Study Finds Optimal Angles for That
Uphill Climb

Runners in extreme environments such as adventure races are always looking for an edge over their co-competitors. To that end, a study from the University of Colorado at Boulder published in February in the Journal of Applied Physiology tried to determine the optimal angle range at which running up a steep hill might be. In other words, given a choice of paths which range in steepness to the top of a hill or mountain, to exactly what extent should we trade steepness for overall length to minimize energy expenditure?

First, it's worth taking a moment to understand the various expressions of slope that can be used when discussing hill steepness or incline.

Generally speaking, incline, pitch, grade, gradient and slope are synonymous. The slope of a hill (rise over run, recalling junior high school algebra) can be expressed either as a percentage grade or as an angle of inclination to the horizontal. In the U.S., percentage grade is the most commonly used unit for communicating slopes on highway signs, land surveying documents and treadmills. To express grade as a percent, multiply the ratio of vertical to horizontal distance (the slope) by 100.

In the UC-Boulder study, the researchers provide results using angles of inclination, which are expressed in degrees. The angle of inclination is simply the angle between the slope line (tangent) and the x-axis, therefore a horizontal line has a  0-degree angle of inclination, and a vertical line has a 90-degree angle.

A slope with a value of 1 indicates that the ratio of rise to run is equal; this gives us an angle of inclination of 45 degrees, and a 100% grade.

Runners who participate in “vertical kilometer” (VK) races—in which participants race short but very steep distances—often have options for how steeply they wish to ascend to the finish at the top. The question of which slope would be optimal in terms of speed vs. metabolic cost intrigued the UC-Boulder researchers.

Imagine that you are standing at a trailhead where the base elevation is 9,000 feet. Your friend challenges you to race to the summit of the mountain, at 12,280 feet. This represents an elevation gain of 1,000 meters (3,280 feet). One trail you can take to the top averages 10 degrees of incline and the sign says it is 3.6 miles long. A second trail averages 30 degrees, but is only 1.25 miles long. A third trail averages 40 degrees, but only 1 mile long. To get to the summit the fastest, which trail should you choose?

VK races have grown in popularity in recent years in mountainous regions of the U.S. and Europe. Athletes run or walk up steep slopes ranging between 10 degrees and 30 degrees in order to ascend 1,000 meters over a distance of less than 5K. Professional runners finish 10K races on the flats in under 30 minutes; the world record for a VK course is 29:42, set in Fully, Switzerland on a course with an average slope of 31 degrees.

Think of it this way: If you were running up a slope at an angle of inclination of just one degree, you’d have to run over 70 miles per hour to ascend 1,000 meters fast enough to beat the VK race world record.

They study unearthed a range of slope angles that would allow an athlete to ascend a mountain the most quickly. This “Goldilocks plateau” of angles lies between 20 and 35 degrees, a range that requires the same energy expenditure and results in the same vertical velocity, or rate of ascent. Among the trails in our hypothetical race example, choosing the second trail (30 degrees) and walking as fast as you can within your aerobic capacity is the fastest way to go.

As reference points, standard gymnasium treadmills only reach a maximum incline of around 9 degrees, while a typical black diamond ski run slopes about 25 degrees.

The study examined 15 competitive mountain runners as they ran and walked on the treadmill at seven different angles ranging from 9 to 39 degrees. (The athletes were unable to balance at angles above 40 degrees on the specially-designed treadmills, suggesting a natural limit on the feasible slope for a VK competition.)

The treadmill speed was set so that the vertical rate of ascent was the same.  Thus, the treadmill speeds were slower on the steeper angles.  The study focused on a vertical rate of ascent of just over one foot per second, a pace that the high-level athletes could sustain aerobically during the testing. At that speed, walking used about 9% less energy than running. So, sub-elite athletes can ascend very steep hills faster by walking rather than running.

That’s heartening news for non-competitive exercise enthusiasts: even if you’re not planning to try a VK race anytime soon, the findings indicate that you can still get a good aerobic workout simply by walking up very steep inclines.

J. App. Phys., 2016, Vol. 120, No. 3, 370-375,  http://jap.physiology.org/content/120/3/370

Online Conversion Forum, Angle to Percent Grade,
http://www.onlineconversion.com/forum/forum_1086616424.htm

The Slope of a Road, Formulas Showing Grade, Ratio & Angle Relationships,