Train Braking Curves (2D)

One formula for every curve

A train that starts to brake after a dead time t (processing plus brake build-up) with a constant deceleration a travels d = v t + v^2 / (2 a) before it stops. Solving for the speed gives the braking curve to a target at distance d:

v(d) = a ( -t + sqrt( t^2 + 2 d / a ) )

The train protection system computes this curve continuously from the end of the movement authority (the stop signal) backwards. Four curves are nested: Permitted (P), Warning (W), Service brake intervention (SBI) and Emergency brake intervention (EBI). They differ only in their dead time and deceleration: P allows for processing + service build-up + the warning interval + the driver reaction allowance; W drops the driver allowance; SBI uses only processing + service build-up; EBI uses the (higher) guaranteed emergency deceleration and aims at the danger point beyond the signal (the overlap). Every curve is capped by the ceiling speed of the line (plus 5, 10 and 15 km/h for W, SBI and EBI).

What the protection does

If the speed exceeds P the driver gets an overspeed indication, above W a warning; above SBI the service brake is applied automatically, above EBI the emergency brake. Once applied the brake stays on until the train stops. A pending brake only takes effect after its dead time, so the train keeps rolling at almost full speed for several seconds - that is why the curves are far earlier than the point where braking is really needed.

Gradient, adhesion and real performance

The system knows the gradient i (per mille): the effective deceleration is a + g i / 1000 (uphill helps). It does not know the real wheel-rail adhesion or how worn the brakes are. The real deceleration is min(a * performance, mu g) + g i / 1000. When the real value is lower than assumed (leaves on the line, tired brakes) the train overruns even though every curve was computed correctly - graph 3 shows the gap between assumed (dashed) and real stopping distance.

Numerical checks on this page

(1) The time-stepped train model is run with the emergency dead time and compared with the closed form v t + v^2/(2a). (2) The energy dissipated by the brakes, summed step by step as m a dx, is compared with 1/2 m v^2. (3) Each curve speed is converted back to a distance and must reproduce the original distance, and the curves must be nested P <= W <= SBI.

Honest simplifications

Constant deceleration with a pure dead time (real brakes ramp up and the deceleration depends on speed and brake blending); no train resistance, no curve resistance; one train, level or constant gradient; the four-curve hierarchy is a simplified version of the supervision logic of European train protection (the real specifications also use dedicated margins, speed measurement inaccuracy, release speeds and many more curve types). The numbers are plausible orders of magnitude for teaching, not the parameters of any real train or signalling product.

• Load Attentive driver brakes in time: the train stays under the green P curve and stops short of the signal. Then Late driver: watch the warning, then the automatic service brake.
• Load Inattentive driver (no braking at all) and follow graph 2: the protection stops the train at the signal without any driver action.
• Load Leaves on the line: adhesion 0.055 means the real deceleration is far below the assumed one - the emergency brake cannot save the stop (see the verdict).
• Raise the speed to 250 km/h in High speed line: the gates slide back by kilometres because the distance grows with v^2.
• Set the build-up times to zero: the four curves collapse towards the pure kinematic curve sqrt(2 a d).
• Try a downhill gradient of -35 per mille: all curves move earlier, because the same brake gives much less deceleration.