72 7272 7272 4.4. Cycling2009-05-04 23:52:14 / rev bb931e4b905e4.4 CyclingThis section discusses cycling as an example of how drag affects the perfor-mance of people as well as fleas. Those results will be used in the analysisof swimming, the example of the next section.What is the world-record cycling speed? Before looking it up, predict itusing armchair proportional reasoning. The first task is to define the kindof world record. Let’s say that the cycling is on a level ground using aregular bicycle, although faster speeds are possible using special bicyclesor going downhill.To estimate the speed, make a model of where the energy goes. It goes intorolling resistance, into friction in the chain and gears, and into drag. At lowspeeds, the rolling resistance and chain friction are probably important. Butthe 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 max-imum speed happens when the power supplied by the rider equals thepower consumed by drag. The problem therefore divides into two esti-mates: the power consumed by drag and the power that an athlete cansupply.The drag power Pdragis related to the drag force:Pdrag= Fdragv ∼ ρv3A.It indeed rises rapidly with velocity, supporting the initial assumption thatdrag is the important effect at world-record speeds.Setting Pdrag= Pathletegivesvmax∼PathleteρA1/3To estimate how much power an athlete can supply, I ran up one flight ofstairs leading from the MIT Infinite Corridor. The Infinite Corridor, beingan old building, has spacious high ceilings, so the vertical climb is perhapsh ∼ 4 m (a typical house is 3 m per storey). Leaping up the stairs as fast as Icould, I needed t ∼ 5 s for the climb. My mass is 60 kg, so my power outputwasPauthor∼potential energy suppliedtime to deliver it=mght∼60 kg × 10 m s−2× 4 m5 s∼ 500 W.73 7373 73Chapter 4. Symmetry and conservation 732009-05-04 23:52:14 / rev bb931e4b905ePathleteshould be higher than this peak power since most authors are notOlympic athletes. Fortunately I’d like to predict the endurance record. AnOlympic athlete’s long-term power might well be comparable to my peakpower. So I use Pathlete= 500 W.The remaining item is the cyclist’s cross-sectional area A. Divide the areainto width and height. The width is a body width, perhaps 0.4 m. A racingcyclist 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:vmaxPathlete∼ Estairs/tstairsEstairs2400 Jm60 kgg10 m s−2h4 mtstairs5 sPdrag∼ ρv3Aρ1 kg m−3v A0.4 m2Now combine the estimates to find the maximum speed. Putting in num-bers givesvmax∼PathleteρA1/3∼500 W1 kg m−3× 0.4 m21/3.The cube root might suggest using a calculator. However, massaging thenumbers simplifies the arithmetic enough to do it mentally. If only thepower were 400 W or, instead, if the area were 0.5 m! Therefore, in thewords of Captain Jean-Luc Picard, ‘make it so’. The cube root becomeseasy:vmax∼∼400 W1 kg m−3× 0.4 m21/3∼ (1000 m3s−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−1or 22 mph.The world one-hour record – where the contestant cycles as far as possiblein one hour – is 49.7 km or 30.9 mi. The estimate based on drag is
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