Alpine Railway Gradient and Adhesion

What is modelled

A train on a gradient needs a tractive force to overcome rolling and air resistance and the component of its weight along the slope. At low speed the wheels can only push as hard as friction allows, adhesion limit F = μ times the weight on driven axles. At higher speed the limit is the power, F = P / v. The train accelerates until the tractive force equals the total resistance.

Equations

• Motion 1.06 M dv/dt = min(P/v, μ f M g cosθ) − M g cr cosθ − k v² − M g sinθ, rolling coefficient 0.003, air term 5 N s²/m².
• Steepest start tanθ = μ f − cr.
• Braking distance with drag ln(1 + kd v0² / a0) / (2 kd).

Checks run in the page

• Steady speed from time stepping against bisection.
• Energy balance of traction work, kinetic energy, climbing and resistance.
• Steepest start by bisection against the adhesion formula.
• Braking distance by integration against the closed form.

Honest simplifications

One uniform gradient, constant power and adhesion, a single resistance law, no curves, tunnels or wind, no wheel-slip dynamics, no rack or regenerative braking, and braking limited by a fixed rate or adhesion. The 3D page shows the whole line; this is an educational train-on-a-slope model, not a timetable or safety calculation.

• Press Wet rail and see the train climb more slowly because adhesion cuts the tractive effort.
• Press Leaves on the line and read how the steepest start drops below your slope.
• Press Heavy freight, low power and watch power rather than adhesion set the steady speed.