Metro Stop-Start Kinematics

What is modelled

A metro train accelerates at a constant rate, cruises at the top speed if the stations are far enough apart, brakes at a constant rate and then waits at the platform. The page follows N trains evenly spaced round a loop line and reports the trip time, the schedule speed (distance divided by trip plus dwell), the headway and the traction energy.

Equations

• Trapezoid trip time t = d / v + (v / 2)(1 / a + 1 / b) when d ≥ dc = (v² / 2)(1 / a + 1 / b).
• Triangular trip (stations too close to reach top speed): vpk = √(2 d / (1 / a + 1 / b)), t = vpk (1 / a + 1 / b).
• Cycle T = n (t + tdwell), headway T / N, trains per hour 3600 N / T.
• Traction energy per stop ½ m v² + crr m g d, reduced by the regenerated share of the braking energy.

Checks run in the page

• A 2 ms stepped brake-at-distance controller against the trip time formula.
• Peak speed of the stepped run against the triangle or top-speed value.
• Mean gap between arrivals at the first station against cycle time over trains.
• Force times speed integrated over the acceleration phase against the energy formula.

Honest simplifications

Acceleration and braking are constant rates with no jerk limit, no gradients, no curves, no signalling blocks or train-to-train interaction, and every dwell is the same length. Real lines have motor torque curves, regenerative braking that depends on the other trains on the same section, and delays that grow.

• Press Short hops and see the trains never reach top speed: the trip becomes triangular.
• Press Express spacing 3 km and compare the schedule speed with the 1.2 km case.
• Press Rush hour and read how many trains per hour a shorter dwell and more trains deliver.