Rotating Pace Line
My cycling friends and I are having a disagreement or maybe just a lively discussion about rotating pace lines. The problem is as follows: If the fast side of a rotating pace line is going 20 mph and the slow side is going 18 mph what is the average speed of the group from point A to point B. My contention is that if the fast side is always going 20 mph then the group will average 20 mph. However, some do not agree since they say that if you are going 18 mph some of the time you cannot average 20 mph, it would be something less than 20 mph (like 19 mph if you were going 20 mph half the time and 18 mph half the time  which is probably not the case). I have not figured out how to prove this one way or the other mathematically yet  too many variables.
Has anyone ever figured this out???????? 
I'm lost on this one too.........actually I'm stumped. But in reality the speed difference within a rotating paceline is much closer than the example you give......more like a fraction of a mile per hour.

just read yoru post briefly, I ithink I got the gist...
It depends on how the lead is being changed. Imagine a pace car starting out even with your group holding 20 mph that stays constant. If the front rider pulls off and drops speed, the new leader is 1 bike length back from the front of the pace car. This leaves two options, either the new leader speeds up for a bit to pull even with the car (goes above 20mph) or he keeps it at 20mph. Case 1. If he pulls even with the pace car, then he has gone over the 20 mph line and bumps up his average speed (which he will bring down when he drops off). The over all avg wil end up being 20 mph in this case. Case 2. If he stays where he is (1 length back from the front of the car) and drops off when he is done, the next leader will now be 2 lengths back from the front of the pace car. This shows that while the leader never drops below 20 mph, the gap on the car is increasing 1 bike length per lead change, then the lower overall average. So, in summary ;), if the "new" leader doesn't speed up to the original leader's spot, the over all avg will be less than 20mph. If he does speed up, it will average out to be 20 over the whole ride. So yes, your friend is right, if the abs. max speed of the group is 20, no deviations....ever, then the overall avg will be less than 20 mph. 
Well if you look at the speed of the group as a whole, it DOES slow down.
If you take a pace line that starts at 20mph with the group and is lined up with the front rider (rider A) in a group of 4, what happens is: As rider A drops back @ 18mph, riders D, C, B are still at 20mph, but the pace line is now 1 person ahead of the riders, rider A rejoins the group at 20mph. As rider B drops back @ 18mph, riders D, C are still at 20mph, but the pace line is now 2 rider lengths ahead of the riders, rider B now rejoins, etc. If everyone goes through 1 rotation, at the end, the pace line will finish at 20mph, a full 4 rider lengths ahead of the group and the group will finish slightly slower than 20mph. EDIT: Arg PCS2 beat me to it. :p Another way of looking at it would be to break down the pace line and look at each individual rider. If there are 5 rotations within the group throughout the ride, then looking at rider A, we see him going 20mph...18mph...20mph...18mph, etc. Therefore, he'll end up finishing @ less than 20mph, as does everyone else. 
Okay...so I just thought about what you said and the answer is darn close to 20 mph. If the guy coming off of the front is going 18 mph to end up at the back, with all of the other riders going 20 mph, he is getting passed at 20 mph. The whole group is always going 20 mph. The time it takes you to go from the front to the back is mere seconds at that lower speed of 18 mph. It would barely reduce that 20 mph average speed since you would be traveling at 20 mph for far longer than the 18 mph.
So each individual's average speeds would be like ~19.9 mph. But, you say how can that be if the whole pace line is always going 20 mph? Well simple.... as the guy in the front comes off, the whole pace line has now in a sense lost about 5 feet(the length of a bike). This tiny distance lost(repeated many times in a rotating pace line) is what reduces the whole 20mph average. Remember that speed is the distance covered in a given time. 
Argh...beaten twice. Oh well, you snooze you lose.

A fully ladened touring bike or unladened bike?
European or american roadie? 
It's much less complicated than all the physics spouted above. The average speed of the group is not determined by averaging the time spent at 18 and/or 20 MPH, it's determined by the time it takes the group to cover the course.
Time your route and do the math; that'll tell you your average speed over a given course  point A to B  regardless of what your computer says. 
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I didn't see anything above than could be classified as physics, not even grade 2 math....... If you call any of that that complicated then.......yikes..... My example, and the others, just gave a visual technique to see the problem without having to do any route timing. Besides, the explanations above give the reason to why it's the case, not just an answer that it is the case. Understanding the why is almost as important as the answer itself. Sorry slvoid & slownsteady, I think my fingers were averageing faster typing than yours :p 
Type slower next time, sheesh!!

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Damn.....well then, that was my 15 secs of glory :D 
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