78 7878 78702010-05-13 00:43:32 / rev b667c9e4c1f1+4.2 Mountain heightsThe next example of proportional reasoning explains why mountains can-not become too high. Assume that all mountains are cubical and made ofthe same material. Making that assumption discards actual complexity,the topic of ??. However, it is a useful approximation.To see what happens if a mountain gets too large, estimate the pressureat the base of the mountain. Pressure is force divided by area, so estimatethe force and the area.The area is the easier estimate. With the approximation that all moun-tains are cubical and made of the same kind of rock, the only parameterdistinguishing one mountain from another is its side length l. The areaof the base is then l2.Next estimate the force. It is proportional to the mass:F ∝ m.In other words, F/m is independent of mass, and that independence iswhy the proportionality F ∝ m is useful. The mass is proportional to l3:m ∝ volume ∼ l3.In other words, m/l3is independent of l; this independence is why theproportionality m ∝ l3is useful. ThereforeF ∝ l3.79 7979 79712010-05-13 00:43:32 / rev b667c9e4c1f1+pressure∝ lforce∝ l3mass∝ l3volume∝ l3area∝ l2The force and area results show that the pressure is pro-portional to l:p ∼FA∝l3l2= l.With a large-enough mountain, the pressure is larger thanthe maximum pressure that the rock can withstand. Thenthe rock flows like a liquid, and the mountain cannot growtaller.This estimate shows only that there is a maximum heightbut it does not compute the maximum height. To do thatnext step requires estimating the strength of rock. Laterin this book when we estimate the strength of materials, Irevisit this example.This estimate might look dubious also because of the assumption thatmountains are cubical. Who has seen a cubical mountain? Try a reason-able alternative, that mountains are pyramidal with a square base of sidel and a height l, having a 45◦slope. Then the volume is l3/3 instead of l3but the factor of one-third does not affect the proportionality between force andlength. Because of the factor of one-third, the maximum height will behigher for a pyramidal mountain than for a cubical mountain. However,there is again a maximum size (and height) of a mountain. In general,the argument for a maximum height requires only that all mountains aresimilar – are scaled versions of each other – and does not depend on theshape of the mountain.4.3 Jumping highWe next use proportional reasoning to understand how high animalsjump, as a function of their size. Do kangaroos jump higher than fleas?We study a jump from standing (or from rest, for animals that do notstand); a running jump depends on different physics. This problem looksunderspecified. The height depends on how much muscle an animal has,how efficient the muscles are, what the animal’s shape is, and much else.The first subsection introduces a simple model of jumping, and the sec-ond refines the model to consider physical effects neglected in the crudeapproximations.80 8080 80722010-05-13 00:43:32 / rev b667c9e4c1f1+4.3.1 Simple modelWe want to determine only how jump height varies with body mass. Eventhis problem looks difficult; the height still depends on muscle efficiency,and so on. Let’s see how far we get by just plowing along, and usingsymbols for the unknown quantities. Maybe all the unknowns cancel.We want an equation for the height h in the form h ∼ mβ, where m is theanimal’s mass and β is the so-called scaling exponent.mmhJumping requires energy, which must be provided by muscles.This first, simplest model equates the required energy to theenergy supplied by the animal’s muscles.The required energy is the easier estimation: An animal of massm jumping to a height h requires an energy Ejump∝ mh. Becauseall animals feel the same gravity, this relation does not containthe gravitational acceleration g. You could include it in theequation, but it would just carry through the equations likeunused baggage on a trip.The available energy is the harder estimation. To find it, divide andconquer. It is the product of the muscle mass and of the energy per mass(the energy density) stored in muscle.To approximate the muscle mass, assume that a fixed fraction of an ani-mals mass is muscle, i.e. that this fraction is the same for all animals. Ifα is the fraction, thenmmuscle∼ αmor, as a proportionality,mmuscle∝ m,where the last step uses the assumption that all animals have the same α.For the energy per mass, assume again that all muscle tissues are thesame: that they store the same energy per mass. If this energy per massis E, then the available energy isEavail∼ Emmuscleor, as a proportionality,81 8181 81732010-05-13 00:43:32 / rev b667c9e4c1f1+Eavail∝ mmuscle,where this last step uses the assumption that all muscle has the sameenergy density E.Here is a tree that summarizes this model:jump height henergy requiredh m genergy availablemuscle massanimal’s mass m muscle fractionenergy densityin muscleNow finish propagating toward the root. The available energy isEavail∝ m.So an animal with three times the mass of another animal can storeroughly three times the energy in its muscles, according to this simplemodel.Now compare the available and required energies to find how the jumpheight as a function of mass. The available energy isEavail∝ mand the required energy isErequired∝ mh.Equate these energies, which is an application of conservation of energy.Then mh ∝ m orh ∝ m0.In other words, all animals jump to the same height.82 8282 82742010-05-13 00:43:32 / rev b667c9e4c1f1+fleaclick beetlelocusthuman101102h (cm)10−410−1102m (g)The result, that all animals jump to thesame height, seems surprising. Our intu-ition tells us that people should be ableto jump higher than locusts. The graphshows jump heights for animals of var-ious sizes and shapes [source: Scaling:Why Animal Size is So Important [26, p. 178].Here is the data:Animal Mass (g) Height (cm)Flea 5·10−420Click beetle 4·10−230Locust 3 59Human 7·10460The height varies almost not at all when compared to variation in mass,so our result is roughly correct! The mass varies more than eight ordersof magnitude (a factor of 108), yet the jump height varies only by a factorof 3. The predicted scaling of constant h (h ∝ 1) is surprisingly