Russell D. Larsen University of Michigan Ann Arbor, MI 48109
II
The Kinetics of Running
The current popularity of running in America (estimates suggest as many as 25 million men and women run for fun and fitness) was bound to inspire the analysis of running as a rate process. Joe Henderson, consulting editor of Runner's World, in a recent commentary' has strongly stated his preference for the all-inclusive word "running" and would like to see the word "jogging" abolished. We will use the former word exclusively in accordance with this attempt to standardize nomenclature. The consideration of running as a rate process has several advantaees for a student studying chemical kinetics for the first time and for other non-k&e&ists who, like the author, lack a feeling for rates of reaction lower than first-order. zero-orde;running-running a t a constant speed-is a good first approximation to actual recreational running as well as long-distance strategy. Clearly, runners rarely maintain a constant rate of speed except over certain intervals of their run. The early portion is usually the slower-paced warm-up; the best running is in the middle portion; and the last part is either a relatively fast sprint or else is run much slower due to fatigue. The familiar chemical kinetic rate equations can be used to show that satisfactory running times are characterized by rate laws that are close to zero-order for each class of runner. An individual runner or runners of similar ability may be defined in terms of a rate constant. For example, a nominal "6-minute miler" can, to achieve this averagepace, run at a constant, zero-order rate using a 6-minute mile rate constant. Orders greater than zerogive longer running times; orders less than %e are probahly acceptable to the goal of most runners. Zero-Order Running Consider, first, the kinetic implications of a constant running rate. I t is convenient to suppose that the running takes place on an indoor %o or 1' s mile track. Serious runners scorn such arenas but during Michigan winters these tracks, like Spring ring-around-the-reservoirin New York City, have certain advantage^.^ As an illustration of a zero-order process, it is convenient to consider running on a '11o mile track (10 laps per mile). Suppose that a runner chooses to run 40 laps or 4 miles on this track. Let L he the number of laps to go to the end. Clearly, L will decrease with time in a manner proportional to some power of the number of laps to be run.
Zero-order (constant) running is defined by the rate expressions -
dL =k dt
-
u
(velocity)
Lo is the initial condition-total laps to he run at t = 0;in this illustration. Lo = 40. " The decikink facing the runner are how far (Lo) and how fast. The decision of how fast may he decided by picking either a rate constant, k, or a half-life, t112. More naturally, arunner considers the total length of running time available (earlv in morning, during lunchhour, after dinner, etc.), then chooses a distance, and then estimates a running rate. Analytically, this is equivalent to solving an IRE such as eqn. (2) when L = 0 (no more laps to be run) and Lo is given. Moreover, the IRE, eqn. (2), may he used to find the number of laps togo, L, given the rate constant, k, and the elapsed time, t. Similarly, one can compare runners with different rate constants, say k l = 1lap min-' and kz = 2 laps min-', by comparing their half-lives. In the above example, the 10-min miler (kl) has a 20 rnin half-life compared to a 10 rnin half-life for the 5-min miler (kz). Generally, such comparisons are not this obvious for more realistic running rates. FirsCOrder Running
Despite being unrealistic, it is instructive to consider the implications of a first-order running rate. The rate equations are dL Rate = -- = kL dt
The number of laps to go, L, as a function of time is an exponential function. ' 10 mi Suppose as illustration, that Lo = 100 = 10 mi on a 1 track. Consider an average running rate of 10 min mi-' for the first 50 laps. Thus, after 50 min, a runner is half finished. There are still 50 lans to eo. The half-life is 50 min: therefore k = 0.0138 min-I. lithatiate constant is maintained from the beginning, it is possible to determine the laps to go after 10 rnin elapsed running time from the second of eqn. (3); viz., there are 82 laps to go after 10 min. Note that in accordance with this exponential first-order attenuation a first-order runner with a 50 min half-life will run at a 1.3 la^ min-I or 7.7 rnin mi-' pace for the first 10 min. The average pace after successive 10 min intervals map be computed easilv. After 1000 min there is still 0.0001 lapto go andthe averagd pace is now 0.10 lap min-' or 100 min mi-'! It is obvious that firstorder runners never complete their goal (exponentially increasing fatigue?). Second-order running is even more discouraging. If the first 50 laps still take only 1 half-life (defined as 50 min), 75 laps takes 3 half-lives (150 rnin). The rate constant k is now given
and L=Lo-kt I t is easy to see that the half-life in this case is just 'Henderson, J., Runner's World, 13. I6 (1978). Van Doom, J., New York, 11.42 (1878).
(2)
1 k=-- 1 = 2 X 10W laps-' min-' Lot1/* (100 laps) (50 min) Note the units of k and the smaller value of k compared to zero- and first-order processes with the same half-life. For second-order running 87.5 laps will take 7 half-lives (350 rnin!!). This corresponds now to a 4 min lap-' or a 40 min mi-' Volume 56. Number 10, October 1979 / 651
pace!! Actual running rates must be somewhat between zeroand first-order processes. Let us briefly consider, therefore, the implication of fractional-order running processes. Fractional-order Running Fractional-order rate processes are, of course, possible where the order n is less than unity. For an n-th order process given by
the IRE is Examination of the general half-life expression3 for order n processes where n # 1,which is given by k t l / z = C,Lol-"
where, C, = 2"-I -1ln
(6)
- 1,shows that for n = 0.54L = 314o
k
t1/2
when, Lo = 100 and t = 50 min, k = 0.341 laps" min-I. The total running time for a '14-order process is, in this case 4
%LO t,,~ = -= 123 min k 114
which corresponds to a 12.3 min mi-' average pace. For a given half-life, which is one way to classify a runner as a zero-order m-minute miler, the rate constant approaches the n = 0 value as the order approaches zero. A runner so classified would not wish to deviate appreciably fiom the zero-order values. For example, a 'Ile-order 6-miu miler is, in reality, about a 6'14 minute miler. Alternatively, a 61' 4 minute miler is a zero-order runner defined in terms of a zero-order law for which the rate constant is a dependent variable given by k =-Lo-L (7) t Utkins, P. W.,"Physical Chemistry," W. H. Freeman andCo., San Francisco, 1978, p. 860. 'Runner's World, 13,122 (1978).
652 1 Journal of Chemical Education
In this case, Lo = 100, L = 0 and tmal = 62.5; thus, a 6I14 min miler obeys a zero-order rate law with k = 1.6 laps min-' = 6% rnin mi-'. Boston Marathon 1978 Hill Rt,dgrrs, the winner of the 1978 Roston Marathon, ran the 2621d73 miles in 2:10:13, a 4.9663 nlin mi-] averare - Dace. . Rodger's zero-order rate constant, given by eqn. (71, is merely the reciprocal of this pace or 0.2013 mi min-'. Note that a 5 min mile zero-order rate cmstant is 0.20 nli min-I. The 10th place finisher, l'om kleminn. ran in ':I444 or 5.1388 min mi-'. Rodger's and Fleming's rate constants differ by 3.3%. The times of the top 158 male runners under 230 are given in Runner's W ~ r l dThe . ~ 158th runner ran a 5.7211 min mile with k = 0.1748 mi min-'. The top woman runner, Gale Barron, finished in 24452 corresponding to a 6.2881 min mile pace with a rate constant k = 0.1590 mi min-'. Clearly, each runner has R different ratr constant and halt-life It is tempting, howe\,er, to arbitrarily ronsider a fixed half-life of 6): min for the 1978 Boston Marathon. This halt-lire is.. oresumably, applicable to the top 10 finishers. The presumption is that this group (and perhaps the top 100 runners) could all run the first 13.1 miles in 65 min. On this assum~tion.Rodeer's time of 130.216 min nearlv follows a order rate law, whereas Fleming3-s tenth place 134.733 rnin is between a '16th and ,,nd order rate.'l'he times for the top 100 male fmish&~all correspond to rate laws below 'hth order. It is important to emphasize that we have modeled running data with certain analytic expressions which characterize these data in terms of simple parameters and variables; viz., duration, distance, speed. Random accelerations and decelerations are averaeed out hv such "deterministic" modeliue. " Actual chemical rate data and running data are implicitly stochastic in that experimental rate laws are determined by appropriate statistical parameter estimation using linear least-squares olots of such data. The reasons for duration and soeed involve other (usually) uncontrolled factors such as altitude of run, humiditv. temnerature. condition of track. and uns~ecified (and us;hly unknownjpsychological and pbysiologkal factors.
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