Fit vs. Fat

Last Monday was the Boston marathon.  Steve and I walked out to see a section of the race close to the finish line.  The marathon includes a wheelchair group, a few of whom we saw go by.  I was surprised by the variety in wheelchair designs, since it seemed like this could have a big impact on a racer's performance.  As I started thinking about it, it occurred to me that even identical wheelchairs could perform differently due to their rider's weight.  Compared to running, wheelchairs preserve momentum much more efficiently, so even though it would be harder for a heavy person to get up to speed, they would stay there much more easily.

Consider a pair of racers, identical except for their weight.  They each have an amount of energy, E, to expend during the race.  They will each race by accelerating to some top speed, vr, where r is 1 for racer 1 and 2 for racer 2.  After reaching top speed, they will remain at that speed for the remainder of the race.  The racers' net force will be given by

where mr is the mass of, ar the acceleration of, and Fr the force exerted by racer r, and Crr is the rolling resistance coefficient.  The energy the racer needs to use then is

With our assumptions about acceleration, we can simplify this to

where xar is the distance spent accelerating and xsr is the distance spent stable of racer r.  However, we're more interested in the time spent accelerating and stable.  The conversion for the stable period is easy,

The conversion for the accelerating part requires a bit of integration, but is pretty standard

So then we have

We want to minimize the racers' total time by varying their top speed, so we differentiate

Since we want

we have

We are still left with a derivative of tsr, but we can set this to zero on the grounds that tsr should be maximized.  Solving for vr gives then

The derivative of E is the same for both racers, so we can just treat it as a constant.  Plugging vr into our equation for E gives

or simplifying,

We're more interested in total time, so we substitute in

Solving for tr then

Clearly, this will be minimized when

Solving for mr,

This is the ideal, and also the maximum, mass under these assumptions.  It does strike me as a little strange that a larger mass would cause tr to become imaginary, but I suspect this is due to mr and E being related.

Also interesting is the implication that, with the ideal mass, the best strategy is actually to accelerate during the entire race, only reaching half maximum speed by the end.  I'm concerned there's a mistake somewhere in here, leading to these strange results, but I don't see it.  Let me know if you notice something amiss.