Vertical Curves
As a train negotiates a curve, the force it exerts on the track changes. Too tight a 'crest' curve could result in the train leaving the track as it drops away beneath it; too tight a 'trough' and the train will plough downwards into the rails and damage them. More precisely, the support force R exerted by the track on a train as a function of the curve radius r is given by
positive for troughs, negative for crests, where m is the mass of the train and v is the speed in m/s. For passenger comfort the ratio of the gravitational acceleration g to the centripetal acceleration v2/r needs to be kept as small as possible, else passengers will feel large 'changes' in their weight.
As trains cannot climb steep slopes, they have little occasion to go over significant vertical curves, however High Speed 1 (section 2) in the UK has a minimum vertical curve radius of 10000m. High Speed 2, with the higher speed of 400 km/h, stipulates much larger 56000m radii. In both these cases the experienced change in 'weight' is less than 1%.
Read more about this topic: Minimum Railway Curve Radius
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