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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