85 8585 85Chapter 5. Proportional reasoning 852009-05-04 23:52:14 / rev bb931e4b905epressure∝ lforce∝ l3mass∝ l3volume∝ l3area∝ l2The force and area results show that the pressure is proportional tol:p ∼FA∝l3l2= l.With a large-enough mountain, the pressure is larger than the maxi-mum pressure that the rock can withstand. Then the rock flows likea liquid, and the mountain cannot grow taller.This estimate shows only that there is a maximum height but it doesnot compute the maximum height. To do that next step requires esti-mating the strength of rock. Later in this book when we estimate thestrength of materials, I revisit this example.This estimate might look dubious also because of the assumption that moun-tains are cubical. Who has seen a cubical mountain? Try a reasonable al-ternative, that mountains are pyramidal with a square base of side l anda height l, having a 45◦slope. Then the volume is l3/3 instead of l3butthe factor of one-third does not affect the proportionality between force and length.Because of the factor of one-third, the maximum height will be higher for apyramidal mountain than for a cubical mountain. However, there is againa maximum size (and height) of a mountain. In general, the argument fora maximum height requires only that all mountains are similar – are scaledversions of each other – and does not depend on the shape of the mountain.5.4 Animal jump heightsWe next use proportional reasoning to understand how high animals jump,as a function of their size. Do kangaroos jump higher than fleas? We studya jump from standing (or from rest, for animals that do not stand); a run-ning jump depends on different physics. This problem looks underspeci-fied. The height depends on how much muscle an animal has, how efficientthe muscles are, what the animal’s shape is, and much else. The first sub-section introduces a simple model of jumping, and the second refines themodel to consider physical effects neglected in the crude approximations.5.4.1 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 using sym-bols for the unknown quantities. Maybe all the unknowns cancel.86 8686 8686 5.4. Animal jump heights2009-05-04 23:52:14 / rev bb931e4b905eWe 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 the energy supplied bythe animal’s muscles.The required energy is the easier estimation: An animal of mass m jump-ing to a height h requires an energy Ejump∝ mh. Because all animals feelthe same gravity, this relation does not contain the gravitational acceler-ation g. You could include it in the equation, but it would just carrythrough the equations like unused baggage on a trip.The available energy is the harder estimation. To find it, divide and con-quer. It is the product of the muscle mass and of the energy per mass (theenergy density) stored in muscle.To approximate the muscle mass, assume that a fixed fraction of an animalsmass is muscle, i.e. that this fraction is the same for all animals. If α is thefraction, 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 store the same energy per mass. If this energy per mass is E, thenthe available energy isEavail∼ Emmuscleor, as a proportionality,Eavail∝ mmuscle,where this last step uses the assumption that all muscle has the same ener-gy density E.Here is a tree that summarizes this model:87 8787 87Chapter 5. Proportional reasoning 872009-05-04 23:52:14 / rev bb931e4b905ejump 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 store roughlythree times the energy in its muscles, according to this simple model.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.FleaClick beetleLocustHuman10−3101105103060Mass (g)h (cm)The result, that all animals jump to the sameheight, seems surprising. Our intuition tellsus that people should be able to jump higherthan locusts. The graph shows jump heightsfor animals of various sizes and shapes [source:Scaling: Why Animal Size is So Important [31,p. 178]. Here is the data:Animal Mass (g) Height (cm)Flea 5 ·10−420Click beetle 4 ·10−230Locust 3 59Human 7 ·1046088 8888 8888 5.4. Animal jump heights2009-05-04 23:52:14 / rev bb931e4b905eThe height varies almost not at all when compared to variation in mass,so our result is roughly correct! The mass varies more than eight orders ofmagnitude (a factor of 108), yet the jump height varies only by a factor of 3.The predicted scaling of constant h (h ∝ 1) is surprisingly accurate.5.4.2 Power limitsPower production might also limit the jump height. In the preceding analy-sis, energy is the limiting reagent: The jump height is determined by theenergy that an animal can store in its muscles. However, even if the animalcan store enough energy to reach that height, the muscles might not be ableto deliver the energy rapidly enough. This section presents a simple modelfor the limit due to limited power generation.Once again we’d like to find out how power P scales (varies) with the sizel Power is energy per time, so the power required to jump to a height h isP ∼energy required to jump to height htime over which the energy is delivered.The energy required is E ∼ mgh. The mass is m ∝ l3. The gravitationalacceleration is independent of l. And, in the energy-limited model, theheight h is independent of l. Therefore E ∝ l3.The delivery time is how long the animal is in contact with the ground,because only during contact can the ground exert a force on the animal. So,the animal crouches, extends upward, and finally leaves the ground.