Trail running: a bioenergetic demonstration against the rule “1000 m ascent = 10 km flat”

Let’s admit it: the equivalence “1000 m of vertical gain = 10 km flat” is a mountain Dahu (a myth). It’s told to novice mountain runners to haze them, and a few apprentice “scientific types” repeat it solemnly, wielding the mechanical talisman below. Except… mountains don’t read rulebooks.

The often-invoked mechanical talisman:

Why it’s a Dahu (myth)

• The energetic cost is neither linear nor symmetric: it depends on slope i. Running downhill at −10% to −20% can cost less than flat, whereas steep uphills blow up the cost per meter.

• The movement pattern is not unique: we run, we hike, we use poles. A “pure running on flat” model is off-base.

• The terrain is not standardized: snow, mud, scree… the surface changes the cost.

Actual speed depends on a simple ratio: available power over cost per meter:

When slope i changes, C(i) changes; therefore, for the same power, you don’t get the same speed. So we replace the back-of-the-envelope rule with a falsifiable measurement: integrate the GPX trace, compute the slope-dependent cost, and convert total energy into a flat-equivalent distance (EFD).

Consequence for rankings

Forcing the Dahu into a “1000 points” ceiling pegged to the marathon world record ends up ranking trail races as if they were marathons in disguise. What you mostly grade is the scoring system’s error: EFD computed from GPX systematically diverges from “10 km/1000 m”, sometimes massively.

Take-home

• The “1000 m D+ = 10 km flat” calculation is a Dahu legend: easy to tell, wrong in practice.

• The mountain imposes real bioenergetics: C(i) varies with i, with surface, and with locomotion mode.

• To compare and rank, measure energy along the actual trace (GPX) and derive EFD—not a mechanical shortcut.

Therefore, the energetic cost of positive and negative slopes—including snow (cf. UTMB 2025)—changes the flat-equivalent distance (EFD), reduces speed for the same trail finishing time, and thus lowers the true flat-equivalent distance. Otherwise, the marathon world record would be… shattered.

Demonstration (with equations and numbers): why the correction “1000 m ascent = 10 km flat” is mechanically tempting but bioenergetically wrong

Realizable speed depends on energy cost per meter, which itself is a function of slope (positive and negative).

1) Mechanical reminder (insufficient)

The classic argument compares gravitational potential to kinetic energy:

Even though this suggests a numerical “equivalence” (since g ≈ 9.81), it ignores locomotor physiology and the dependence of energetic cost on slope and muscle contraction type (concentric/eccentric).

2) Energy cost of running as a function of slope

The mass-specific energy cost per meter of running (level: 3.6 J·kg⁻¹·m⁻¹) varies nonlinearly with slope i (decimal; e.g., +0.10 = +10%):

The curve has a minimum around −17% (moderate downhill), and rises steeply for large magnitudes of slope—both uphill and downhill (eccentric braking).

3) Realizable speed and “Equivalent Flat Distance” (EFD)

For a given athlete, sustainable speed on slope i relates to the sustainable fraction of VO₂max and to the cost:

On a GPX trace, integrate segment energy to obtain EFD:

4) 2025 application (men’s winners) — Sierre‑Zinal & UTMB

Public profiles and cautious average slopes for uphill/downhill segments provide a lower bound on the divergence from the “10 km/1000 m” rule.

The “10 km/1000 m” rule overestimates EFD by +22% (Sierre‑Zinal) and +34% (UTMB) compared with the bioenergetic calculation. Conditions like snow/mud would further increase actual EFD.

• “1000 m = 10 km” rule: 31 + 2.2×10 = 53.0 km → overestimation ~25%.

UTMB 2025 (shortened + snow): three illustrative scenarios:

Naïve comparison ignores (i) the downhill cost minimum around −10% to −20% and (ii) the surface penalty (snow) when present.

Abbreviations & notation

• D+, D−: positive/negative elevation gain (m).

• i: slope in decimal (e.g., +0.10 = +10%).

• C_r(i): mass-specific cost per meter at slope i [J·kg⁻¹·m⁻¹].

• EFD: Equivalent Flat Distance — flat distance with identical total energetic cost.

• VO2max (\(\dot{V}O_{2}^{max}\)): maximal oxygen uptake (rate).

• f: sustainable fraction of VO2max (depends on duration/pacing).

• GPX: track file (distance/elevation/time points).

• UTMB: Ultra‑Trail du Mont-Blanc; ITRA: International Trail Running Association.

Summary

Comparing a trail to a standardized flat race via “1000 m D+ = 10 km” is physiologically unfounded. Running cost depends on slope (up/down) and terrain conditions, so average speed and points derived from a naïve flat‑equivalent distance are biased. EFD from integrating C_r(i) along the trace provides a robust basis for evaluation.

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