYU) for starting me thinking about this one.

accelerating a train

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.