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5 Proportional reasoning 35 Since A is the cross sectional area of the animal Ah is the volume of air that it sweeps out in the jump and Ah is the mass of air swept out in the jump So the relative importance of drag has a physical interpretation as a ratio of the mass of air displaced to the mass of the animal To find how this ratio depends on animal size rewrite it in terms of the animal s side length l In terms of side length A l2 and m l3 What about the jump height h The simplest analysis predicts that all animals have the same jump height so h l0 Therefore the numerator Ah is l1 the denominator m is l3 and Edrag Erequired l2 l 1 l3 So small animals have a large ratio meaning that drag affects the jumps of small animals more than it affects the jumps of large animals The missing constant of proportionality means that we cannot say at what size an animal becomes small for the purposes of drag So the calculation so far cannot tell us whether fleas are included among the small animals The jump data however substitutes for the missing constant of proportionality The ratio is Edrag Erequired Ah l2 h m animal l3 It simplifies to Edrag Erequired animal h l As a quick check verify that the dimensions match The left side is a ratio of energies so it is dimensionless The right side is the product of two dimensionless ratios so it is also dimensionless The dimensions match Now put in numbers A density of air is 1 kg m 3 The density of an animal is roughly the density of water so animal 103 kg m 3 The typical jump height which is where the data substitutes for the constant of proportionality is 60 cm or roughly 1 m A flea s length is about 1 mm or l 10 3 m So Edrag Erequired 1 kg m 3 1m 1 3 3 10 kg m 10 3 m The ratio being unity means that if a flea would jump to 60 cm overcoming drag would require roughly as much as energy as would the jump itself in vacuum Drag provides a plausible explanation for why fleas do not jump as high as the typical height to which larger animals jump 5 4 3 Cycling 6 055 Art of approximation 36 This section discusses cycling as an example of how drag affects the performance of people as well as fleas Those results will be used in the analysis of swimming the example of the next section What is the world record cycling speed Before looking it up predict it using armchair proportional reasoning The first task is to define the kind of world record Let s say that the cycling is on a level ground using a regular bicycle although faster speeds are possible using special bicycles or going downhill To estimate the speed make a model of where the energy goes It goes into rolling resistance into friction in the chain and gears and into drag At low speeds the rolling resistance and chain friction are probably important But the importance of drag rises rapidly with speed so at high enough speeds drag is the dominant consumer of energy For simplicity assume that drag is the only consumer of energy The maximum speed happens when the power supplied by the rider equals the power consumed by drag The problem therefore divides into two estimates the power consumed by drag and the power that an athlete can supply The drag power Pdrag is related to the drag force Pdrag Fdrag v v3 A It indeed rises rapidly with velocity supporting the initial assumption that drag is the important effect at world record speeds Setting Pdrag Pathlete gives vmax Pathlete A 1 3 To estimate how much power an athlete can supply I ran up one flight of stairs leading from the MIT Infinite Corridor The Infinite Corridor being an old building has spacious high ceilings so the vertical climb is perhaps h 4 m a typical house is 3 m per storey Leaping up the stairs as fast as I could I needed t 5 s for the climb My mass is 60 kg so my power output was potential energy supplied time to deliver it mgh 60 kg 10 m s 2 4 m 500 W t 5s Pauthor Pathlete should be higher than this peak power since most authors are not Olympic athletes Fortunately I d like to predict the endurance record An Olympic athlete s long term power might well be comparable to my peak power So I use Pathlete 500 W The remaining item is the cyclist s cross sectional area A Divide the area into width and height The width is a body width perhaps 0 4 m A racing cyclist crouches so the height is maybe 1 m rather than a full 2 m So A 0 4 m2 Here is the tree that represents this analysis 5 Proportional reasoning 37 vmax Pdrag v 3 A Pathlete Estairs tstairs tstairs 5s Estairs 2400 J m 60 kg g 10 m s 2 1 kg m 3 v A 0 4 m2 h 4m Now combine the estimates to find the maximum speed Putting in numbers gives vmax Pathlete A 1 3 500 W 1 kg m 3 0 4 m2 1 3 The cube root might suggest using a calculator However massaging the numbers simplifies the arithmetic enough to do it mentally If only the power were 400 W or instead if the area were 0 5 m Therefore in the words of Captain Jean Luc Picard make it so The cube root becomes easy vmax 400 W 1 kg m 3 0 4 m2 1 3 1000 m3 s 3 1 3 10 m s 1 So the world record should be if this analysis has any correct physics in it around 10 m s 1 or 22 mph The world one hour record where the contestant cycles as far as possible in one hour is 49 7 km or 30 9 mi The estimate based on drag is reasonable 5 4 4 Swimming The last section s analysis of cycling helps predict the world record speed for swimming The last section showed that vmax Pathlete A 1 3 To evaluate the maximum speed for swimming one could put in a new and A directly into that formula However that method replicates the work of multiplying dividing and cube rooting the various values Instead it is instructive to scale the numerical result for cycling by looking at how the maximum speed depends on the parameters of the situation In other words I ll use the formula for vmax to work out the ratio vswimmer vcyclist and then use that ratio along with vcyclist to work out vswimmer The speed vmax is vmax Pathlete A 1 3 6 055 Art of approximation 38 So the ratio of swimming and cycling speeds is Pswimmer vswimmer vcyclist Pcyclist 1 3 swimmer cyclist 1 3 Aswimmer Acyclist 1 3 …


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MIT 6 055J - Proportional reasoning

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