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?
   

Mystery Wheels

Bengt has made a new invention. It's a randomiser.
It has a number of wheels, wheels which are partially conductive. The wheels spin, and there is a contact brushing against each wheel. All the wheels are connected in one circuit, so each wheel can either let the current pass or break the circuit. It acts as a switch, switching on and off once per turn.
Then there is a little device which sends a short electrical pulse through the circuit and registers whether it makes it through or not. There are n wheels, and the fraction of the wheel which is conductive is 1/(nth root of 2), so the probability of the pulse going through is 1/2.
The wheels all look the same, but they spin at different (constant) speeds. Bengt figures that since the ratios between the speeds are not rational numbers, the same configuration of wheels will never occur twice, making the device a really great randomiser - you will never be able to predict the result of a pulse. That's the idea, anyway.
One way to use the machine is to send pulses periodically, with a certain frequency, to get a randomised bitstring. This frequency can be assumed to be significantly lower than the rotation speed of the wheels.

1. Will you eventually be able to guess the next bit in the bitstring, that is, the outcome of the next pulse? Will you be able to guess it only with a limited degree of certainty, or will you ever be able to predict it with 100% accuracy?
   
2. If it is possible to reach enough predictability to make the randomiser unreliable - that is, the number of correct guesses become significantly larger than the number of false guesses - how does the time it takes to get there depend on the number of wheels?
   
3. How does it depend on the speed of the wheels? Will faster wheels make the randomiser better? How much better?
   
4. Is it possible to calculate an actual upper bound for how long it takes to reach predictability, as a function of the number of wheels, the upper bound of the speeds of the wheels, and the pulse frequency? If it's possible, please do.
   
5. Suppose you are capable of sending pulses at any time, as long as there is a certain minimum delay between each pulse. Sending pulses periodically with exactly that delay will give you a certain bitstring. Now, in order to predict that bitstring, is there any other pattern of pulses that would do it faster? That is, is there any pattern of pulses that would give you the information needed to predict the periodic pulses faster than the periodic pulses themselves?
   
6. Provided you have the added advantage of knowing the outcome of the previous pulses, is there any faster way of "cracking" the machine than sending pulses at a predetermined pattern? (No, you're not allowed to physically crack it open.)