A theory of competitive runningJoseph B. Keller

Citation: Physics Today 26, 9, 43 (1973); doi: 10.1063/1.3128231View online: https://doi.org/10.1063/1.3128231View Table of Contents: https://physicstoday.scitation.org/toc/pto/26/9Published by the American Institute of Physics

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A theory of competitive runningUsing simple dynamics one can correlatethe physiological attributes of runners with world trackrecords and determine the optimal race strategy.

Joseph B. Keller

World records for running provide dataof physiological significance. In thisarticle, I shall provide atheory of run-ning that is simple enough to be ana-lyzed and yet allows one to determinecertain physiological parameters fromthe records. The theory, which isbased on Newton's second law and thecalculus of variations, also provides anoptimum strategy for running a race.

The records give the shortest time Tin which a given distance D has beenrun. Our theory determines this func-tion theoretically in terms of the fol-lowing physiological quantities: themaximum force a runner can exert, theresistive force opposing the runner, therate at which energy is supplied by theoxygen metabolism, and the initialamount of energy stored in the runner'sbody at the start of the race. By fit-ting the theoretical curve to four ob-served records, these quantities can bedetermined and the other records canbe predicted. Alternatively, by mea-suring these quantities by independentphysiological studies, one can use thetheory to predict all the track recordsto which it applies.

The theory accounts for the mainfeatures of the records at distancesfrom 50 meters to 10 000 meters. How-ever, it does not account for the rec-ords at larger distances. These rangeup to 59 days for 5560 miles, the dis-tance from Istanbul to Calcutta andback, set by M. Ernst (1799-1846).1

Other physiological factors must be in-cluded to account for these records.

Optimal running strategy

A runner's speed varies during arace; we assume that the speed, r(f), ischosen in the way that minimizes thetime T required to run the distance D,subject to physical and physiologicallimitations. We shall determine thisoptimal speed variation v{t) by formu-lating and solving a mathematicalproblem in optimal control theory.

The theory predicts that the runnershould run at maximum accelerationfor all races at distances less than a

critical distance Dc = 291 meters.Thus the races at distances less than291 meters should be classified togeth-er as "short sprints" or "dashes." ForD greater than 291 meters the theorypredicts maximum acceleration for oneor two seconds, then constant speedthroughout the race until the final oneor two seconds and finally a slightslowing down. This result confirmsthe accepted view that a runner shouldmaintain constant speed to achieve theshortest time, and refines that view byfitting the constant speed to appropri-ate variable speeds during the initialand final seconds.

The author is professor of mathematics atthe Courant Institute of Mathematical Sci-ences, New York University.

The mathematical solutionfor the optimal velocity

For D < Dc the critical distance wehave f(t) = F. and equations 2 and 3yield

1/(0 = FT{\ - e " ) (ADThen equation 1 becomes

O = FT -(J + err* - l) (A2)

This gives the relation between D and 7"for D < Dc. To find E(t) we use equa-tion Al for v{t) in equations 5 and 6 toobtain

E(t) = £„ + at -

F'T-V- + e ' • - lj (A3)

By setting E(TC) - Owe get the largestvalue of t for which the assumption / =

F is consistent with equation 7. ThenDc is the value of D given by equationA2 with T = Tc.

For D > Dc, v(t) is given by A2 for 0< t < t-\. and by v(t) = v{tO = con-stant for fi < t < t2. In the interval l2< t < T, v(t) is obtained by setting E(t)= 0 in equation 5, which yields t =a/v. With this value of F, equation 2can be solved with the result

M M = OT + [('-(?,) - UT]I>-- '

(A4)

The times ^ and tz can be found bycomputing D from equation 1, using thethree expressions for v(t) just given,and then maximizing D with respect tofi and ti The fact that v{t) equals aconstant in the middle interval can beproved by the methods of the calculusof variations.

PHYSICS TODAY SEPTEMBER 1973 43

oo

The 220-yard dash. The optimal velocity v(t) is plotted versus t.f(t) = F is maximum throughout the race.

T

The propulsive forceFigure 1

10 20TIME (sec)

The 400-meter run. The optimal velocity v{t) is plotted versus f. The propulsive forcefit) = F is a maximum during the initial 1.78 seconds. After this initial acceleration, vremains constant until 0.86 seconds before the end of the race when the oxygen supply Ebecomes zero. Finally E{t) remains zero during the last 0.86 seconds. This can be un-derstood by thinking about a car with a limited amount of gas traveling over a distance Din the shortest time; the fuel should be used up shortly before the end. Figure 2

Runners at distances greater than291 meters often finish with a kickrather than with the negative kick ofthe optimal solution. This discrepan-cy indicates either that they are notdoing as well as they could or that thetheory is inadequate. Presumablytheir goal is to beat competitors ratherthan to achieve the shortest time, andthat goal influences their strategy.But if they ran at the optimal speeddetermined by the theory, they might

do even better at beating competitors.Some trials in which runners attemptto follow the optimal strategy mightdetermine which is the correct expla-nation, and whether the optimal solu-tion is better than the usual strategy.

In our theory we assume that the re-sistance to running at speed u is pro-portional to v. Another assumption,suggested by measurements of R. Ma-garia and his collaborators,2-3 is thatthe resistance is a constant indepen-

dent of the velocity. We nave wontedout the theory in this second case forcomparison and found that it leads to aquite unsatisfactory prediction, whichindicates that it must be rejected if theother assumptions of the theory arecorrect.Formulation of the theory

The length D of a race is related tothe time T required to run it by theequation

D = /Jo

v(t)dt (1)

The velocity vit) is determined by theequation of motion, which we assumeto be

(2)

In this equation fit) is the total propul-sive force per unit mass exerted by therunner, part of which is used to over-come the internal and external resis-tive force V/T per unit mass. It is anassumption that the resistance is a lin-ear function of v and that the dampingcoefficient r is a constant. Initiallythe runner is at rest; so

r(0) =0 (3)

The force fit) is under the control ofthe runner; so we may think of it asthe control variable. The runner mustadjust it so that T, determined byequation 1, is as small as possible whenv(t) is the solution of equations 2 and3. There are two restrictions on fit).First, there is a constant maximumforce per unit mass F that the runnercan exert; so / must satisfy the inequa-lity

fit) < F (4)

Second, the rate fv of doing work perunit mass must equal the rate at whichthe body supplies energy. This rate islimited by the availability of oxygen forthe energy-releasing reactions, whichwe shall now consider.

Initially there is a certain quantity ofavailable oxygen in the muscles, andmore oxygen is provided by the respi-ratory and circulatory systems. It isconvenient to measure the quantity ofavailable oxygen in units of the energyit could release upon reacting. Thuswe denote by E(t) the energy equiva-lent of the available oxygen per unitmass at time r, by Eo the initialamount, and by the constant a theenergy equivalent of the rate at whichoxygen is supplied per unit mass in ex-cess of the non-running metabolism.Then the equations of energy or oxy-gen balance can be written in the form

dE , (5)

In addition E satisfies the initial con-dition

44 PHYSICS TODAY/SEPTEMBER 1973

£(0) = £„ (6) 12

Because the energy equivalent of theavailable oxygen can never be negative,E also must satisfy the inequality

E(t) > 0 (7)

This is, indirectly, the second restric-tion of f(t).

Now the runner's problem and oursis to find v(t), f(t) and E{t) satisfyingequations 2 through 7 so that T, de-fined by equation 1, is minimized.The four physiological constants T, F,a and Eo are given, and so is the lengthof the race D. In other words, theproblem is to find the rate of consump-tion of the initial oxygen supply inorder to run the distance D in theshortest time.

My solution to this problem givesthe result that for D not greater thanDc, where Dc is the critical distancementioned above, fit) = F, and v in-creases monotonically. For D greaterthan Dc, u{t) increases for t less thant\, v is constant for t between ti and(2. and v decreases for t greater than tiuntil the end of the race, T. Figure 1shows the optimal velocity v(t) as afunction of t for the 220-yard dash,which is a short sprint. In figure 2 theoptimal v(t) is shown for the 440-meterrun, (D greater than Dc).

Comparison of theory and observation

A. V. Hill first pointed out the phys-iological significance of track records.3

Since then, many investigators havetried to extract physiological informa-tion from these records,1 but they werehampered by a lack of any theory ofrunning that correlates the data.Twenty-two world records for dis-tances from 50 yards to 10 000 metersare shown in Table 1. The first fourare taken from data published by B. B.Lloyd,4 and the others are from theReader's Digest Almanac 1972, page980. We have determined the twoconstants T and F to yield a least-squaresfit of the times given by the theory tothe record times for the first eightraces, which we assume to be shortsprints. Then we determined a and Eoto give a least-squares fit of the timesgiven by the theory for the remaining14 races. In both cases we minimizedthe sum of the squares of the relativeerrors. The values of the four physio-logical constants obtained in this way,appear in Table 2. Also shown there isthe value of Dc computed from thetheory with these constants. We seethat Dt. = 291 meters is between 220yards and 400 meters; so the first eightraces are short sprints and the other 14are not. The ratio of the initial oxygensupply, Eo. to the rate of oxygen sup-ply a is E0/a = 58 seconds. Thus theinitial supply is equivalent to the oxy-gen that would be supplied by respira-

500 1000

DISTANCE OF RACE (meters)

1500 2000

The average velocity for running a race. The theoretical curve is seen to agree with av-erage velocities calculated from world records. Figure 3

Distance D

50 yd50 m60 yd60 m

100 yd100 m200 m220 yd400 m440 yd800 m880 yd

1000 m1500 m

1 mile2000 m3000 m

2 miles3 miles

5000 m6 miles

10000 m

Time T(record)min:sec

5.15.55.96.59.19.9

19.519.544.544.9

1:44.31:44.92:16.23:33.13:51.14:56.27:39.68:19.8

12:50.413:16.626:47.027:39.4

Table 1. Track Records

Time T(theory)min:sec

5.095.485.936.409.29

10.0719.2519.3643.2743.62

1:45.951:46.692:18.163:39.443:57.285:01.147:44.968:20.82

12:44.8913:13.1125:57.6226:54.10

Error(per cent)

-0.2-0 .4

0.5-1 .5

2.11.7

-1 .3-0 .7-2 .8-2 .9

1.61.71.43.02.71.71.20.2

-0.7-0 .4-3.1-2 .7

Averagevelocity

D/T(theory)m/sec

8.999.129.269.389.859.93

10.3910.399.249.227.557.547.246.846.786.646.456.436.316.306.206.20

'1 sec

1.781.771.071.060.980.880.870.84

.8080.77.77.75.75

T-t2

1.1.1.1.1.1.1.1.1.1.2.2.

sec

8686080816313443606380821012

tion and by circulation is 58 seconds.By using the constants in Table 2,

we have computed the time given bythe theory for each race. The resultsare shown in Table 1, together with theaverage velocity D/T given by thetheory and the values of ti and T - f2for the races with D greater than Dc.The error in time between the theoreti-cal value and the record is also shown inTable 1 as a percentage. We see thatfor the short sprints the error is at

most 2.1%. However, for the longerraces, it reaches 3.1% for the six-milerace. The average velocity given bythe theory is plotted against distancein figure 3, which also shows the aver-age velocities computed from the rec-ord times. Note that the initial in-crease and ultimate decrease of the av-erage velocity is predicted by the theo-ry quite satisfactorily. In comparingthe theory with the actual records, itmust be borne in mind that the theory

PHYSICS TODAY SEPTEMBER 1973 4 5

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Table 2. Physiological Constants

r = 0.892 secF = 12.2m/sec2

a = 9.93 calories/kg sec£o= 575 calories/kgDc = 291 m

uses a single set of physiological con-stants at all distances, while the recordholders at these distances undoubtedlyhad somewhat different constants fromone another. The theory I have pre-sented is too simple, because it omitsvarious important mechanical andphysiological effects. It ignores theup-and-down motion of the limbs; itfails to distinguish between internaland external resistance; it does nottake into account the depletion of thefuel that uses the least oxygen and thetransfer to the use of less efficientfuels; it ignores the accumulation ofwaste products and the mechanisms ofremoving them, and it probably ignoressome other effects as well. A bettertheory incorporating some of these ef-fects might be able to account for therecords at longer distances, as well asthose considered here. In support ofsuch a new theory, measurements ofthe resistive force and the other physi-ological parameters are needed.

Nevertheless, this theory yields somedefinite results, and it would be of in-terest to adapt it to other types ofraces. Ice skating, swimming andbicycle races, for example, could bestudied, taking into account the specialfeatures of each type of race. Anotherinteresting problem would be to deter-mine the influence of hills and valleyson the optimal velocity in longer races;this would require only a modificationof the present theory.

This work was supported in part by the Na-tional Science Foundation. I thank CesarLevy and Clyde Kruskal for performing thecalculations, and Thomas J Osier, who n onthe A.A.I'. 50-km race in 1967 and who isnow professor of mathematics at GlassboroState College, New Jersey, for his commentson the manuscript.

References

1. B. B. Lloyd, Advancement of Science 22,515(19661.

2. R. Margaria, P. Cerretelli. P. Aghemo, (iSassi, J. Appl. Physiol. IN, 367 (1963).

3. R. Margaria, P. Cerretelli. F. Mangilli, P.E. Di Pramero, "The energy cost of

. sprint running," ler Congres Europeen deMedicine Sportive, Prague, (19631.

4. A. V. Hill, Muscular Activity: HerterLectures, 1924, Williams and Wilkins.Baltimore (1926). °

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