The Biomechanist Went Over the Mountain

The best way up a hill is steeper than the best way down.

hiking
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I turned forty recently. Of all the gifts I received, the most surprising and rewarding birthday present came from my wife: a three-day hike in the Sierra Nevada. My declaration of midlife vigor included a scamper up to 11,000 feet—a chance to go over a literal mountain rather than the metaphorical hill. But the thin air made me take plenty of time to rest on trailside rocks, which inevitably led the biomechanist in me to reflect on the mechanics of hill climbing.

It seemed that the steeper sections of the trail quickly exhausted my reserves, but that the less steep sections did not get me up the hill fast enough to make satisfactory progress. As it turns out, I’m not the first to have noticed this: studies of human locomotion—not to mention numberless hikers before me—have come to the same conclusion.

Some of the first serious investigations of human walking were done in the late 1930s by Rodolfo Margaria, an Italian physiologist, who measured its metabolic costs at various inclines. His data showed that the optimum grade was, unsurprisingly, downhill. At an incline of about minus ten degrees, roughly the same angle as the wheelchair-access ramp at many curbs, people use the least amount of oxygen to walk a given distance with their natural gait.

What may be surprising is that the energy expenditure isn’t minimized at a steeper incline. After all, isn’t it obvious that getting downhill faster should also be metabolically easier? Well, it’s not, and the reason the ten-degree incline is the easiest lies in the mechanics of walking.

Walking is a cyclic movement, and in any cyclic movement there is the possibility of energy storage and reuse. Your center of gravity (which lies approximately two inches in front of the small of your back) rises and falls with each step. It is highest when one foot is planted firmly on the ground directly beneath you. Then it drops as your body swings forward with the next step and rises again as your other leg passes through the vertical. Each stride represents the sweep of an energy-saving, inverted pendulum.

Imagine that your center of gravity is a steel ball attached to a steel rod, which can represent your planted leg, because during that part of the cycle your knee hardly bends. Your planted foot is the pivot of the pendulum. If you trace the path of the steel ball through a stride, it follows the arc of a single sweep of a pendulum.

walking

Video sequences of a man walking show the bobbing course of his center of gravity (white circle along red curve). As he walks on flat ground (first sequence), his center of mass acts like the mass at the end of an inverted pendulum (broken lines), storing some of the energy of motion of each step as the mass rises and releasing the energy as it falls. Some of the energy of motion remains available for subsequent steps as the slope he climbs increases to a five-degree angle (second sequence), but all the energy is used up in finishing the current step when the angle is fifteen degrees (third sequence).

Tom Moore

In a pendulum system the kinetic energy of the ball as it passes through the lowest part of its swing is traded for the gravitational potential energy of height as it rises to the top of its arc. As the ball falls, the potential energy is then converted back into kinetic energy. Your planted leg, because it acts as the radius of a circle, translates the downward force of gravity into a force pushing you forward. As you might expect, your forward speed is lowest when your planted leg is directly below your center of gravity, and highest when you are striding. At that moment, your forward leg, now planted, acts as the radius of a new circle and translates some of the momentum of your center of gravity into upward motion, preparing you for another gravity-aided step forward.

About 60 percent of the energy spent changing the altitude of a walker’s center of gravity is conserved. Walking on a slight downhill grade adds energy to the system; that energy offsets some of the energy lost to heat generated in the muscles. (On steeper downhill grades the pendulum motion is lost. The cost of each step rises, though not by much.)

But what about uphill? Alberto E. Minetti, an Italian biomechanist at Manchester Metropolitan University in England, recast Margaria’s data in real-world terms. Minetti noted that walkers don’t usually choose the slope at which each step costs the least because they are usually glad to trade a quick gain in altitude for the cost of a little extra fatigue.

In my case, according to my topographic maps, I needed to climb 1,108 feet on the final push to the summit. The shortest route would be the “directissima”—a vertical ascent, much as a spider might do it. Such a route—because metabolic rate rises rapidly with increasing slope—would be hugely expensive. What about a gentler slope—say an angle of about five degrees? That grade would barely rise, but I would have to walk nearly two-and-a-half miles to gain those 1,100 feet. The net cost of taking the “easy” way would be high simply because I would have to walk so far to reach the summit.

Minetti discovered the best solution. When the slope of the route is about fifteen degrees, a rise of about a foot for every four feet of horizontal travel, the energy required to gain a certain altitude is minimized. Hence to minimize the energetic cost of climbing, hikers should proceed directly up slopes no steeper than fifteen degrees, but should take switchbacks on steeper grades, keeping the climbing angle at fifteen degrees.

Minetti’s conclusion is borne out in a somewhat surprising and satisfying way: from data gleaned from topographic maps of trails connecting mountain towns, in mountain ranges from the Dolomites to the Himalaya. Sure enough, as I panted by the side of the trail in the Sierra Nevada, a Park Service sign informed me that the 1,100 foot climb to Twin Peaks would cover a mile, making an incline of about twelve degrees, near enough the optimum slope but emphatically not metabolically free.

It’s nice to know that, metabolically, there is a best way to climb higher. But I think there’s another practical implication of Margaria’s and Minetti’s work. If the best grade up is fifteen degrees, and the best grade down is only ten, the most efficient round trip can’t take you down the same way you came up. So you get two hikes every time you climb a slope. That rule notwithstanding, the Park Service would prefer that you stick to the path.

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