Congratulations to all the Olympic athletes who have competed thus far, especially Michael Phelps, Nastia Liukin, and Shawn Johnson. Gymnastic events are all great to watch, and I don’t think you could find a more colorful analyst than Bela Karolyi. The balance beam is probably the most classic and thrilling gymnastic event – the Summer Olympics equivalent of figure skating – but the men’s high bar and women’s uneven bars are just absolutely jaw-dropping. It looks like magic.
Now the track meets take center stage. My younger brother ran track in high school, and he tells me the track event he has the most respect for is the 800 meter race. As he puts it, “It’s long enough that you can’t run the entire race at full speed, but it’s short enough that you have to run the entire race at full speed.”
Sounds brutal. I’ve never done anything more than casual jogging, and I expect my 800 meter time would be more like 4 minutes than the Olympic standard of somewhere in the 1:40s. I’m in reasonable shape, but I’m definitely no sprinter.
Anyway, enough Olympics. While we’re talking about speed though, we have a good excuse to do a toy calculation and look at some numbers.
Temperature is a measure of the average kinetic energy of the molecules of a substance. The actual kinetic energies of each molecule will vary about that average. We do know that the way to convert from temperature to energy is to multiply the temperature by Boltzmann constant k. Room temperature might be taken to be 25 C, so convert that to Kelvin and multiply by k and you’ll get E = 0.0257 eV. We know how much a nitrogen molecule weighs, so we can use the kinetic energy relationship to find the velocity:
Plugging in the mass of the nitrogen molecule, I get v = 445 m/s. Now this is a ludicrously loose calculation. I’ve completely ignored the Maxwell-Boltzmann distribution, issues involving degrees of freedom, and the fact that air is not entirely nitrogen. But you might notice nonetheless that this is not too far from the speed of sound, which at that temperature is about 346 m/s. You might expect the speed of sound to be about the same as the speed of the air molecules, since it’s the motion of the air molecules that carry the sound waves.
It further turns out that if we replace the 2 in the above equation with 7/5 (the adiabatic index) to correct for the degrees of freedom of the diatomic nitrogen, we get a speed of 373 m/s which is actually a pretty darn good approximation considering the roughness of the calculation.
It’s one of those things that make you feel good when you’re figuring out thermodynamics – yes, all these weird concepts and calculations do in fact give you numbers which aren’t too far removed from what we actually see in everyday life.