2009-03-12

Too Fast for Physics

Bengt and Sture are cruising down the motorway. Bengt is driving.
"Why are you driving so slowly?" says Sture.
"I'm not driving slowly."
"Oh, I get it. You're going to say that since you're driving east, the rotation of the Earth is added to our velocity, so we're actually going really fast."
"No, that would be silly. Also, in this place at this time of day, the rotation of the Earth around its axis is completely counteracted by the rotation around the sun, so the speedometer shows our actual speed in the inertial reference frame of the sun. No, I'm just saying I'm driving as fast as the speed limits allow."
"Which is really slow. QED. And by QED I mean 'quick easy driving'. I never thought of you as the kind to follow laws."
"I always follow the laws of physics."
"Yeah, and I'm not asking you to drive faster than 3*10^8 m/s. Just a little bit quicker than this."
"Well, I'm just being an environmentalist. Mostly because it contains the word 'mentalist'. You know, like a mind-reader or something."
"What do you do when people find out that you can't read their minds?"
"I just claim that I meant that I'm part of the philosophical school which is also called 'mentalism'. And I look at them as if they were really stupid."
"Ooh, good one."
"Thanks. Anyway, speaking of air friction..."
"We were?"
"Yes: with higher speed, the air friction is higher, which increases the fuel consumption, which is bad for the environment, which is why environmentalists don't drive fast. Try to keep up."
"Right."
"So I was wondering: How much does the air friction increase with the velocity? In first approximation, it's probably a power of the velocity. So is it v, v^2, v^3, or what? If you drive twice as fast, is the air friction twice as big, four times as big, eight times as big?"
"Hm, good question. I think we can solve that with dimension analysis. The force should reasonably be directly proportional to the density of the air, and to the area of the front of the car, right? Then we'll see which power of v fits, to give the dimension of force."
"Sure, but there's a quicker way. You hit twice as many air molecules, and they travel twice as fast, so it should be at least v^2, maybe v^3, but hardly more."
"Slightly vague, but okay. How do you know which one?"
"It has to be v^3. Because v^2 is a scalar. If it was v^2, you would get the same result if the car was going backwards. Replacing v with -v, we get the same force, but it should be negated."
"Good point. But anyway, you drive too slow. Next time I want to drive."
"Then you'll have to stay sober."
"Oh."
Sture looks disappointed. They sit quietly for a while, thinking. It's hard to tell whether they're thinking about air friction or about beer.
"You know, it's probably not good for traffic safety, doing physics while driving. You know what they say, 'don't drink and derive'." says Sture.
"Yeah, but we always do that anyway, don't we? And this would just be a case of drive and derive, except you don't need derivatives to do dimension analysis, so your whole point is moo."
"You mean 'moot'?"
"No, I mean 'moo'. I mean it's really stupid, like something a cow would say."
"Or we can take the train. Trains are good." says Sture.
  1. What result does the dimension analysis give?
  2. If either of the arguments was wrong, where did it go wrong?
  3. Bengt claims that the rotation of the Earth around its axis is completely counteracted by the rotation around the sun. Ignoring the actual speeds involved, what time of day would it have to be?
  4. Not ignoring the speeds, is it actually possible?
  5. If it is: How far from the equator are they? If not: How many times bigger would the planet have to be to make it possible?
  6. If you are at the equator, how fast would you have to go to stand still in the sun's reference frame?