On thinking in extremes
Many physicists like to think in extremes first. Give you an example, often I can see that some vector is a projection of some other vector, then often my first thought is “what's the answer if $\theta = 0$? What about $\pi/2?$” and those are usually extreme cases: something is rolling “down” an un-inclined plane, vs that thing being in free-fall. But based on asking myself those questions I, with my experience, can write down something like $-\sin\theta.$ I knew it was a component of a rotation so it was a sine or a cosine or something like that, it was zero on the flat plane and $-1$ in free-fall, this is the only function which satisfies all of my requirements.
So in this case what is the most extreme way to get yourself from point A to point B? Maybe “baby steps” vs “I clear the whole distance in one amazing jump.”
Now if your hunch is right, these two should probably have the exact same energy requirements, but if your hunch is wrong they're probably very different. You probably don't need to run the actual experiment. That is, you probably have enough intuition to not have to time yourself walking to city blocks with the smallest baby steps you can muster, then measure your distance of the biggest standing jump you can perform, and then try to space those jumps over the same time interval to see if they lead to the same exhaustion level. Your intuition probably tells you that one of those is going to hurt the next day and the other one you could keep doing indefinitely.
So then like good physicists we ask, why does the jumping hurt more? Well, I have these muscle fibers and they get damaged whenever I jump. Okay, but why does that happen? Well, as I land I have a considerable amount of surplus kinetic energy, and energy is conserved so that needs to be dissipated. And it is getting dissipated in those muscle fibers, unless I wanted to be dissipated by the breaking of my bones or the friction of my skin against the pavement. The bigger issue here is that my muscle fibers can convert ATP to ADP and use that to power a muscle contraction, but they do not use muscle extensions or contractions to do the reverse and convert ADP back to ATP. We are not capable of “regenerative braking” so any kinetic energy we generate must be dissipated. Sometimes we use friction with external systems—think of shoes sliding on pavement or so—but in this case we are dissipating the force internally.
This is actually very interesting because it is a behavior of a thermodynamic system that is far from equilibrium. In thermodynamic equilibrium processes tend to be reversible, and also slow. So in some sense your muscles only work because they are being constantly cooled. And it's likely the evolutionary reason for this is that we want these processes to happen very fast, faster than our historical predators’ muscles and faster than our historical prey’s muscles.
Back to the physics. We understand the energy waste of jumping and landing after jumping, now. But jumping is not a perfect model for the energy waste in running, for reasons we can explore in the next section. But we have one clue, our muscles are not reversible. Here are some other clues:
- It is possible to run on a treadmill, and this is not significantly harder than running on a flat surface. This indicates that the energy involved is involved in maintenance of the running state, not in the actual forward kinetic energy.
- It is possible also to run—at least, to do a dance that on Earth would have caused running—in zero gravity on the International Space Station. But, this appears to have been an ineffective means for exercise, because they have a treadmill, and they strap you into a harness with springy bands holding you down to the treadmill. So it's certainly partly resetting the stride, but at least half of it has something to do with our weight.
-People can attain much higher speeds with “kangaroo boots” or heelsprings that act as good elastic energy storage mechanisms.
On resetting the stride: your foot needs to be going a certain speed backwards when it contacts the ground so that your shoe does not slide and you can effectively transfer force that propels you forwards. The problem is that this leaves your foot at the end of the stride with a lot of energy that needs to very rapidly return to the front of your body to be prepared for another step, and while some of it can be redirected, most of this appears to be soaked up just like in a jump. But, again, if that were all of it, why ship the treadmill up to the ISS and not just some bands that would suspend you in one place?
What energy transfers does gravity induce?
I claimed above that jumping is a poor model and this became a bit questioned in comments so I wanted to check myself since biophysics so often challenges my common sense. To that point one interesting paper I found was Gullstrand et al. (2009) “Measurements of vertical displacement in running, a methodological comparison.” Gait & Posture 30: 71-75 (link), which is mostly about a different topic, basically whether you can use a reflector or accelerometer in lieu of a sophisticated center-of-mass model to find vertical displacement during running. Figure 4 of that paper is:
Fig. 4. The relation between step duration (s) and CoM Vdisp (m) for each subject at all running velocities.
This is a really fascinating figure and I had to stare at it for a little while. The first exciting thing to look at is how close all of the step durations are. Almost all of the data is between 300 ms and 375 ms, or 200 steps/minute at fast speeds to 160 steps/minute at slow speeds. We double the speed but the step rate only increases by 25%, I would have expected more! So this means that running faster is actually a function of increasing the stride length moreso than moving your legs faster, but moving your legs faster is certainly part of it.
But to our question, the vertical displacement of the center of mass is a direct measurement of gravitational energy during a step, and so if I divide by step duration I get a power exerted to fight “jumping” motion. So on the fast side I see the point (310 ms, 75 cm) as being in the middle of the cluster of high-speed runs, that ratio is something like 24 W/kg while maybe (370 ms, 100 cm) is more distinctive of the slower-speed, something more like 28 W/kg.
So based on these measurements, I have kind of two conclusions. First one is, “that is a lot!” ... These runners are presumably at least 50 kg so the power exertions in fighting gravity are around a kilowatt of power! Just for comparison, the baseline metabolism is an order of magnitude lower: 2000 kcal/day is about 100 W, some advice in the internet says to get an hour of exercise in the 50-150 W range, so exercise is usually lower than this as well.
But the other observation is, it seems that at higher speeds you are actually fighting gravity less per unit of time, something like 15% less power exerted at double the speed. Now, a caveat, the central purpose of the paper in including this figure is to argue that runners are more sloppy at slower speeds: so some amount of this is due to individual variability rather than some physical constraint of the problem. So I don't feel comfortable saying “we know that you fight gravity less at higher speeds” as some sort of statement of the biophysics of the problem, I can certainly imagine that trained runners are essentially 15% more sloppy when running at slower speeds than at high speeds.
But either way, it's not 100% higher or whatever, like you might expect if this vertical displacement mechanism were to explain our exhaustion during running. Flat or decreasing seems supported by the evidence, steeply increasing is required for it to be a candidate explanation. So what this points to instead is a huge amount of power being stored and released elastically in our leg muscles and joints.
So, what is the answer?
I think what happens is that our strides become less efficient. It is clear that there is so much power being exerted back and forth against gravity that we must be incredibly elastic in our running, just these numbers of 1 kW energy transfers in exercise that burns 100 watts of calories, means we must have something like 90% efficiency.
So I am kind of thinking of the energy transfers as sort of a leaky hose going in a circle. There is this constant flow of energy between the springs of our legs and the gravitational potential energy, and it's a lot of energy sloshing back and forth—but it actually does not get much larger or smaller as you travel faster or slower. That flow is more or less fixed. But, as we take these slightly faster strides and also make them significantly longer, we are pushing our muscles more into an inelastic regime, and so they lose more and more energy. And this energy cannot be reclaimed by our systems because we are nonequilibrium systems.