# How fast can a train go?

As high speed rail takes over from short hop air travel there will be demand for increased performance. MagLev technology may be useful for this, but how fast can steel wheels on steel rails go?

The TGV has been taken up to almost 580km/hour. The AGV will almost certainly be able to go faster. This is the same as the top maglev speed (I wonder if this means that power output is the limiting factor?).

There is no technical reason why railed vehicles can't go faster than planes (For example, the land speed record is 10000km/hour by a railed rocket sled, and the highest speeds measured have all been in evacuated guideways on magnetic levitation).

The main requirements are:

a) The ROW is suitably straight . At 27000km/hour the tangential acceleration from the earth's curvature alone is sufficient to lift the train off the track. Far below that tilting trains become necessary for passenger comfort on corners. Spain's Talgo passive, damped tilt trains already demonstrate a practical cornering at high speed on sharp corners. You can scale tilt and speed quadratically - quadruple the radius and you can double the speed for a given tilt, but adhesion improves dramatically with superelevation, allowing passive tilt trains to operate on cant excess and dramatically increase their top speed.

b) Sufficient power. The short contact time of pantographs allows a maximum current of about 2000A, which means that a 12MW train needs at least 6kV to operate. In practice a lower current (say 500A) and higher voltage is beneficial to reduce erosion and transmission losses. The practical voltage of a conventional pantograph is probably about 100kV (beyond which the diameter of the conductor to minimise dielectric breakdown becomes an issue), allowing a maximum power output of say 50MW (roughly the forwards power output of a 747). Planes on the other hand are fundamentally limited by something akin to the rocket equation - as their speed increases the fuel increases dramatically and it gets hard to carry enough without fundamental improvements to aerodynamics or energy storage. Planes can go higher to reduce some drag, but doing so increases the energy cost of climbing, which (discounting incredible improvements in energy storage) is lost on landing.

c) Aerodynamics. Even at just 350km/hour great care is taken to minimise drag and provide good adhesion and prevent ballast damage. This can be solved with evacuated tubes, which given the tunneling efforts in Europe, may be practical today (once you can build a tunnel, it is easy to line it and evacuate it - the difference in structural loading is negligible. The Large Hardon Collider is an example of such a design). Any improvement in plane aerodynamics can be transferred to trains, and without problems of induced drag and weight tradeoff. Ground effect may become useful for further improvements.

d) Braking. The current state of the art is inductive eddy braking, which works by turning the rail into a shorted motor. The rail heats up from this. Consider a 1000tonne 600km/hour (166m/s) train slowing at 0.3g in an emergency (brakes are never used in normal operation, much better to use regenerative braking). This generates an instantaneous power of

dE/dt = 1/2 m v*dv/dt = mva = 1000t * 166m/s * 0.3m/ss = 0.5GW

this sounds a lot, but consider that it is dissipating into 100kg/m rails at 166m/s. That means that the energy per metre of rail heats the rail by 0.5GW / ((166m/s * 100kg/m) * (0.45 J/g K)) = 67K ( = 67C)

e) Travelling waves in the catenary. A significant issue for high speed travel is the fact that as the pantograph pushes against the wire it lifts it slightly. As long as the wave generated moves away from the pantograph faster than the train is moving the wire doesn't move much. But just like a shockwave in the air, once the transverse wave is travelling at the same speed the wire will lift up until something gives (the wire snaps for example, or the train loses power and slows down). The speed of the traverse wave is proportional to the mass, tension, and stiffness of the wire, which is in turn proportional to the young's modulus and the bending area (second moment). We can thus increase the maximum operating speed of the wire by increasing the diameter, or the wire tension. These give a limit of about 700km/hour with copper clad steel. Another strategy is use a stiffer material such as glass fiber, or perhaps in the near future, carbon nanotubes (which can also be superconducting). Carbon nanotubes have 4 times the stiffness of steel and 7 times less dense too, allowing for perhaps 20 times the maximum velocity. Another intriguing idea, which I haven't confirmed with a suitable engineer, is to realise that just like with air, the problem only occurs very near the speed of wave propagation. Once you are traveling faster than the wave, you leave the bump behind. Unlike in air, we can change the speed of wave quite easily in the overhead wire simply by changing the tension. Thus, a train might accelerate to 600km/hour on low speed catenary, then cross onto 'high speed wire' tensioned to a speed of say 500 km/hour allowing it to accelerate to 1000km/hour. The pantograph might be retracted in the case that the train is about to drop below 500km/hour.

(additional thought:) f) wheels and rails. At high speed I suspect a similar problem occurs with waves in the wheel and rails. The speed of wave in these must be tremendously high (perhaps 4000m/s), but even so, they do cause a problem similar to those in the catenary.

I suspect that given sufficient demand existing HSR routes can be brought up to 400mph average, which is about the same speed as a plane over the LA SF route. At that point it is basically game over for intracontinental and especially short haul planes. The Japanese have developed dual use tracks with both rails and maglevs. Although their existing vehicles do not operate this way, it is conceivable to imagine a dual use vehicle which operates on steel up to say 350km/hour, then hops onto hyperlinks consisting of evacuated tunnels and maglev operation for extremely high speed. The maglev technology is fairly innocuous aerodynamically, and wouldn't affect the steel on steel running, and the maglev would need to enter an evacuated tube to really have a benefit, at which point the steel wheels have no aerodynamic cost. Weight it unlikely to be a problem.