Effort & pacing

Grade adjusted pace

What your hilly pace would have been on the flat, from the energy cost of running on a gradient.

Peer-reviewed

The Minetti cost-of-running curve is peer-reviewed lab science, but it was measured on elite mountain runners on a treadmill, so it encodes elite downhill economy and no air resistance.

Grade Adjusted Pace (GAP) answers a simple question: if this stretch of trail were flat, how fast would that same effort have carried you? Running uphill costs more energy per metre, so a slow uphill pace can hide a hard effort. GAP converts your pace on a hill to the equivalent flat pace using the measured metabolic cost of running on a gradient, so you can compare a climb, a descent, and a flat straight on one honest scale.

The idea in one line

Every gradient has a known energy cost per metre. Divide your actual pace by how much more expensive the slope is than flat ground, and you get the flat-equivalent pace: the speed the same energy would have produced on the level.

GAP = pace_actual ÷ (Cr(i) ÷ Cr(0))

Step 1: the energy cost of a gradient

Minetti and colleagues (2002) measured the oxygen consumed by runners on a treadmill set to a range of slopes, then fit the energy cost of running to a fifth-order polynomial (i is the gradient as a fraction, rise over run):

Cr(i) = 155.4i⁵ − 30.4i⁴ − 43.3i³ + 46.3i² + 19.5i + 3.6

The result is in joules per kilogram per metre. On the flat (i = 0) every term with i vanishes and the cost is the constant 3.6 J/kg/m. That flat value is the denominator we divide by.

The energy cost of running is roughly U-shaped. It bottoms out on a moderately steep downhill (around -18%, where gravity still helps more than braking costs; Minetti's paper reports the running minimum near -20%) and climbs steeply on both steep uphills (lifting your body against gravity) and steep downhills (where eccentric braking becomes expensive).

Step 2: divide out the slope

The adjustment factor at any gradient is just its cost relative to flat, Cr(i) ÷ Cr(0). Because a climb costs more than 1, dividing your uphill pace by that factor gives a faster flat-equivalent number: the level pace the same effort would have produced.

Worked example

You run a 6% uphill (i = 0.06) at 5:00/km. First the energy cost of that gradient:

Cr(0.06) = 155.4(0.06⁵) − 30.4(0.06⁴) − 43.3(0.06³) + 46.3(0.06²) + 19.5(0.06) + 3.6
= 0.0001 − 0.0004 − 0.0094 + 0.1667 + 1.1700 + 3.6 = 4.927 J/kg/m
Cost relative to flat: 4.927 ÷ 3.6 = 1.369 (the climb is about 37% more expensive)
GAP: 5:00 ÷ 1.369 = 3:39/km

So grinding up that hill at 5:00/km was really a 3:39/km effort on flat ground. GAP gives you credit for the climb instead of punishing your average pace for it.

From adjusted pace to effort pace

Gradient is not the only thing that makes a given pace harder. On top of GAP we apply two more corrections, so a flat run in the cool and a mountain run in the heat land on a comparable effort scale.

heat = 1 + 0.005(T − 15), capped at 1.15 altitude = 1 + 0.01((m − 600) ÷ 300), capped at 1.12

Here T is temperature in degrees Celsius and m is elevation in metres. Warmth above 15C and thin air above 600m each nudge the effort figure up, and both are capped so a single scorching or very high run cannot dominate the picture. These two adjustments are cadennce's own, not part of Minetti's curve, and in mild conditions they sit near 1 and barely move the pace.

Honest caveats

This is peer-reviewed lab science with limited external validity, and it is worth knowing where the curve comes from before you trust it to the decimal.

Minetti's subjects were 10 elite male mountain runners on a treadmill. The curve therefore encodes elite downhill economy and, because it is a treadmill, no air resistance at all.
The fitted level cost of 3.6 J/kg/m is slightly above the value actually measured on the flat (about 3.40), an artifact of forcing one smooth polynomial through the whole range.
It is valid only for gradients from -0.45 to 0.45 (-45% to +45%). Outside that band the polynomial is extrapolation, not measurement.
It is purely metabolic. It rewards a steep descent by the energy it would cost at a matched speed, but a real runner cannot safely take a steep downhill at that metabolically-equivalent pace, so GAP can over-credit steep descents.

We read GAP as what it is: a well-grounded estimate of the flat-equivalent effort behind your pace, most trustworthy on moderate gradients and least so on the steepest descents.