28 2828 286.055 / Art of approximation 282008-01-14 22:31:34 / rev 55add9943bf1Next estimate the force. It is proportional to the mass:F ∝ m.In other words, F/m is independent of mass, and that independence is why the proportion-ality 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 the proportionalitym ∝ l3is useful. ThereforeF ∝ l3.pressure∝ lforce∝ l3mass∝ l3volume∝ l3area∝ l2The force and area results show that the pressure is proportional to l:p ∼FA∝l3l2= l.With a large-enough mountain, the pressure is larger than the maximum pressurethat the rock can withstand. Then the rock flows like a liquid, and the mountaincannot grow taller.This estimate shows only that there is a maximum height but it does not compute themaximum height. To do that next step requires estimating the strength of rock. Laterin this book when we estimate the strength of materials, I revisit this example.This estimate might look dubious also because of the assumption that mountains are cu-bical. Who has seen a cubical mountain? Try a reasonable alternative, that mountains arepyramidal with a square base of side l and a height l, having a 45◦slope. Then the volumeis 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 be higher for a pyrami-dal mountain than for a cubical mountain. However, there is again a maximum size (andheight) of a mountain. In general, the argument for a maximum height requires only thatall mountains are similar – are scaled versions of each other – and does not depend on theshape of the mountain.5.3 Animal jump heightsWe next use proportional reasoning to understand how high animals jump, as a functionof their size. Do kangaroos jump higher than fleas? We study a jump from standing (orfrom rest, for animals that do not stand); a running jump depends on different physics.This problem looks underspecified. The height depends on how much muscle an animalhas, how efficient the muscles are, what the animal’s shape is, and much else. The firstsubsection introduces a simple model of jumping, and the second refines the model toconsider physical effects neglected in the crude approximations.29 2929 295 Proportional reasoning 292008-01-14 22:31:34 / rev 55add9943bf15.3.1 Simple modelWe want to determine only how jump height varies with body mass. Even this problemlooks difficult; the height still depends on muscle efficiency, and so on. Let’s see how farwe get by just plowing along, and using symbols for the unknown quantities. Maybe allthe unknowns cancel.We want an equation for the height h in the form h ∼ mβ, where m is the animal’s mass andβ is the so-called scaling exponent.mmhJumping requires energy, which must be provided by muscles. This first, simplest modelequates the required energy to the energy supplied by the animal’s muscles.The required energy is the easier estimation: An animal of mass m jumping to a heighth requires an energy Ejump∝ mh. Because all animals feel the same gravity, this relationdoes not contain the gravitational acceleration g. You could include it in the equation,but it would just carry through the equations like unused baggage on a trip.The available energy is the harder estimation. To find it, divide and conquer. It is theproduct of the muscle mass and of the energy per mass (the energy density) stored in mus-cle.To approximate the muscle mass, assume that a fixed fraction of an animals mass is muscle,i.e. that this fraction is the same for all animals. If α 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 the same: that they storethe same energy per mass. If this energy per mass is E, then the available energy isEavail∼ Emmuscleor, as a proportionality,Eavail∝ mmuscle,where this last step uses the assumption that all muscle has the same energy density E.Here is a tree that summarizes this model:jump height henergy requiredhm genergy availablemuscle massanimal’s mass m muscle fractionenergy densityin muscle30 3030 306.055 / Art of approximation 302008-01-14 22:31:34 / rev 55add9943bf1Now finish propagating toward the root. The available energy isEavail∝ m.So an animal with three times the mass of another animal can store roughly three times theenergy in its muscles, according to this simple model.Now compare the available and required energies to find how the jump height as a functionof 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.FleaClick beetleLocustHuman10− 3101105103060Mass (g)h (cm)The result, that all animals jump to the same height, seemssurprising. Our intuition tells us that people should be ableto jump higher than locusts. The graph shows jump heightsfor animals of various sizes and shapes [source: Scaling: WhyAnimal Size is So Important [4, 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 isroughly correct! The mass varies more than eight orders of magnitude (a factor of 108), yetthe jump height varies only by a factor of 3. The predicted scaling of constant h (h ∝ 1) issurprisingly accurate.5.3.2 Power limitsPower production might also limit the jump height. In the preceding analysis, energy isthe limiting reagent: The jump height is determined by the energy that an animal can storein its muscles. However, even if the animal can store enough energy to reach that height,the muscles might not be able to deliver the energy rapidly enough. This section presentsa simple model for the limit due to limited power generation.Once again we’d like to find out how power P scales (varies) with the size l Power is energyper time, so the power required to jump to a height h