The study energy expenditure of walking and running concludes that running spends more energy than walking.
My understanding is that although running makes one feel more tired, that only indicates that the power was higher (since the time of displacement was shorter), but at the end of the day the total energy dispensed to move oneself forward by friction should be the same.
Given the study shows otherwise, what could be the flaw in my reasoning?
The basic reason is that an ideal object does work in the physics sense, but a biological entity has so many, many, ways it doesn't behave like an ideal object.
One can estimate and calculate, but other answers attempt that. I'm just going to try and summarise some of these big not-ideal-object behaviours
This answer may be completely incorrect. See discussion in comments. The formula for power halfway down the page is definitely incorrect. If a moderator would like to delete the answer that would be fine, but I'll leave it up until then since I think the discussion below has value. I think my formula for average power should be 1/8 of what I used, which would no longer qualify as most of the energy expenditure, even accounting for muscle inefficiency.
Most of the extra energy expenditure for running is the component jumping, not the component spent on horizontal propulsive force. Once you're at your top speed, it doesn't take much more horizontally applied power to keep you going than it takes to walk. But to maintain that speed you need to spend much of the time airborne, and the up and down motion has a high power cost.
Addendum with actual calculation:
A typical runner spends about half of every stride (about 0.15 s) airborne when running at speeds above 6m/s.$^1$ That is to say, gravity is doing work on them for 0.15 seconds out of every 0.3 seconds. If we assume that the runner's collision with the ground every stride is perfectly inelastic, then to not fall flat, they must therefore apply a time-averaged vertical thrust with power half of that which gravity is applying to them during each 0.15s period of falling.
from $Pt = T = 1/2 mv^2, a = dv/dt$
we have:
$$\langle P \rangle = m\Delta ta^2$$
so:
$$P_{\text{runner}} = \frac{\Delta t_{\text{air}}}{\Delta t_{\text{stride}}} P_{\text{gravity}} = 0.5\cdot0.15\mathrm s\cdot g^2\cdot m_{\text{runner}}$$
That is about 7 watts per kilogram of the runner. Extrapolating from the data in (1) it would be significantly less if you were jogging slowly (more ground contact time and less air time per stride), and slightly less if you were sprinting very fast (less ground time and air time per stride, but about the same ratio of ground and air time). Of course, muscle power isn't 100% efficient, so I'd expect the real value to be something considerably higher.
The people in the study were actually "running" very slowly - 2.8m/s, which is more of a lazy jog and includes almost no air time per stride. So it's not surprising that their difference in power output vs walking is small (245 watts).
Inputting 0.05s airtime and .35s of ground time per stride (from (1)) into the above formula gets, for a 70kg runner, $P \approx 45\mathrm W$, leaving about 200W as wasted power.
1:http://gsnider.blogspot.com/2014/01/running-physics-redux-part-1-running.html
I made a math error the first time I posted this addendum. $P = 2m\Delta ta^2$ as I stated but we want time-averaged power, while the power applied by gravity increases with falling time as per the formula. The formula I should have used (now fixed) is $\langle P \rangle = m\Delta ta^2$
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:
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?
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.
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.
When running, muscles require a higher consumption of oxygen, so it's expected that a significant part of the ATP synthesis takes the fermentation route, which is knowingly less efficient than the usual cellular respiration.
Jumping by itself, accurately predicts all the results found in the paper as will be show below.
The impact of friction is almost negligible in the energy expenditure from walking and running. For starter consider ground friction: It only applies when your foot is touching the ground during your gait. At this point, there is no relative motion between your feet and the ground, and therefore (unless you slip) work done is $\approx 0$. Second air friction: although small corrections need to be consider depending on how fast you are running, it usually is not the main cause of energy expenditure as you can prove by running in a treadmill. Look at the paper and observe that running on the threadmill: $481 \, \mathrm{J}$ and on track: $480 \, \mathrm{J}$, i.e., no difference at all.
The main reason for energy expenditure on running and walking is work done against the force of gravity, and since horizontal displacement is perpendicular to gravitational force it costs you nothing and is completely irrelevant. This is the flaw in the paper's authors reasoning: They are moving a “specific mass” perpendicularly to the relevant force.
Cause 1: The fundamental reason for the difference observed: Jumping
Contrary to the wheeled motion, motion with legs do have a vertical component. Let $r$ be the length of your leg measured from hip to ankle. If you start straight up with your legs stretched, your center of mass (approximately at one’s belly button) will be at height $h_0$. At the moment your front ankle touches the ground with your feet maximally separated, your center of mass will “fall” to the position $h_1$ by the amount
$$\Delta h=h_0 – h_1=r(1-\mathrm{cos (\alpha)}) \tag{1}$$
Where $\alpha$ is the angle formed by your legs and the normal line to the ground. This fall costs you nothing. It is at gravity’s expenses. However, to recover your height at the middle of your gait you need to use muscle force and “climb” back to $h_0$. This procedure repeats itself at each step, and due to this, the motion of one’s center of mass looks-like an up-and-down wave (see dotted line in the image below). Even thought you still at ground level, after $n$ steps you have climbed (and felled) an equivalent height of
$$h=n \Delta h \tag{2}$$
Motion with legs includes a vertical up-and-down, wave-like component which is the basic responsible for walking energy consumption: See dotted line. From: “Three-Dimensional Gait Analysis Can Shed New Light on Walking in Patients with Haemophilia“, Sebastien Lobet et al, No changes made to the image
And performed a work of
$$W=n \times mg \times \Delta h \tag{3}$$
Where $mg$ is the person weight and $\Delta h$ is the average vertical displacement of the person’s belly button at each step. From $(3)$ you can see that the energy expenditure depends on the total number of steps and on person's weight (as expected). This dependence on weight explains why women accounted for less energy expenditure after the exercise in the paper. It happened because they were lighter, notice that the ratio between the average weight for man by the average weight for women ($76.6 \, \mathrm{kg} / 63.9 \, \mathrm{kg} \approx 1.2$) is the same as the ratio of average energy expenditure for man by the average energy for women ($520.6 \, \mathrm{kJ}/ 441.1 \, \mathrm{kJ} \approx 1.2$).
During a run, this pattern (the vertical motion) considerably increases, becoming little jumps. The ratio of the work done running vs walking is
$$\frac{W_{ walk }}{W_{run }}=\frac{n}{N} \frac{\Delta h_{walk }}{\Delta h_{ run}} \tag{4}$$
Where $N$ is the total number of steps done running.
From this, this and this papers we get $\Delta h_{walk} \approx 5 \, \mathrm{cm}$, $\Delta h_{run} \approx 12 \, \mathrm{cm}$ , $n \approx 2352^1$, $N \approx 1882^1$ therefore $$W_{ walk } \approx 0.52 \, \times W_{ run} \tag{5}$$ In other words, walking accounts for only $52\%$ of the energy expended by running the same distance. This is exactly what was found in the paper (males: $54\%$, females: $52\%$). As you see equation $(3)$ fully explains all the difference found in the paper, including the difference in energy expenditure for males and females. Still, in the paper the run only took approximately $10$ minutes. For longer runs a second effect will probably start make a big difference. This effect is
Cause 2: Heat dissipation.
As you noticed, when running there is a higher power in the game. This increased power would make a person over-heat, which is a potentially life-threatening condition. To prevent such a catastrophic event, our body starts sweating to create a natural air cooling system, because liquid water will require an astonishing $2260 \, \mathrm{kJ/kg}$ of latent heat of vaporization to vaporize. If such a process would prevent a regular $65 \, \mathrm{kg}$ jogger from overheating by a dangerous $2 \,^{\circ} \mathrm{C}$, this would account for an energy expense of $170 \, \mathrm{kcal}$. This effect would pile up as the running session becomes longer, to the point of having a considerable impact in a half- or in a marathon.
Cause 3: Metabolic efficiency:
Due to the increased need of oxygen, your cells will start some anaerobic shortcuts to the glycose metabolism and build up lactic acid, resulting first, in a less energy efficient process and second, since this lactic acid will start causing problems to the runner, like soreness for example, some of it will be excreted from your body without further metabolization and its energy will be wasted.
Cause 4: Wind drag:
Wind resistance increases with the speed.
Cause n: Non-ideal machine in an non-ideal environment:
As a non-ideal machine moving in non-ideal environment, changes in speed may cause different kind of energy loss. For example, the intensity of sound waves created when smashing the ground, ground friction, muscle and tendon elasticity,… you name it…
$^1$Steps for $1607 \, m \approx 1$ mile.
One difference is drag, which has a power requirement scaling with the third power of speed.
I couldn't easily find an assessment of drag for running, but this is commonly calculated for cycling, and a cyclist is also an upright human. Of course the ground losses will be different (rolling is more efficient than the intermittent movement of running) so this isn't a complete picture, just an estimate of one element. Putting the figures from the abstract linked in the question (1.41 or 2.82m/s, 1600m distance) into a bike power calculator with a runner of 70kg and a weightless bike gives an expenditure of 27kJ at walking pace, and 37kJ running just to overcome air resistance. In power terms this is 7W for nearly 17 minutes walking or 16W for nearly 10 minutes walking.
This, by the way, is why long distance running records use a team of pace-runners. They're obviously going faster so drag has a bigger effect.
The flaw in your reasoning is simply that most systems do not behave as ideal and simplified systems. This is especially true for biological creatures.
In the ideal case, the energy needed for movement is 0. Energy is only needed for an increase in potential energy or speed. However, this is markedly not the case for people. Walking is the a process of something simply following Newton's first law, but rather a person falling forward and catching themselves.
Clearly, some method of movement is going to be more efficient than others. Leapfrogging is less efficient than running, but more efficient than rolling on the ground.
Why running is faster than walking is a complicated question, and is probably better suited for a biologists. However, here are a few reasons.
Drag
Faster speed means more drag. That means you do more work to counter the effect of air friction. This energy is dissipated as heat.
Pendulum motion
Your legs behaves sort of like a pendulum. At the ideal walking speed, your legs swing at the natural frequency. At running speeds, you need to use energy to accelerate your legs back and force.
Vertical motion
Your center of mass goes up and down more when running. When your center of mass is going down, gravity is doing work. However, you don't recover this energy. In fact, you need to use energy to stop your downward velocity. You also need energy to push yourself back up.
Balance
You need to spend more energy to keep balance when running. For example, you need to move your arms back and forth more, and once again, not at their natural frequency.
Practice
Running and Walking are both skills. You get better at them with practice. Better, in this case, includes more efficient. Because a person usually spends more time practicing walking than running, they are better at walking.
A common stumbling block pointed out in the other answers and the comments is the inappropriateness of modeling a runner as a point-like object (a spherical runner in vacuum), as is done in simple classical mechanics. Indeed, kinematic or dynamic (in terms of Newton laws) description of a runner may require more complex models. On the other hand, energetic and thermodynamic arguments do not suffer from this limitation.
A rather general answer given in basic mechanics (and applicable to objects that are not point-like) is that power is proportional to force times velocity: $$ P=\mathbf{F}\cdot\mathbf{v}, $$ that is, moving with the higher velocity, given the same force, results in more energy spent.
Whether the force is constant or increases with velocity depends on how we model running. It is our everyday experience that, if we do not apply any force (e.g., if we stop moving our legs), we stop. The likely reason is the air drag. We thus could write the Newton's equation as $$ m\dot{v}=-F_{drag}(v) + F. $$ The simplest choice for drag force is $F_{drag}(v)=m\nu v$, where $\nu$ is the drag coefficient. However, we may limit ourselves to a general statement that thsi force is directed against the direction of our motion and increases with velocity, i.e., $$ F_{drag}(v)>0,\\ \frac{d}{dv}F_{drag}(v) >0. $$ The Newton equation gives $$ F=F_{drag}(v) \Rightarrow P(v)=F_{drag}(v)v,\\ $$ and the power grows with velocity: $$ \frac{dP}{dv} >0. $$
Remark
As a runner for several decades, I can confirm that a significant amount of body heat is generated, and the faster one goes, the more heat. Unless it's being done in relatively cold weather, this means that the body has to dispose of the heat, which costs even more energy.
Racewalking: when walking uses more energy than running at the same speed.
Other answers deal with both running and walking at the usual speed of the corresponding activity.
One can run at any speed between 0 and his maximum and walk at any speed between 0 and their walking maximum.
And there is a range of speeds where walking is possible, but running is more efficient.
Humans (as well as other animals, most notably horses that can move at quite a few different ways) instinctively switch between different modes at more or less the proper speed.
Why do different modes exist in the first place?
Human mucsles can do two types of work:
By increasing the speed, walking requires more and more static work just in order to keep the feet on the ground. At some point, running becomes easier - even with the additional work needed for jumping higher (some of the jumping energy is recovered by bouncing on the next leg).
As an extreme example: kangaroos. Their muscles have a third type of action: elastic. (Human muscles can do the same, but less efficiently.) They can use their muscles as springs and use this ability for their famous jumping.
Biomechanic efficiency.
Walking has a higher efficiency, i.e you generate more units of useful displacement per unit of energy spent. Consider wheeled, self powered movement, cycling: It is more efficient still, and that is why you can travel further and faster for the same effort.
