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# Does 100 m equal 1 km?

Climbing or not climbing?

Photo : Alexis Martín

In trail running or orienteering people say that if you have to run 100 m of elevation up and down it would take the same time as running flat for 1000 m. We can find a similar old rule of thumb and more recently, some researchers (Davey, Hayes & Norman, 1994 ; Scarf, 2007) added a little science to this vernacular knowledge but with few data points (2 athletes and 300 race results, respectively).

Could we add some modern massive data to check this saying ? Using our dataset scraped from ITRA on 16 949 race results from 2 802 runners, we can fit a basic non linear model to estimate the parameters :

```> (model <- results %>%
>   nls(heures ~ (dist_tot + deniv_tot / k) / v, data = .,
>      start = list(k = 100, v = 8)))

Nonlinear regression model
model: heures ~ (dist_tot + deniv_tot/k)/v
data: .
k      v
87.239  9.565
residual sum-of-squares: 313542

Number of iterations to convergence: 3
Achieved convergence tolerance: 1.292e-06

> confint(model)

Waiting for profiling to be done...
2.5%     97.5%
k 81.672165 93.296957
v  9.332102  9.809812
```

So we see that the average flat speed sustainable over a long period of our sample (which is biased towards elite runners) is around 9.6 km⋅h-1 and that 1 km flat is equivalent to 87 m [82 – 93] of height gain, not far from the old 100 m. Of course these values will vary according the athlete shape, the total race length and profile and many other parameters…

## References

Davey, R. C., Hayes, M., & Norman, J. M. (1994). Running uphill : An experimental result and its applications. The Journal of the Operational Research Society, 45(1), 25. https://doi.org/10.2307/2583947

Scarf, P. (2007). Route choice in mountain navigation, Naismith’s rule, and the equivalence of distance and climb. Journal of Sports Sciences, 25(6), 719‑726. https://doi.org/10.1080/02640410600874906