ICE test train reaches speeds of up to 405.0 km/h

Also, the passenger miles per unit of energy drops geometrically, as speed increase linearly. Most of the loss is aerodynamic, so you either need a hyperloop tunnel or wings to take you into the literal stratosphere, to avoid high fuel burn at high speeds, so even if you could run the train at 400 km/hr along the whole track, it would be unlikely that any operator would do so, on a recurring basis.

I don't think cost is too much of a concern for ICEs. They are already a higher-priced option that competes with plane travel for many destinations.

You can take RE trains at virtually no cost between any two cities you can dream of, yet people en-masse still happily fork over a hundred bucks to save 3-4 hours with an ICE train.

> "geometrically"

It's only quadratic. Aerodynamic drag force ∝ v^2, so aerodynamic power dissipation ∝ v^3—but travel time ∝ v^{-1}, so that cancels out back to ∝ v^2 energy per km.

(I don't know if this was your intent or the opposite, but "geometric" is a synonym for "exponential", which this isn't).

There is a distinct difference in quality of ride in ICE trains when compared to RE trains:

- For some reason, the luggage compartments in many regional RE commuter trains are an example of drug induced circus design. People can not fit larger commuter bags in there

- The seating is less comfortable

- ICE trains have priority in terms of railway usage

- The passenger group is more friendly and respectful on the ICE trains

- Great restaurant service

- ICE are faster. I can travel in 1h from Frankfurt to Cologne.

If you need the configuration for work (speed of travel, working space), then ICEs are clearly superior and worth the cost. I cringe every time I have to get into RE trains. Not because I can not handle it, I am well travelled, but because I know what I can in the other trains.

(Full Disclaimer: I spend a lot of money for the BC100 every year. It is an essential component of my life).

It may not be just 3-4 hours difference, it could be a very significant difference in convenience. For Berlin to Munich, I'm pretty sure there's no single RE alternative, so you're probably transferring 3 or 4 times as well.

I'm sure in some cases, an RE is a bit more competitive with an ICE, but in plenty of others you're asking for a pretty unpleasant time. (And this is from someone with a Deutschland card, making RE trains "free".)

TIL

> So in your example (1^2,2^2,3^2,…), the growth is quadratic (a type of polynomial growth), not exponential. The term "exponentially" is often misused in everyday language to mean "very fast," but mathematically, it specifically requires the exponent to be changing.