Keller's runner

Keller (1973) wrote a runner as two equations: a propulsive force f against a resistance that grows with speed, and an energy store E fed at a fixed aerobic rate σ and drained by the work f·v.

dv/dt = f − v/τ, dE/dt = σ − f v, f ≤ F, E ≥ 0

For distances beyond a sprint the optimal policy is: full force to a cruise speed, hold the cruise the store can afford, and arrive with the store exactly empty. The platform's Split Second game runs it live: go too early and the last bend is very long.

The optimal 1500 m: a hard acceleration, a long flat cruise, and an anaerobic store that runs out exactly at the line.

What the constants mean

F is the maximum force per unit mass, τ the time constant of the resistance, σ the aerobic supply and E₀ the anaerobic store. Heat, altitude, shoes and drafting all enter through σ: altitude lowers it by about 6.3% per 1,000 m above 300 m; carbon shoes raise it by 4% economy, which becomes about 3% of speed through the cost curve.

What the constants mean

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