The mathematics of hill walking

Life-and-death estimations of how fast people in danger can run up a slope are based only “a random Scottish dude from the 1890s and some data from the 1950s”, according to geographer Michael Campbell of Fort Lewis College in the US. In a paper published in the journal Applied Geography, Campbell and colleagues have used material collected by a huge crowdsourced fitness-tracking database to refine calculations regarding the time taken to climb hill slopes of varying degrees of steepness.

At first blush, perhaps, such considerations might seem to be of interest only to curious hikers and mathematicians with too much time on their hands, but in fact they are of critical importance in several areas of human endeavour.

Fire fighters battling forest blazes, for instance, or infantry soldiers in battle zones may find themselves at lethal risk if they underestimate the time it will take them to get to the top of a hill in an emergency.

Of course, several obvious factors influence the time taken to climb a slope of any given degree, not the least of them personal fitness and whether the chosen gait is walking or running.

Nevertheless, generalised models for speed and energy cost for scaling hills are used by people as diverse as hike organisers, army officers and emergency planners.

The problem, Campbell and his colleagues realised, is that the two most commonly used models contain multiple problems, not the least of which is that they do not sufficiently reflect real-world conditions.

The first is called Tobler’s hiking function. It is named after geographer Waldo Tobler, who in 1993 constructed a mathematical model to describe the relationship between walking, hill slope and time.

It is used today for planning evacuations in the event of a tsunami, wilderness search-and-rescue missions, and several other activities. The issue, though, is that Tobler had access only to a small dataset, which comprised numbers provided by hikers in the 1950s.

Its shortcomings are rather obvious, but even so Tobler’s contribution is still many times more useful than the second most used model, known as Naismith’s Rule.

This dates from 1892, and is based entirely on notes and calculations made by a solitary Scottish mountaineer, William Naismith, who wrote things down, did some sums, and published the results. In effect, Naismith’s Rule makes predictions based on a sample size of one.

In their new approach, Campbell and colleagues use a sample size of just shy of 30,000, all of them people active around Salt Lake City in Utah, US. The dataset comprises information uploaded via fitness tracker or GPS unit as the cohort variously ran, jogged or walked a combined 130,000 kilometres.

“Calculating how quickly people move through the environment is a problem more than a century old,” says co-author Philip Dennison.

“Having data from such a large number of people moving at all different speeds allowed us to create much more advanced models than what’s been done before. Any application that estimates how fast people walk, jog, or run from point A to point B can benefit from this work.”

The new model produces some handy – and accurate – rules of thumb.

The results reveal that a leisurely walk of 1.6 kilometres along flat ground takes an average 33 minutes. Walking at the same speed over the same distance but with a 30 degree upward incline takes 97 minutes. Running in the two scenarios takes six and 13 minutes, respectively.

And – surely a prime contender for most recondite dinner party nugget of the year – walking down a 30 degree slope takes the same amount of time as walking up a 16 degree one.

Campbell has started working with fire crews in Utah, Idaho, Colorado and California in a bid to better tune the model for emergency services applications.

“From a firefighter perspective, under normal conditions a fire crew may have ample time to hike to a safety zone,” he says, “but if the shit hits the fan, they’re going to have to sprint to get there.

“We tried to introduce predictive flexibility that can mimic the range of conditions that one might need to consider when estimating travel rates and times.”

Related reading: Lessons from climbing the highest mountains

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