The rider goes up the hill at 5mph and down the hill at 30mph. At first glance, it would be tempting to say that the average speed at the end of this 10 mile ride is 17.5mph.

Well that is correct in one sense but absolutely wrong in the sense that anybody cares about. 17.5mph is the correct average if you are taking a "distance average" Measuring the speed at every position along the way, you would get half the data points saying 5mph and half saying 30mph.

The distance-averaged speed is 17.5mph in this case. Unfortunately, I can't think of a single application for this kind of average. Nobody cares what the distance-average speed is. Everybody cares about what the "time-averaged" speed is. That's the one that tells you how fast you get to the finish line. That is the one that is

Here we go. The time it takes to go the first half is t1=d1/v1 and the time it takes to go the second half is t2=d2/v2.

That's much more complicated than the distance average. So that means that if you went up the hill at 5mph and down the hill at 30mph, then your average speed would be 8.57mph, which is just a bit more than the uphill speed. I think that this is really annoying. I'd much rather have my average speed be faster than that. So what's the best way to improve your average speed on a hill like this? This graph might help. The colors (from purple to white) represent the average speed at the end of the ride. Purple is the lowest average speed and white is the maximum average speed.

I've put a point that corresponds to an uphill speed of 5mph and a downhill speed of 30mph like we've been talking about. In order to increase your average speed, you could increase your downhill speed by 4mph or increase your uphill speed by 4mph. I've represented these two choices by the arrows that go out from the point. If you increase your uphill speed by 4mph (vertical arrow), then you cross over lots of contour lines and your average speed is significantly better than before--13.85mph. If you take the other choice and increase your downhill speed by 4mph (horizontal arrow), then you barely improve your average speed at all. The arrow doesn't even make it to the next contour line. This choice would give you a very slight improvement in average speed--8.72mph. Remember the original average speed was 8.57mph!

Hopefully this all makes sense. It basically shows that your uphill speed more important than your downhill speed. You can make a significant difference by making a small change in uphill speed. So now it makes sense why the mountain stages of the Tour are the most important. If you can just make a slight improvement in speed over the rest of the field, it will pay off significantly and if somebody has a less than average day on the mountains, it will hurt them horribly.

This should also prove to you that if you are running late to work on your bike, you should push really hard on the uphills and take it super easy on the downhills. The downhills will make very little difference in the time it takes you to get to work.

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