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Let's say you are already walking at a maximally efficient combination of pace and stride (or $\omega$ and $X_0$ I guess) but you need to reach your destination faster. Should you increase/decrease your pace and/or stride to achieve this goal (assuming that any adjustment will result in a net increase in speed)? How can I interpret this question as a question about kinetic energy of oscillators?

I have tried solving the equation $KE'=0$ to find a minimum $KE$ but this seems like the wrong approach, for all that seems to tell me is the if you do not walk (ie $w$ or $X_0$ is $0$) then yes, of course you have kept your kinetic energy at a minimum. Where does this idea about "efficiency" come into play?

Edit: Pace is the frequency of steps, stride is the length of steps, and average speed is the product of pace and stride.

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The right approach here will depend somewhat on exactly what assumptions have been made. Your text suggests that you are suppose to treat walking as some kind of harmonic process, but unless we know what model you are expected to use it is hard to know how to proceed. Can you say a little more about what model is being discussed. – dmckee♦ Nov 17 '12 at 18:48
@dmckee I have added definitions for pace, stride, and effective average speed, but to the point of a specific model, I guess that is what I am asking - what is a reasonable (and appropriate for an elementary mechanics student) way of modeling walking using the idea of a harmonic oscillator? – tacos_tacos_tacos Nov 17 '12 at 18:55
Well, the things we can start with include stride is proportional to the amplitude of oscillation, and pace to the frequency (as you have already noted). Speed is then pace * stride, but I still don't quite see where to go with efficiency. Possibly we should be thinking about the $Q$ of a damped-driven oscillator. – dmckee♦ Nov 17 '12 at 19:13
Do large ratios of $Q$ correspond to very efficient walking? (Apologies in advance if that is a thick question.)... and if so, does this mean since we are "already" at $\omega_r$ we should keep that pace up and just increase the length of our stride to what it needs to be to achieve the higher speed? – tacos_tacos_tacos Nov 17 '12 at 19:37
$Q$ represents roughly the inverse fraction of energy given up on each oscillation, so yes high values of $Q$ represent small fractions of the energy given up on each cycle. – dmckee♦ Nov 17 '12 at 19:41
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