I still remember the first physics problem I tried to solve, though at the time I didn't even know what the word physics meant. I spent many a family road trip pondering this question, but my 6 year old brain could never quite find an answer. It popped back into my head during a recent road trip from Santa Fe to Salt Lake City. Finally I think my much older brain managed to solve the mystery.
Whenever I would travel across a state line with my mom she would tell me to hold my breath and bring one state's air into the next. In the beginning I would do just that, but then I realized I was just putting car air back to into the car, not Maryland air into Delaware. So then I started to stick my head out the window when I saw the state line approaching. Needless to say my mom was not thrilled with her 6 year old's head hanging out a window while traveling 60 mph (well, probably more like 85 mph). To minimize the maternal worry, I wanted to know the minimum time I would have to stick my head out the window to really and truly take one state's air into another's. I knew it had to be longer than the average air molecule traveled in a reasonable amount of time. Back then I had no idea how to calculate it, though I had a rough idea what must be involved.
As I drove from New Mexico to Colorado two weeks ago I stuck my head out the window and held my breath as usual but also swore I would finally figure out the problem that had been bugging me for more than 25 years. So here goes:
The distance an average air molecule travels depends on how hot it is, the pressure of the air, how many times it hits another molecule and how the molecules like to wander.
When looking at one particular air particle and how it travels it can be treated like a "random walk" problem. In a random walk system, the particle has an equal likelihood of moving in every direction. In this type of system it is the "diffusion constant" that is most important. This says how big and fast the "steps" are in a random walk. It is dependent on temperature and pressure. For air at 77 degrees Fahrenheit, the diffusion constant is 0.214 cm^2/s. The percentage of particles that have traveled a certain distance from their original spot in a certain amount of time is given by a negative exponential function which you can find here.
For my particular problem, I just picked a time of 10 minutes, figuring that would be a decent amount of time to "look" at the particles. Then I started looking at different distances and finding the percentage of particles that had traveled that far. I decided that if less than 1% of particles had traveled a certain distance, that would be a small enough number to count as not mixing over a state line. That magic distance was 0.5 meters. So after 10 min, only .7% of particles had traveled 0.5 meters or more from their original location.
So, after many years, I have finally learned that I only need to stick my head out of the window for a meter, 0.5 meters on each side of the state line. If mom was driving 85 mph that would mean less than half a second with my head out the window. I think even a really overprotective mom could handle that. Or at least I could get my head back in the window before she noticed.
Wednesday, December 07, 2011
Posted by The Mathlete at 12/07/2011 05:20:00 PM