Walking uphill is harder than walking on level ground. Walking downhill is easier—until it’s not, as anyone who has walked down a long, steep mountain slope eventually discovers. The precise relationship between how fast you walk, how steep your trail is, and how much energy you burn turns out to be less obvious than you might assume, which is why researchers at the United States Army Research Institute of Environmental Medicine decided to develop an equation that captures these nuances.

Lots of researchers have tackled this question over the years, most notably the Italian physiologist Rodolfo Margaria back in the 1930s, so the basics are well understood. When you head uphill, your energy expenditure is directly proportional to the steepness of the grade. When you go downhill, in contrast, your energy expenditure initially decreases but at grades of about -10 percent it reaches a minimum then starts to increase again.

But finding a single equation that captures this pattern has proven to be difficult. In the 2019 paper, which is published in Medicine & Science in Sports & Exercise, the military researchers decided to take an existing equation that works on level ground—the Load Carriage Decision Aid—and add a term to adapt it for uphill and downhill calculations. They pulled existing data from 11 different studies and used that data to fine-tune their equation, with the result that it outperformed four previous calorie estimation equations.

For the record, the equation gives you energy expenditure (EE) in watts per kilogram of body mass, as a function of walking speed (S) in meters per second and gradient (G) in percent:

EE = 1.44 + 1.94*S^0.43 + 0.24*S^4 + 0.34*S*G*(1-1.05^(1-1.11^(G+32)))

In case that’s a little hard to visualize, let’s take a look at the equation’s basic behavior. With a few adjustments to express the energy expenditure in calories and the walking speed in miles per hour, and assuming a hiker weight of 150 pounds, here’s a basic graph for level ground (G = 0):

In theory, backpackers have similar interests, but few bother to model the energy requirements of their routes in that much detail. Instead, the new equation is mainly interesting to me as a way of getting a rough estimate of the relative difficulty of hiking at different slopes. The horizontal line on the graph above tells me that if I’m used to hiking at 4 miles per hour on level ground, it’ll take a similar effort to go 3 miles per hour up a 5 percent slope, and 2 miles per hour up a 10 percent slope.

*Alex Hutchinson is the author "Endure: Mind, Body, and the Curiously Elastic Limits of Human Performance."*