lacticturkey wrote:
According to this video 180km would take as long 40km if it's all downhill and start at the same elevation?
That is not generically true, but would be true if the course was along a circular shaped profile. The guy from the video was trying to make some cool points about pendulums. And one of the super cool things about a pendulum is that the period (to a very good approximation) is purely a function of the length of the pendulum (which is why they are good in clocks). A ball (or bicycle) riding along a circular profile is just like a pendulum.
Let's play with a toy problem. The formula for a pendulum's period is T = 2pi sqrt(L/g), where L is the length of the pendulum arm, g is gravity and T is the period. We get our smallest T by having the shortest L. Conveniently for our 40km/25mi TT course a half-circle profile with a radius of ~8mi works out almost exactly to 25mi and would fit our criteria of shortest L that doesn't go past vertical at the ends. So how long does this course take in our ideal no friction world. L=8 mi. g=32 ft/s^2. T=2pi*sqrt(8mi * 5280ft/mi / (32 ft/s^2)) = 228 seconds. Not bad. And if you started halfway down for a 20km TT, it would also take 228 seconds, cause of the whole pendulum thing having a constant period. If we picked a "flatter" course, like 25mi section at the bottom of a circle with a 16 mile radius, the period of the pendulum would be longer, so we would be slower. In the limit, a flat course corresponds to a pendulum with infinite radius and our time goes to infinity as well (no pedaling of course).
How fast would you be going at the bottom? Well, from conservation of energy, the PE at the top is the same as the KE at the bottom. So we have mgh=1/2 mv^2. Simplifying, we have v=sqrt(2gh). h here is our radius L. So v=sqrt(2 * 32 ft/s^ * 8mi * 5280 ft/mi) = 1644 ft/s. That is fast, 1121 mph.
Lets see why friction sucks. If we were really riding the course I described, we wouldn't hit 1121 mph. We'd be pretty lucky to top out at 80 mph if you have a really good aero tuck on your bike. So that downhill segment instead of taking us 228s/2=114s, takes us optimistically 12.5mi/80mph, or around 10 minutes, not that bad. Now we have to ride up a 12.5 mile quarterpipe starting at <80mph instead of rolling in at 1121 mph. Even discounting riding up the cliff face at the top, I'm going to say doing more than say 10 mph avg (even at say 5 w/kg) is gonna be tough on that second half (8 mi of elevation in 12.5 mi of riding), which is going to mean the second half takes more than 75 minutes, for a total of 85 minutes. Quite a bit slower than the 228 seconds without friction. And certainly slower than a flat TT would have been given the effort required to climb the halfpipe.