# How to calculate maximum altitude difference between metro stations?

For story purpose I am trying to design a mass transit system for a perfect city. Right ... maybe not so perfect, if you look carefully into the details, but actually the mass transit system is supposed to be highly efficient.

Nevertheless, the city is built from scratch with heavy inspiration from New Urbanism (mass transit, high buildings, mixed zoning, nice parks, very little cars). In RL the metro suburban rail system is able to waste even 45% of used energy for braking (and engineers hope to recover a small fraction for regenerative breaking). In RL there are also attempts to reduce wasted energy by working with slopes - after leaving the station the train goes down a bit to go up just before the next one, thus saving all this accelerating /decelerating stuff.

I want to go one step further - I want to position the stations clearly higher than the track, thus effectively no breaking would be needed. (Yeah, the metro would resemble a bit like a roller coaster, I know). The system would work perfectly, if all stations were on the same height, just a bit energy would be used to compensate for rolling and air friction. If the altitude difference would be minimal, then train would need to use energy only in this slightly uphill direction, while on the opposite, everything would be provided by gravity.

Is there any idea how to calculate (rule of thumb, whatever) what the maximum height difference between stations would be, under which such a system may indeed work?

The tech level is comparable to contemporary Earth.

(Clarification: I know that you have to include rolling friction, air friction, etc... The issue is just how to either find data to put into such a formula, or how to make a very rough adjustment based on some RL life example, to derive calculations, that would not make any engineer cry)

• Don't forget to account for the length of the carriages. There should be a maximum possible curvature of the rails in order to assure that the carriages are not hitting each other or hitting the ground and that the wheels stay on the rails when the train bends in the vertical around the connections. – Pahlavan Jul 20 '17 at 9:33
• Did you try to ask that question on physics.stackexchange.com? I believe people there may have knowledge required to help you. – running.t Jul 20 '17 at 9:50
• You don't need stations to be at varying heights; you need the platforms to be at varying heights. The rest of it: concourse, ticket counters, food stalls etc. can be at whatever height the local terrain prescribes. The advantage: if you have separate tracks for either direction (you should), you can force trains to roll uphill when entering the station, and downhill when leaving. Since both platforms are independently designed, you won't have a problem with trains entering from one direction facing downhill if the other direction is pointing uphill. – nzaman Jul 20 '17 at 10:42
• There might be some information you find useful here specifically check out in text note #7 – bendl Jul 20 '17 at 13:05

The resistance of the train is dependent on a lot of things. I've tried to bring all those things together as best as I can for you here. You can play with the value x (in radians) and the output is the maximum velocity the train will reach at that slope. The coefficients of rolling and air resistance were taken from here and some of the math borrowed from here. Solving for x lets you play with V to get the appropriate slope for a given speed.

Finally, the difference in height will be proportional to the loss in energy do to friction, which itself depends on the slope of the rail, the velocity of the train and the distance it travels between stations. You can calculate that here.

These are all rough calculations and I haven't practiced my physics in a long time. There's also a good chance that I used the wrong unit of air density so please take this answer with a grain of salt, and I'd definitely recommend checking in with our friends at physics.stackexchange.com