Learn & Understand

Why You Slow Down: The Mathematics of Endurance

A runner cannot hold their five-kilometer pace over a marathon — everyone slows as the distance grows. But the slowing follows a remarkably predictable mathematical pattern, one that lets a race time at one distance predict performance at another. Behind this pattern lies the physiology of fatigue and a simple, powerful formula.

Pace Fades With Distance

The longer a race, the slower the sustainable pace — a universal truth of endurance. A sprinter's speed cannot be maintained for a mile, nor a miler's pace for a marathon. As distance increases, the body must ration its effort, and the pace it can hold declines. This fading is not weakness but the inevitable consequence of how fatigue accumulates over time.

A Predictable Decline

Remarkably, this decline is regular enough to be captured in a formula. In the 1970s, Peter Riegel proposed a relationship that predicts race time at a new distance from a known time at another, using an exponent that encodes how much pace typically slows per unit of added distance. It has proven a reliable rule of thumb across a wide range of racing.

More Than Double

The formula's key insight is that doubling the distance more than doubles the time, because pace slows along the way. A ten-kilometer time is a little more than twice a five-kilometer time, not exactly twice, and the gap grows for longer distances. The exponent quantifies exactly this — the small, steady penalty distance exacts on pace.

The Limits of Prediction

The prediction works best between similar distances with comparable training, because the physiological demands stay alike. Extrapolating across very different distances — a short race to a marathon — is less reliable, since the marathon adds challenges of fueling and endurance a short race never tests. The formula predicts the pattern of fading well, but cannot foresee the unique demands of every distance.

Predicting Your Time

To predict a race time across distances, use the Riegel Race Time Predictor. Turn a target into splits with the Marathon Pace Calculator, and relate it to fitness with the VO2 Max Calculator.