A mathematical analysis of the bioenergetics ofhurdlingA.J. WARD-SMITH - PDF document

A mathematical analysis of the bioenergetics ofhurdlingA.J. WARD-SMITHDepartment of Sport Sciences, Brunel University, Osterley Campus, Borough Road, Isleworth, MiddlesexTW7 5DU, UKAccepted 11 February 1997A mathematical model of the bioenergetics of running has been extended to apply to hurdling. This has beenachieved by two principal modi®cations. First, a new term has been added to the energy equation to account action of the trailing leg during hurdling is rather differ-ent from that in normal sprinting. Whereas in sprintingit passes vertically beneath the body, in hurdling thetrailing leg is abducted away from the running plane toavoid contact with the hurdle. As a consequence of thisaction, the centre of mass is raised by a smaller amountthan would otherwise be the case. By minimizing thevertical movement of the centre of mass, the amount ofpotential energy that has to be generated at each hurdleis minimized.It is known from steady-state treadmill tests thatthere is an essentially linear relationship between run-ning speed and oxygen consumption (Margaria, 1976;Davies, 1980). It follows (Ward-Smith, 1984) that therate of increase of energy dissipated as heat in runningis directly proportional to velocity under steady-stateconditions. Ward-Smith (1984, 1985a) has shown, byextrapolation, that this relationship can also be appliedto the acceleration phase at the start of sprinting. Inhurdling, the normal sprinting action is compromised,adjustments being made by the athlete to both thestride length and vertical displacement of the centre ofmass to accommodate the height and spacing of thehurdles. Because the stride pattern is compromised, itfollows that the rate of degradation of mechanicalenergy into thermal energy, or, expressed another way,the rate of heat production, is greater in hurdling thanin sprinting.Summarizing the above discussion, from an energystandpoint there are two principal differences betweenhurdling and normal sprinting. In hurdling, potentialenergy has to be supplied to the centre of mass at eachhurdle over and above the amount associated with nor-mal running. Also, the stride pattern is distorted relat-ive to that in running. Ultimately, both of these factorscontribute to an increase in the energy dissipated asheat.Mathematical analysisThe preceding discussion has shown that hurdling canbe regarded as a specialized form of sprinting adaptedto the clearance of obstacles. Here we begin the ana-lysis of hurdling by using the mathematical model ofrunning, based on energy considerations, establishedby Ward-Smith (1985a). Modi®cations appropriate tohurdling will then be introduced. The analysis con-siders energy changes associated with the motion of thecentre of mass of the runner; for distances greater thana few stride lengths, the cyclical variations associatedwith the stride pattern are ignored on order-of-magnitude grounds. The energy exchanges are sum-marized in Fig. 1.Energy balance during runningThe chemical energy released, C, is equal to the sum ofthe external work done on the centre of mass of thehurdler, W, plus the mechanical energy converted intothermal energy, H. The overall energy balance is givenby:C5H1W(1)Differentiation of the energy balance with respect totime, t, results in the corresponding power equation,given by:dC dt5dH dt1dW dt(2)The individual contributions to the power equation areas follows. Figure 1Representation of energy exchanges occurring dur-ing running.518Ward-Smith The rate of chemical energy conversion is, followingWard-Smith (1985a):dC dt5Pmaxexp(2lt)1R[12exp(-lt)](3)where Pmaxrepresents the maximum power availablefrom the anaerobic mechanism, Ris the maximumaerobic power and lis a parameter governing the rateof release of chemical energy.The sum of the four contributions H1, H2, H3andH4shown in Fig. 1 is denoted by H. The overall rate ofdegradation of mechanical energy into heat is givenby:dH dt5Av(4)where vis velocity and Ais a parameter governing therate of dissipation of mechanical energy.The work done by adding kinetic energy to thecentre of mass will be denoted by W1and the workdone against aerodynamic drag by W2. Then, the rate atwhich the horizontal component of kinetic energy isadded to the runner of mass mis given by:dW1 dt5mv dv dt(5)The rate of working against aerodynamic drag, for stillair conditions, is:dW2 dt5Dv51 2rv3SCD51 2rv3AD(6)where Dis the aerodynamic drag experienced by thebody, ris the air density, Sis the projected frontal areaof the body, CDis the drag coef®cient and AD(5SCD)is the drag area.Substitution of the individual components into equa-tion (2) yields the full form of the power equation forrunning, which is:Pmaxexp(2lt)1R[12exp(2lt)]5Av11 2rv3SCD1mv dv dt(7)In this formulation of the power equation, the relativelysmall effects of the vertical movement of the runner'scentre of mass in the earth's gravitational ®eld areabsorbed into the dissipation term, Av, as explained inFig. 1.The energy equation is derived by integration ofequation (7) with respect to tbetween the limits t50,x50 and t5T, x5X, and isPmax l[12exp(2lT)]1R[T21 l(12exp(2lT))]5AX1KET0v3dt11 2 mV2(8)whereK5rSCD 2(9)Modi®cations for hurdlingFor hurdling, the analysis developed for running needsto be modi®ed in two ways. First, it is useful to con-sider explicitly an additional component in the powerand energy equations which accounts for the changedvertical movement of the body's centre of mass associ-ated with the negotiation of the hurdles. Just as in run-ning, over the race as a whole, the net verticaldisplacement of the centre of mass is almost zero; thisterm is therefore ultimately manifested as a contribu-tion to the degradation term. However, it is convenientto represent the term explicitly in the analysis of hurd-ling. Secondly, it must be recognized that, to negotiatethe hurdles, athletes are forced to depart from theiroptimal stride pattern adopted in running, and as aconsequence there is an increased degradation ofmechanical energy into heat associated with thiseffect.By conservation of energy, the vertical component ofkinetic energy when taking off to clear the hurdle isconverted to potential energy because of the gain inheight of the centre of mass of the athlete. Thus:1 2 mVV25mgh(10)where his the vertical displacement of the centre ofmass relative to its nominal horizontal level, and VVisthe vertical velocity component at take-off. If the num-ber of hurdles is denoted by NH, then the additionalenergy transferred to potential energy in clearing thehurdles is NHmgh. Whereas the terms in equation (7) -the power equation for running - are continuous, thecontributions to the power and energy equations fromthe hurdling action are discontinuous. To establish thepower equation for hurdling, we average the hurdlingterms over the entire race. A power term associatedwith the potential energy used to clear the hurdles maybe de®ned by dW3/dtand this is given by:dW3 dt5NHmgh T(11)A mathematical analysis of the bioenergetics of hurdling519 The dissipation term for hurdling will be differentfrom that for running and can be derived in the follow-ing way. It is inevitable that, to a greater or lesserextent, a hurdler has to modify the stride pattern notonly in those strides used to clear the hurdles but inothers in the approach and landing as well. As a basisfor developing the theoretical work, we assume that ahurdler progresses through a race: (1) by advancingusing a normal running action, and (2) by modifyingthe normal running action in a certain number ofstrides that is proportional to the number of hurdlescleared. Let Aand AHrepresent the rate of degradationof mechanical energy into thermal energy for normalrunning strides and hurdling strides, respectively. Wealso de®ne NTand N, respectively, as the total numberof strides and the number of normal running strides ina hurdle race. Hence:NT5N1NH(12)We introduce the symbol ato represent the pro-portion of the total number of strides used for negotiat-ing the hurdles: Thus:a5NH NT(13)Then:N NT5NT2NH NT5(12a)(14)An average effective rate of energy degradation forthe entire race Aeffcan then be de®ned to satisfy therelation:Aeff5(12a)A1aAH(15)We can relate Aand AHby the expression:AH5bAwhere b5f(a,h)(16)The parameter bwill have a magnitude greater than 1and in principle depends, to a greater or lesser extent,upon the vertical displacement of the centre of mass ofthe athlete and upon the stride pattern required tonegotiate the hurdles.Hence for hurdling the energy equation becomes:Pmax2R l[12exp(2lT)]1RT5AeffX1KET0v3dt11 2 mv21NHmgh(17)The corresponding power equation is:Pmaxexp(2lt)1R[12exp(2lt)]5Aeffv1Kv31mv dv dt1NHmgh T(18)Equation (18) can be rewritten as:dv dt51 v[(P*max2R*)exp(2lt)1R*2A*effv2K*v32NHgh T](19)where values shown with an asterisk (*) are values nor-malized with respect to body mass. Thus R*5R/m, etc.Furthermore, x, vand tare related by:dx dt5v(20)The total energy contributions from the anaerobicand aerobic mechanisms, Canand Caerrespectively, canbe obtained by integration of the corresponding powerterms with respect to time. Thus the total chemicalenergy converted at time tfrom rest is given by:C5Can1Caer(21)whereCan5Pmax l[12exp(2lt)](22)andCaer5R[t21 l(12exp(2lt))](23)Application of analysis to hurdlingData relevant to the solution of the derived equationsare assembled in this section. Attention will be focusedon the performance of elite athletes.Position of the hurdles on the trackOver the past century, there have been a number ofdifferent race distances. Currently, the recognized out-door hurdle events are the men's 110 m and 400 m andthe women's 100 m and 400 m. The men's andwomen's indoor 50 m and 60 m hurdles events haverecently been recognized by the International AmateurAthletic Federation (IAAF). Although the 3000 m520Ward-Smith steeplechase requires hurdles to be cleared, this eventhas additional features and, because it is radically dif-ferent from the other events considered here, it hasbeen excluded from the analysis. The number ofhurdles in an event, and the positions of the hurdles onthe track, are given in Table 1.Typical stride patterns of elite athletesThe quantitative data on stride patterns reported herewere obtained by observing videos of elite hurdlers inaction; additional unpublished data were provided byPaul Grimshaw.In the men's 110 m hurdles, most elite hurdlersadopt a stride pattern of eight strides to the ®rst hurdle,although some exceptional athletes use seven; threestrides are used between hurdles, with the fourth usedto negotiate the hurdle, and the race is completed withsix strides to the ®nish. The race therefore consists ofabout 51 strides, 10 of which are exaggerated by hurdleclearance, yielding a representative value for aof 10/51.In the men's 400 m hurdles, most hurdlers use about22 strides to the ®rst hurdle, 13 or 15 strides betweenhurdles - although some athletes are able to alternatetheir leading leg and thereby use 14 strides - and about25 strides from the last hurdle to the ®nish. As fatiguesets in, the natural stride shortens and so the stridepattern in the early part of the race may be differentfrom that adopted later in the race. For a total of 192strides, ais evaluated as 10/192. In the men's 60 mhurdles, athletes usually adopt the same stride patternas in the 110 m hurdles, with eight strides to the ®rsthurdle and, subsequently, three strides between eachhurdle. For this event, ais about 5/28. In the 50 mhurdles, the stride pattern is the same as in the 60 mevent, with the exception of the number of strides usedin the ®nal distance. An approximate estimate for ais4/24.In the women's 100 m hurdles, the stride pattern isthe same as that in the men's 110 m hurdles, with eightstrides to the ®rst hurdle and three strides in between,the fourth stride being used to negotiate the hurdle,and six strides to the ®nish, yielding a representativevalue for aof 10/51. In the women's 400 m event, thestride pattern usually consists of about 28 strides to the®rst hurdle, either 17 or 19 strides between the hurdles,depending on the physique and natural stride of theathlete, followed by about 26 strides to the ®nishingline. As in the men's event, the number of strides mayincrease towards the end of the race due to fatigue. Arepresentative value for ais 10/221. In the women's60 m hurdles, about eight strides are used to the ®rsthurdle and three strides in between. A typical value forais 5/28. Over 50 m, the stride pattern adopted is thesame as for the 60 m hurdles, with the exception of thenumber of strides at the ®nish. A representative valuefor ais 4/25.Biophysical dataRepresentative biophysical data describing an elite maleathlete capable of running the 100 m in 10.23 s havebeen established by Ward-Smith (1985a). Marar andGrimshaw (1993) have reported that the main factorwhich determines performance at hurdling is sprintingspeed. Therefore, the representative biophysical datapreviously derived for a male sprinter are directlyapplicable to the case of a male hurdler. Correspondingvalues for an elite female runner capable of sprintingthe 100 m in 10.93 s have been evaluated by Ward-Smith and Mobey (1995). Again, it is appropriate touse biophysical data for elite sprinters to evaluatehurdling performance.Tanner (1964) showed that the average height ofmale hurdlers in the 1960 Olympics was 183 cm. ATable 1Positions of hurdles EventNumberofhurdlesDistancefrom startto ®rsthurdle(m)Distancebetweenhurdles(m)Distancefrom lasthurdleto ®nish(m) Men's 50 m413.729.148.86Men's 60 m513.729.149.72Men's 110 m1013.729.1414.02Men's 400 m1045.0035.0040.00Women's 50 m413.008.5011.50Women's 60 m513.008.5013.00Women's 100 m1013.008.5010.50Women's 400 m1045.0035.0040.00 A mathematical analysis of the bioenergetics of hurdling521 representative value of 173 cm for the height of world-class female hurdlers has been obtained from unpub-lished data provided by Paul Grimshaw. A summary ofthe representative biophysical and other data for sub-stitution in the computer calculations is given inTable 2.Centre of massMeasurements of the vertical displacement of thecentre of mass of several athletes have been made fromGrimshaw's data. Information was obtained by meas-urement from three-dimensional digital images createdfrom videos taken at various athletics meetings. Thewhole body centre of mass positions were obtained bythe usual segmentation method. Data on the clearanceheight of a number of athletes were available for themen's 110 m and the women's 110 m hurdle eventsonly, and these are presented in Table 3. The represent-ative clearance height used in the present calculations isthe average of the data in Table 3, and these data wereused for all events except the 400 m hurdles.As no measurements had been made for either themen's or women's 400 m events, an estimate of theclearance was made; the estimated value of the clear-ance was in close accord with data reported by Kauf-mann and Piotrowski (1976) for hurdlers ofintermediate standard. The vertical displacement of thecentre of mass of the hurdler was then calculated byadding the clearance height to the height of the hurdleand subtracting the height of the centre of mass abovethe ground during sprinting. Page (1978) quoted theresults of earlier work he had undertaken which showedthat the centre of mass of an adult male in a normalupright stance lies about 2.5 cm below his navel, orapproximately 57% of his full height from the ground.A female's weight is distributed differently; she has awider pelvis and, usually, narrower and lighter shoul-ders. Her centre of mass is nearer the ground and isabout 55% of her full height from the ground (Page,1978).The height of the hurdles varies according to theevent. It is therefore necessary to consider each eventseparately in evaluating the mathematical model. Theheight of the hurdles, the average clearance height andthe vertical displacement of the centre of mass for eachevent are summarized in Table 4.Effective rate of energy degradationAt this stage, the one unknown quantity required tosolve equations (19) and (20) is a measure of thedegradation of mechanical energy to thermal energyassociated with hurdling. In principle, the modi®edstride pattern, the number of hurdles and the verticalmovement of the centre of mass are factors that affectA*eff. Here it will be assumed that the effect of the ver-tical movement is much smaller than the effect associ-ated with the changed stride pattern, and can beignored on order-of-magnitude grounds. This assump-tion can be tested when predicted and actual results areTable 2Magnitudes of parameters for male andfemale hurdlers ParameterMale hurdlerFemale hurdler l0.03 s-10.03 s-1P*max50.5 W kg-147.2 W kg-1R*23.5 W kg-121.8 W kg-1A*3.9 J kg-1m-13.98 J kg-1m-1K*3.3310-3m-13.6310-3m-1H1.83 m1.73 mAD0.385 m20.358 m2m70 kg60 kg Table 3Performance data for athletes incompetition (from Marar and Grimshaw, 1993) EventTime (s)Clearance (cm) Men's 60 m7.7727Men's 60 m8.4028Men's 60 m8.2024Men's 110 m13.6429Men's 110 m13.8628Men's 110 m13.3325Men's 110 m13.2325Women's 100 m13.2440Women's 100 m13.3336Women's 100 m13.7234Women's 100 m13.0835Women's 100 m13.1537Women's 100 m13.4137 Table 4Estimated vertical displacement of centre of mass EventHeight ofhurdle (m)Clearance(m)Displacementof centre ofmass (m) Men's 50 m1.0670.2680.291Men's 60 m1.0670.2680.291Men's 110 m1.0670.2680.291Men's 400 m0.9140.3180.188Women's 50 m0.840.3650.254Women's 60 m0.840.3650.254Women's 100 m0.840.3650.254Women's 400 m0.7620.3650.176 522Ward-Smith compared. It is evident that A*effmust exceed A*andthe assumption will be made that the excess is directlyproportional to a. Thus A*effis relatred to A*by theexpression:A*eff5(11a)A*(24)This relationship corresponds to a value for bof 2;expressed another way, for the strides directly affectedby the hurdling process, the ef®ciency of locomotion ishalf that for normal running.Results and discussionEquations (19) and (20) were solved to obtain v5v(t)and x5x(t) using a numerical scheme based on thefourth-order Runge-Kutta method. A time-step of0.01 s was adopted, and programs previously writtenfor male sprinting (Ward-Smith, 1985a) and femalesprinting (Ward-Smith and Mobey, 1995) were used asa basis for the program development.The program made use of representative values ofthe biophysical data contained in Table 2; data for hand aderived above and summarized in Table 5 werealso used. Zero wind and a nominal sea level density forair of 1.21 kg m-3were assumed. For all events, currentworld record times were used as a basis of comparisonwith the mathematical model.Comparisons of predicted and actual running timesare shown in Table 6, for both outdoor and indoorevents. It is perhaps helpful to mention that several fac-tors can contribute to the differences between the pre-dicted and actual times. First, there is the basicvariability in performance between one individual ath-lete and another, with the consequence that worldrecords do not ®t some precise correlating equation.Secondly, the physiological parameters of individualathletes may depart to some extent from the represent-ative data incorporated into the present calculations.Thirdly, how well the mathematical model works as awhole depends on how well the individual contribu-tions to the energy balance are modelled. For example,there is evidence that a tail wind enhanced the worldrecords in the men's 110 m event (wind 10.5 m s-1)and the women's 100 m event (wind 10.7 m s-1). Thedata in Table 6 are not adjusted for this effect and weshall return to this matter later. Overall, the differencesin Table 6 between predicted and actual times weregenerally less than 2.5%, with the exception of thewomen's 100 m event, where the predicted time wassome 6% greater than the world record. Bearing inmind the three broad sources of possible discrepanciesdiscussed above, the general level of correlation for theother seven events is considered to be very good. Wenote at this stage that, in deriving an expression forA*eff, the effect of the vertical movement of the centre ofmass was ignored compared to the effect of the modi-®ed stride pattern on order-of-magnitude grounds.This assumption is seen to be justi®ed, as there is nosubstantial disparity between the actual and predictedtimes which can be systematically correlated with thisfactor.A number of supplementary calculations has beenmade. The energy contributions from the anaerobicTable 5Input data used in the program tocalculate hurdling performance Eventh(m)NHa Men's 50 m0.2940.17Men's 60 m0.2950.18Men's 110 m0.29100.20Men's 400 m0.19100.052Women's 50 m0.2540.16Women's 60 m0.2550.18Women's 100 m0.25100.20Women's 400 m0.18100.045 Table 6Comparison of predicted hurdling times with world record times EventActual time,tA(s)Predicted time,tP(s)Dt(tA2tP)(Dt/tA) % Men's 50 m6.256.1010.1512.4Men's 60 m7.307.2410.0610.8Men's 110 m12.9113.1920.2822.2Men's 400 m46.7847.0120.2320.5Women's 50 m6.586.4410.1412.2Women's 60 m7.697.7020.0120.2Women's 100 m12.2112.9420.7326.0Women's 400 m52.6151.6310.9811.9 A mathematical analysis of the bioenergetics of hurdling523 and aerobic mechanisms were evaluated by substitutingthe world record times in equations (21) and (23). Theresults are given in Table 7. These show that the powerexerted throughout a hurdling event is predominantlyfrom the anaerobic source, a result which is consistentwith previous results for running (Ward-Smith, 1985a).These calculations are also consistent with datareported by strand and Rodahl (1986), who havetabulated the approximate contributions from the aer-obic and anaerobic processes under general conditionsof maximal effort in exercise. strand and Rodahl givethe anaerobic-to-aerobic ratio as 85% at 10 s and65-70% after 60 s, diminishing to 1% after 2 h.Indoor hurdling events are held under conditionswhere air movements are small; in the 400 m outdoorevents, the effects of wind tend to be self-cancelling andare quite small. However, performance in the men'soutdoor 110 m hurdles and the women's outdoor100 m hurdles is signi®cantly in¯uenced by wind con-ditions. To take account of wind effects, equation (6)was modi®ed. The rate of working against aerodynamicdrag is then given by:dW2 dt5Dv51 2 rv(v2VW)2SCD51 2 rv(v2VW)2AD(25)where VWrepresents the wind velocity, positive for afollowing wind.Calculations of the effects of both head and followingwinds were made, and the results are given in Table 8.These calculations indicate that a following wind of2 m s-1confers an advantage of just over 0.2 s for boththe men's 110 m and the women's 100 m events. These®gures are very similar to those predicted for sprinting;for example, an advantage of 0.18 s has been computedfor the men's 100 m sprint (Ward-Smith, 1985b).Table 7Contributions from the aerobic and anaerobicmechanisms EventActualtime (s)Anaerobic(%)Aerobic(%) Men's 50 m6.2595.74.3Men's 60 m7.3095.05.0Men's 110 m12.9191.28.8Men's 400 m46.7871.428.6Women's 50 m6.5895.54.5Women's 60 m7.6994.85.2Women's 100 m12.2191.88.2Women's 400 m52.6168.631.4 Table 8Predicted in¯uence of head and tail winds onhurdling performance (a following wind is positive; ahead wind is negative) EventWind speed(m s-1)Predicted time(s) Men's 110 m2213.462113.32013.19113.07212.97Women's 100 m2213.222113.07012.94112.83212.73 Table 9Predicted in¯uence of vertical movementof centre of mass hon hurdling time Eventh(m)Predicted time (s) Men's 50 m0.240.290.346.076.106.13Men's 60 m0.240.290.347.207.247.28Men's 110 m0.240.290.3413.0913.1913.29Men's 400 m0.140.190.2446.8747.0147.15Women's 50 m0.200.250.306.406.446.47Women's 60 m0.200.250.307.667.707.75Women's 100 m0.200.250.3012.8312.9413.06Women's 400 m0.130.180.2351.4851.6351.79 524Ward-Smith Returning again to the conditions under which theworld records were set and using the method on whichTable 8 is based, the mathematical model for the men's110 m hurdles, with VW50.5 m s-1, predicts a time of13.13 s; for the women's 100 m hurdles, withVW50.7 m s-1, a time of 12.86 s is predicted. Thecorresponding percentage differences between the pre-dicted and actual world record times (see Table 6) arethereby reduced to 21.7% and 25.2%, respectively.These calculations do suggest that the world record of12.21 s for the women's 100 m hurdles, established byYordanka Donkova on 20 August 1988, is quite excep-tional when compared against all the other worldrecord hurdling performances, even after the effects ofwind assistance are taken into account.The results of the computation depend to someextent upon the representative values of the vertical dis-placement of the centre of mass, h. The values used areshown in Table 4, and are based on the assumed posi-tion of the centre of mass of the runner as well as theaverage results for clearance height set out in Table 3.Although in Table 3 there is no direct correspondencebetween clearance height and running times, the timescover a range of performances, some of which fall sig-ni®cantly short of world standard. As it is useful to havesome insight into how variations in haffect the pre-dicted time, calculations at 1 cm intervals in hweremade; a selection of the results is set out in Table 9. Adetailed inspection of the calculations showed that,over the range investigated, the predicted time variedlinearly with h; small but signi®cant bene®ts arise fromcontrolling this effect during hurdling.The effects on hurdling performance of a number ofvariables, including body mass (m), projected frontalarea of the athlete (S), air density (r) and drag coef®-cient (CD), are taken into account by the parameter K*.Rather than investigate the separate effects of these arevariables, particularly bearing in mind that mand Sareinterrelated, calculations were made to investigate theoverall effect of K*on hurdling time. The results areshown in Table 10. The effect of changing K*on pre-dicted time is small; typically, a 20% change in K*pro-duced a change in predicted time of 1% or less.ConclusionIt has been shown that a mathematical model of run-ning can be successfully adapted to the analysis ofhurdling. This has been achieved by two principalmodi®cations. First, a new term has been added to thepower (or energy) equation to account for the verticaldisplacement of the centre of mass required to nego-tiate the hurdles. Secondly, the term expressing the rateof degradation of mechanical energy to thermal energyhas been increased in magnitude to account for theeffects of the adjustments in stride pattern to negotiatethe hurdles. Good correlations of actual and predictedtimes for outdoor and indoor hurdles events wereachieved.AcknowledgementsThe author wishes to thank Dr P.N. Grimshaw forreleasing unpublished information. Contributions tothe early stages of the work were undertaken by A.Mobey as a ®nal-year project.Referencesstrand, P.-O. and Rodahl, K. (1986). Textbook of Work Physi-ology. New York: McGraw-Hill.Davies, C.T.M. (1980). Effect of wind assistance and resist-ance on the forward motion of a runner. Journal of AppliedPhysiology, 48, 702-709.Kaufmann, D.A. and Piotrowski, G. (1976). Biomechanicalanalysis of intermediate and steeplechase hurdling tech-nique. In Biomechanics V-B (edited by P.V. Komi), pp.181-187. Baltimore, MD: University Park Press.Marar, L. and Grimshaw, P.N. (1993). A three-dimensionalbiomechanical analysis of sprint hurdles. Journal of SportsSciences, 12, 174-175.Margaria, R. (1976). Biomechanics and Energetics of MuscularExercise. Oxford: Clarendon Press.Page, R.L. (1978). The Physics of Human Movement. Bath:Wheaton.Tanner, J.M. (1964). The Physique of the Olympic Athlete.London: George Allen and Unwin.Table 10Predicted in¯uence of K*on hurdlingperformance times (s) K*(m-1) Event (m)0.00300.00330.0036 Men's 50 m6.086.106.11Men's 60 m7.227.247.26Men's 110 m13.1413.1913.24Men's 400 m46.7547.0147.27 K*(m-1) 0.00330.00360.0039 Women's 50 m6.426.446.45Women's 60 m7.687.707.72Women's 100 m12.9012.9412.98Women's 400 m51.3951.6351.88 A mathematical analysis of the bioenergetics of hurdling525 Ward-Smith, A.J. (1984). Air resistance and its in¯uence onthe biomechanics and energetics of sprinting at sea leveland at altitude. Journal of Biomechanics, 17, 339-347.Ward-Smith, A.J. (1985a). A mathematical theory of running,based on the ®rst law of thermodynamics, and its applica-tion to the performance of world-class athletes. Journal ofBiomechanics, 18, 337-349.Ward-Smith, A.J. (1985b). A mathematical analysis of thein¯uence of adverse and favourable winds on sprinting.Journal of Biomechanics, 18, 351-357.Ward-Smith, A.J. and Mobey, A.C. (1995). Determination ofphysiological data from a mathematical analysis of therunning performance of elite female athletes. Journal ofSports Sciences, 13, 321-328.526Ward-Smith