We present a toy model of anaerobic glycolysis that utilizes appropriate physiological and mathematical consideration while remaining useful to the athlete. The toy model produces an optimal pacing strategy for 400-m and 800-m races that is analytically calculated via the Euler-Lagrange equation. The calculation of the optimum is presented in detail, with an emphasis on intuitive arguments in order to serve as a bridge between the basic techniques presented in undergraduate physics textbooks and the more advanced techniques of control theory. Observed pacing strategies in 400-m and 800-m world-record races are found to be well-fit by the toy model, which allows us to draw a new physiological interpretation for the advantages of common weight-training practices.
REFERENCES
1.
Progression of IAAF World Records, 2011 Edition
, edited by Imre
Matrahazi
(International Association of Athletics Federations
, Monaco
, 2011
).2.
J. B.
Keller
, “A theory of competitive running
,” Phys. Today
26
(9
), 42
–47
(1973
).3.
J. B.
Keller
, “Optimal velocity in a race
,” Am. Math Monthly
81
(5
), 474
–480
(1974
).4.
T. D.
Noakes
, M. I.
Lambert
, and R.
Hauman
, “Which lap is the slowest? An analysis of 32 world mile record performances
,” Br. J. Sports Med.
43
, 760
–764
(2009
).5.
R.
Tucker
, M. I.
Lambert
, and T. D.
Noakes
, “An analysis of pacing strategies during men's world record performances in track athletics
,” Int. J. Sports Physiol. Perform.
1
, 233
–245
(2006
).6.
W.
Woodside
, “The optimal strategy for running a race (a mathematical model for world records from 50 m to 275 km)
,” Mathl. Comput. Modelling
15
, 1
–12
(1991
).7.
H.
Behncke
, “A mathematical model for the force and energetics in competitive running
,” J. Math. Biol.
31
, 853
–878
(1993
).8.
H.
Behncke
, “Optimization models for the force and energy in competitive running
,” J. Math. Biol.
35
, 375
–390
(1997
).9.
R.
Margaria
, Biomechanics and Energetics of Muscular Exercise
(Clarendon Press
, Oxford
, 1976
), pp. 53
–55
.10.
A. J.
Ward-Smith
, “A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world-class athletes
,” J. Biomech.
18
(5
), 337
–349
(1985
).11.
G. J.
van Ingen Schenau
, R.
Jacobs
, and J. J.
de Koning
, “Can cycle power predict sprint running performance?
,” Eur. J. Appl. Physiol.
63
, 255
–260
(1991
).12.
G. J.
van Ingen Schenau
, J. J.
de Koning
, and G.
de Groot
, “The distribution of anaerobic energy in 1000 and 4000 metre cycling bouts
,” Int. J. Sports Med.
13
, 447
–451
(1992
).13.
G. J.
van Ingen Schenau
, J. J.
de Koning
, and G.
de Groot
, “Optimisation of sprinting performance in running, cycling and speed skating
,” Sports Med.
17
(4
), 259
–275
(1994
).14.
O.
Meyerhof
, “Zur Verbrennung der Milchsäure in der Erholungsperiode des Muskels
,” Pflügers Arch. Ges. Physiol.
175
, 88
–93
(1919
).15.
A. V.
Hill
and H.
Lupton
, “Muscular exercise, lactic acid, and the supply and utilization of oxygen
,” Q. J. Med.
16
, 135
–171
(1923
).16.
400 m: Andrew Rock, 400-m US Champion 2006 (private communication).
17.
800 m: Suzy Favor Hamilton, 800-m US Olympian 1996 (private communication).
18.
19.
Results of the 7th World Championships in Athletics, Seville, 1999 (day 6), <http://www2.iaaf.org/sevilla99/results>.
20.
W.
Baumann
, “Kinematic and dynamic characteristics of the sprint start
,” in Biomechanics VB (Int. Ser. Biomech., vol. Vb)
(University Park Press
, Baltimore, MD
1976
), pp. 194
–199
.21.
M.
Čoh
, B.
Jošt
, B.
Škof
, K.
Tomažin
, and A.
Dolenec
, “Kinematic and kinetic parameters of the sprint start and start acceleration model of top sprinters
,” Gymnica
28
, 33
–42
(1998
).22.
C.
Hanon
and B.
Gajer
, “Velocity and stride parameters of world-class 400-meter athletes compared with less experienced runners
,” J. Strength Cond. Res.
23(2)
, 524
–531
(2009
).23.
Reaction time of 0.3 s and 200-m split determined by the author from analysis of race video.
24.
400-m split was given by Track and Field News as 48.9 s, in “Lap times in 800 m world records since the first sub-1:50,” available online at <http://www.trackandfieldnews.com/index.php/display-article?arId=48269>.
25.
600-m split time and finish time are from Ref. 1.
26.
B.
Gajer
, C.
Hanon
, J.
Marajo
, and J.-C.
Vollmer
, Le 800 mèters: Analyse descriptive et entraînement
(Institut National des Sports et de l'éducation Physique
, Paris
, 2000
).27.
C.
Hanon
, C.
Thomas
, J.-M.
Le Chavalier
, B.
Gajer
, and H.
Vandewalle
, “How does VO2 evolve during the 800m?
,” New Stud. Athletics
17
(2
), 61
–68
(2002
).28.
Andrew Rock (private communication).
29.
K.
Sahlin
, A.
Alvestrand
, R.
Brandt
, and E.
Hultman
, “Intracellular pH and bicarbonate concentration in human muscle during recovery from exercise
,” J. Appl. Physiol.
45
(3
), 474
–480
(1978
).30.
D. L.
Costill
, A.
Barnett
, R.
Sharp
, W. J.
Fink
, and A.
Katz
, “Leg muscle pH following sprint running
,” Med. Sci. Sports Exerc.
15
(4
), 325
–329
(1983
).31.
D.
Wilkes
, N.
Gledhill
, and R.
Smyth
, “Effect of acute induced metabolic alkalosis on 800-m racing time
,” Med. Sci. Sports Exerc.
15
(4
), 277
–280
(1983
).32.
G. J.
Kemp
, M.
Roussel
, D.
Bendahan
, Y.
Le Fur
, and P. J.
Cozzone
, “Interrelations of ATP synthesis and proton handling in ischaemically exercising human forearm muscle studied by 31P magnetic resonance spectroscopy
,” J. Physiol.
535.3
, 901
–928
(2001
).33.
G.
Walter
, K.
Vandenborne
, M.
Elliott
, and J. S.
Leigh
, “In vivo ATP synthesis rates in single human muscles during high intensity exercise
,” J. Physiol.
519.3
, 901
–910
(1999
).34.
E.
Hultman
and K.
Sahlin
, “Acid-base balance during exercise
,” Exerc. Sport Sci. Rev.
8
, 41
–128
(1980
).35.
J.
Hirvonen
, A.
Nummela
, H.
Rusko
, S.
Rehunen
, and M.
Härkönen
, “Fatigue and changes of ATP, creatine phosphate, and lactate during the 400-m sprint
,” Can. J. Sport Sci.
17
(2
), 141
–144
(1992
).36.
D. G.
Allen
, G. D.
Lamb
, and H.
Westerblad
, “Skeletal muscle fatigue: Cellular mechanisms
,” Physiol. Rev.
88
, 287
–332
(2008
).37.
W.
Yourgrau
, A.
van der Merwe
, and G.
Raw
, Treatise on Irreversible and Statistical Thermophysics
(Dover
, NY
, 1982
), pp. 1
–54
.38.
E.
Schrödinger
, What is Life?
(Cambridge U.P.
, Cambridge
, 1992
), Chapter 6, p. 71
.39.
H.
Barcroft
and J. L. E.
Millen
, “The blood flow through muscle during sustained contraction
,” J. Physiol.
97
(1
), 17
–31
(1939
).40.
F. A.
Gaffney
, G.
Sjøgaard
, and B.
Saltin
, “Cardiovascular and metabolic responses to static contraction in man
,” Acta Physiol. Scand.
138
(3
), 249
–258
(1990
).© 2013 American Association of Physics Teachers.
2013
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
