Newtonian Mechanics – Forces and Energy Expenditure During Running

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I would like to understand which forces do work during running, to estimate the related energy expenditure. For a bird flying at a constant speed $V$, it's clear that lift $L$ doesn't do any work, and all the energy expenditure is due to drag $D$, which has to be overcome by thrust $T$, which does work at a rate $P=TV$:

However, things seem more complicated for running:

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Here, I'm not sure if the energy expenditure is due mainly to drag $D$, since $D$ for a runner ought to be much smaller than for a flyer. As a matter of fact, $V$ is quite lower, and $D$ is approximately proportional to $V^2$:

$$D=\frac 1 2 \rho_{air}C_D A V^2 $$

wher $C_D$ is the drag coefficient. On the other hand, the cross-sectional area $A$ is larger than for a flyer, and $D$ must be equal to the friction $F$ on the foot (which is obviously in the forward direction, since the foot pushes backward), if the runner is not accelerating horizontally.

However, there's something I don't understand: $D$ is doing work on the runner at a rate $P=DV$. However, the foot on the ground is at rest in the ground reference frame, otherwise the runner would slip. This means that $F$ does no work on the runner! How can then the kinetic energy of the runner be approximately constant? Which other forces are doing work on the runner?

Best Answer

You're right, Frictional $F$ does no work on the runner in the ground frame. Running is a more complex motion.

During the thrust phase of your step, the muscles in your leg are doing work on your moving torso. ($F$ holds your foot still so this work can be done). Then during the recovery phase, your leg muscles do work on your foot to lift and accelerate it ahead of your torso.

Unlike the airplane where much of the loss is in drag, the runner will also have losses due to the inelastic collisions with the ground. Every step that lands will have vibrational/heat losses that are not recovered with the next step. There's no simple "proportional to $v^2$" type equation that I'm aware of to estimate these biomechanical inelastic losses.

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