Keep on Truckin'

Apologies to anyone frantically refreshing the page looking for a new post; this was my birthday weekend, and my girlfriend, Jen, came out for a visit.

Due to my chemotherapy treatment, I've been spending a lot of time in bed, so Nurse Steve has been diligent in taking me on forced marches through Boston to keep me in shape.  The wheelchair calculations I did earlier got me thinking about the mechanics of walking.  Specifically, it occurred to me that though a longer stride allows you to cover more ground, it causes your body to move up and down more – as your leg goes out, you drop slightly, and as you bring your back leg forward, you return to your full height.

Let's say your legs are a length l and you take strides of length s:

Then your lowest height is

and with every step your change in height is

so every time you lift yourself up, you expend an energy of

Clearly if you wanted to expend the least energy, you'd just take infinitesimal steps, but you'd never get anywhere, so let's define a step quality, Q, as the energy over the stride to some power a (a scaling factor):

As you can see from the plot of this function, the results are a little disappointing. Depending on how much you value stride over energy, it's either best to stretch all the way to the ground, or to take the tiniest steps you can. Oh well, you can't get interesting results every time...

[This conclusion has an error.  You can see my correction here.]