air resistance, again

I should stop complaining about air resistance, but I can't help myself! I am teaching this semester from Kibble & Berkshire, and in Chapter 3 there are problems about air resistance that use speeds of around 100 meters per second and an atmospheric drag law that is proportional to velocity to the first power. I don't think there is any physical system that could have these properties: If you are small enough to have viscosity matter, you can never go 100 meters per second. Well, I guess molecules can go that fast, but (a) that isn't what the authors have in mind, and (b) molecules aren't really well described by continuum mechanics!

bad physics of Total Recall

On the airplane home from Spitzer Science Center and LIGO I made the mistake of watching the new Total Recall (2012 remake). (I also made the good decision to watch Men in Black 3 but that is not relevant to this post.) Central to the story is a tunnel through the Earth through which a train called The Fall goes from Britain to Australia in 17 minutes. Important to the plot is a gravity reversal on the journey in the core of the Earth, where the riders on The Fall are briefly weightless, and their chairs rotate from one orientation to the opposite so they are reoriented for arrival.

As everyone in first-year physics (well, for Physics Majors anyway) ought to calculate, the no-air-resistance, free-fall time for this journey (indeed on any straight chord through the Earth) is about 45 minutes (yes, identical to the time to go half-way around the Earth ballistically). That means that if riders were weightless for the entire journey the crossing would take 45 minutes.

The fact that The Fall does the journey faster than that would mean not one gravity reversal but three gravity reversals, because you would have to start the journey accelerating faster than gravity, and end the same. So the whole thing, which was obviously so highly thought out and worked out for the story, was just straight-up wrong.

I think if we ever do dig a chord-like tunnel through the Earth (the Chunnel is getting close to the relevant scale), we probably should run the trips at 45 minutes, because I think for pretty deep reasons this will be very close to minimum-effort travel times. In thinking about this, I have also thought about the relevant engineering gains and safety losses incurred if the Chunnel were operated evacuated of air or at low pressure. There were also idiotic things in the movie related to the implied air pressure, temperature, and air flow in the tunnel outside The Fall but these are more subtle. I digress.

Actually, the excellent MIB3 was even more physically unrealistic than Total Recall (bad time travel and so on), but the tone of the movie made it absolutely clear that you were expected to cut them physics slack.

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.

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.

numerical homework

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.

golf and math

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.

golf and air resistance

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!

cars and energy

I worked out a page of dimensional analysis and order-of-magnitude estimation to compare automobile energy expenditure in the form of acceleration with energy expenditure in the form of battling air resistance (ram pressure). After putting it together I realized the obvious: The air resistance losses exceed the acceleration/braking losses when the journey is long enough that the car has swept up its own mass of air! This means that for typical US cars, acceleration/braking dominates for journeys much less than 1 km (or city journeys in which there are stops much more frequently than once every km), and battling air resistance dominates for journeys that are uninterrupted by stops for distances much longer than 1 km.