75 7575 752010-05-13 00:43:32 / rev b667c9e4c1f1+4Proportional reasoning4.1 Flight range versus size 684.2 Mountain heights 704.3 Jumping high 714.4 Drag 76Symmetry wrings out excess, irrelevant complexity, and proportional rea-soning in one implementation of that philosophy. If an object moves withno forces on it (or if you walk steadily), then moving for twice as longmeans doubling the distance traveled. Having two changing quantitiescontributes complexity. However, the ratio distance/time, also known asthe speed, is independent of the time. It is therefore simpler than distanceor time. This conclusion is perhaps the simplest example of proportionalreasoning, where the proportional statement isdistance ∝ time.Using symmetry has mitigated complexity. Here the symmetry operationis ‘change for how long the object move (or how long you walk)’. This op-eration should not change conclusions of an analysis. So, do the analysisusing quantities that themselves are unchanged by this symmetry opera-tion. One such quantity is the speed, which is why speed is such a usefulquantity.Similarly, in random walks and diffusion problems, the mean-square dis-tance traveled is proportional to the time travelled:hx2i ∝ t.So the interesting quantity is one that does not change when t changes:76 7676 76682010-05-13 00:43:32 / rev b667c9e4c1f1+interesting quantity ≡hx2it.This quantity is so important that it is given a name – the diffusion con-stant – and is tabulated in handbooks of material properties.4.1 Flight range versus sizeHow does the range depend on the size of the plane? Assume that allplanes are geometrically similar (have the same shape) and therefore differonly in size.Since the energy required to fly a distance s is E ∼ C1/2Mgs, a tank of fuelgives a range ofs ∼EtankC1/2Mg.Let β be the fuel fraction: the fraction of the plane’s mass taken up byfuel. Then Mβ is the fuel mass, and MβE is the energy contained in thefuel, where E is the energy density (energy per mass) of the fuel. Withthat notation, Etank∼ MβE ands ∼MβEC1/2Mg=βEC1/2g.Since all planes, at least in this analysis, have the same shape, their mod-ified drag coefficient C is also the same. And all planes face the samegravitational field strength g. So the denominator is the same for allplanes. The numerator contains β and E. Both parameters are the samefor all planes. So the numerator is the same for all planes. Therefores ∝ 1.All planes can fly the same distance!Even more surprising is to apply this reasoning to migrating birds. Hereis the ratio of ranges:splanesbird∼βplaneβbirdEplaneEbird CplaneCbird!−1/2.Take the factors in turn. First, the fuel fraction βplaneis perhaps 0.3 or0.4. The fuel fraction βbirdis probably similar: A well-fed bird having77 7777 77692010-05-13 00:43:32 / rev b667c9e4c1f1+fed all summer is perhaps 30 or 40% fat. So βplane/βbird∼ 1. Second,jet fuel energy density is similar to fat’s energy density, and plane en-gines and animal metabolism are comparably efficient (about 25%). SoEplane/Ebird∼ 1. Finally, a bird has a similar shape to a plane – it is not agreat approximation, but it has the virtue of simplicity. So Cbird/Cplane∼ 1.Therefore, planes and well-fed, migrating birds should have the samemaximum range! Let’s check. The longest known nonstop flight byan animal is 11, 570 km, made by a bar-tailed godwit from Alaska toNew Zealand (tracked by satellite). The maximum range for a 747-400is 13, 450 km, only slightly longer than the godwit’s
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