Naruto running

"Naruto running" is running with your arms behind you (pictured above). Many characters in the Naruto series are superhuman and can run way faster than a normal human. I thought that they run like this because they run so fast that swinging their arms will add drag instead of making them go faster. Ignoring whether that's true or not, how fast does a person have to run to create enough lift on their body to keep it always leaning forward and not tip over? (I imagine the body is like the wing of an airplane that never actually takes off because the thrust comes from the feet pushing the ground).

EDIT: By leaning forward I meant having enough air resistance to push your body upwards instead of just falling to the ground when you lean forward. Basically like standing in strong winds, but you are the one moving instead of the air:

enter image description here

(As I'm typing this I just realized that the answer might be "as fast as the speed of the wind it takes to make a person lean that far".)

  • $\begingroup$ Why would lift keep the body leaning forward? $\endgroup$ – probably_someone Jul 22 '19 at 8:03
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    $\begingroup$ I think it goes like this: as you're running forward, some air is pushing back on your torso. If you try to lean forward while running at normal speeds, the air pushing back isn't strong enough to do anything, so you'll either balance it by putting your foot further ahead or just fall to the ground. But what if you're running so fast that the air is pushing your torso strong enough that your body doesn't fall but continuously pushed upwards by the air? $\endgroup$ – M Arif Rahman Winandar Jul 22 '19 at 8:46
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    $\begingroup$ Doing your homework for that Area-51 raid, eh? $\endgroup$ – Nat Jul 22 '19 at 9:12
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    $\begingroup$ I wonder how much of this running style is attributed to the ease of drawing a stationary arm (in other words, just how lazy are animators)? $\endgroup$ – Kyle Kanos Jul 24 '19 at 23:26

If you wanted to get an estimate of how fast you'd need to be moving in order to lift yourself with air deflection from running, you need to establish a couple of important parameters:

  1. Body weight $p$
  2. Torso angle $\theta$ (defined such that $0^\circ$ would be running upright)
  3. Speed $s$
  4. Torso length $\ell$
  5. Torso width $w$
  6. Air density $\rho$

Afterwards, all you need to do is a little bit of fluid mechanical force balancing. You can estimate that your torso deflects all of the incoming air you come into contact with straight downwards, which is only a decent approximation if your torso angle is high, in which case the lift force you would feel is simply the change in vertical momentum of the air coming at you in your frame of reference:

$$ \begin{align} L &= v_{\text{down}} \dot{m}_{\text{down}} \\[5px] &= v_{\text{down}} \dot{m}_{\text{front}} \\[5px] &= v_{\text{down}} \left(s w \ell \cos{\left(\theta\right)} \, \rho \right) \\[5px] &= \left(s\frac{\cos{\theta}}{\sin{\theta}}\right) \left(s w \ell \cos{\theta} \right) \\[5px] &= s^2 \cot{\theta} \, \cos{\theta} \, w \ell \rho \end{align} $$

You can see that the approximation is bad for small torso angles since it predicts that you'd shoot off into space if you ran normally $\left(\theta = 0^\circ \right) ;$ that's because air would most certainly not deflect downwards if you ran this way.

To get liftoff, you want this lift to be equal to your body weight:

$$L = pg$$

Let's plug in some numbers. If you weigh $p = 175$ pounds, are running with a torso angle of $\theta = 45^\circ$, have a torso length and width of $\ell = 0.5$ meters and $w = 0.4$ meters respectively, you can plug this in along with an air density of $\rho = 1.225$ kilograms per meter cubed and $g = 9.81$ meters per second squared to find what minimum speed you need to run at:

$$s^2 \, \cot{\theta} \, \cos{\theta} \, w\ell\rho = pg$$

$$s^2 \times \frac{\left(0.2 \times 1.225\right)}{\sqrt{2}} \frac{\mathrm{kg}}{\mathrm{m}} = 778.45\ \mathrm{N}$$

$$s = 67.0332 \frac{\mathrm{m}}{\mathrm{s}} \approx 150\ \text{mph}$$

There are a great deal of things I haven't considered here, like the amount of body weight your feet are holding up by virtue of running, pressure changes along the vicinity of your body, etcetera. But you can always just slap on some safety factors and run three or four times as fast, since your speed would reduce back to the liftoff value (if not less) if your feet were lifted off the ground.

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