On the performance of Usain Bolt in the 100 metre sprint
aa r X i v : . [ phy s i c s . pop - ph ] J u l On the performance of Usain Bolt in the 100 metre sprint
JJ Hern´andez G ´omez, V Marquina and RW G ´omez
Facultad de Ciencias, Universidad Nacional Aut ´onoma de M´exico.Circuito Exterior C.U. M´exico D.F., 04510, M´exicoE-mail: jorge [email protected] , [email protected] , [email protected] Abstract.
Many university texts on Mechanics deal with the problem of the effect of the air dragforce, using as example the slowing down of a parachute. Hardly no one discuss what happenswhen the drag force is proportional to both u and u . In this paper we deal with a real problem toillustrate the effect of both terms in the speed of a runner: a theoretical model of the performanceof the 100 m world record sprint of Usain Bolt during the 2009 World Championships at Berlin isdeveloped, assuming a drag force proportional to u and to u . The resulting equation of motion issolved and fitted to the experimental data obtained from the International Amateur of AthleticsFederations that recorded Bolt’s position with a LAVEG (laser velocity guard) device. It is worthto note that our model works only for short sprints.PACS numbers: 01.80.tb, 01.55.tb, 45.20.D, 07.10-h, 06.30.Bp Keywords : Usain Bolt, mechanical model, hydrodynamic drag, sport physics.
Submitted to:
Eur. J. Phys.
1. Introduction
In June 21, 1960, at Z ¨urich, Switzerland, the German Armin Harry astounded the sports worldachieving what was considered the physiological and psychological barrier for the 100 m dash:10 s flat. It was until June 20, 1968, at Sacramento, California, that Jim Hines ran 100 m in 9.9s, breaking this barrier. In following years many sprinters had run this distance faster than 10s, but 31 years were needed to lower Harry’s record by 0.14 s (Carl Lewis, August 25, 1991,at Tokyo, Japan). The current world record of 9.58 s was established by Usain Bolt (who alsoheld the 200 m world record of 19.19 s up to 2012) in the 12th IAAF World Championships inAthletics at Berlin, Germany (2009).The performance of Usain Bolt in the 100 meter sprints is of physical interest since he canachieve, until now, accelerations and speeds that no other runner can. Through time, severalmathematical models to fit the position, the velocity, or both, of a sprinter have been proposed[1, 2, 3, 4, 5, 6]. Recently, Helene et al [6] fitted Bolt’s performance during both the summerOlympics in 2008 at Beijing and the world championships in 2009 at Berlin, using a simpleexponential model for the time dependence of the speed of the runner.
2. Theoretical model
The important forces acting during the race are the horizontal force that Bolt exerts and adrag force that depends upon the horizontal velocity (speed). Other factors, such as humidity,altitude above sea level (36 m), oxygen intake and that he turns his head to watch other runners,affecting the mechanics of his motion, are not taken into account. Based on the fact that Bolt’s200 m time is almost twice the one for 100 m, our main assumption is that in the 100 m dash, heis able to develop a constant horizontal force F during the whole sprint. The drag force, D ( u ) ,is a function of Bolt’s horizontal speed respect to ground u ( t ) , with or without wind. This forcecauses a reduction of his acceleration so his speed tends to a constant value (terminal speed).Thus, the equation of motion is m ˙ u = F − D ( u ) . (1)This equation can be readily casted as a quadrature, t − t = m Z uu du ′ F − D ( u ′ ) . (2)The integral above does not have an analytical solution for a general drag function; howeverthe drag force can be expanded in Taylor series, D ( u ) ≃ D (0) + dD ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) u + 12 d D ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) u + O (cid:0) u (cid:1) . (3)The constant term of the expansion is zero, because the runner experiments no drag when atrest. The second and third terms must be retained. While the term proportional to the speedrepresents the basic effects of resistance, the term proportional to the square of the speed takesinto account hydrodynamic drag, obviously present due to the highly non-uniform geometryof the runner. In general, for relatively small speeds, it suffices to take only the first three termsof the expansion.Renaming the u and u coefficients as γ and σ respectively, the equation of motion (1) takes theform m ˙ u = F − γu − σu , (4)whose solution follows straightforward from Equation (2), u ( t ) = AB (cid:0) − e − kt (cid:1) A + Be − kt , (5)where the coefficients are related by σ = km/ ( A + B ) , F = kmAB/ (2 A + 2 B ) and γ = km ( A − B ) / ( A + B ) .The position can be obtained by integrating Equation (5), x ( t ) = Ak ln (cid:18) A + Be − kt A + B (cid:19) + Bk ln (cid:18) Ae kt + BA + B (cid:19) , (6)while the acceleration can also be calculated by deriving Equation (5), a ( t ) = ABk ( A + B ) e − kt ( A + Be − kt ) . (7)
3. Experimental data fitting
The experimental data we used were from the 12th IAAF World Championships in Athletics(WCA), which were obtained from [7], and consist of Bolt’s position and speed every / s. Tocorroborate the accuracy of the data obtained from [7], we reproduced with them the velocityvs position plot given in [8], which was obtained by the IAAF, by means of a LAVEG (laservelocity guard) device. The parameters A , B and k were fitted by a least-squares analysis, withthe Origin 8.1 c (cid:13) software, in both position and speed data sets, considering a reaction time of0.142 s [6]. In figure 1(a) and figure 1(b) we show such fittings, together with the experimentaldata. x ( m ) t (s) Measured data Theoretical position fit (a) v ( m / s ) t (s) Measured data Theoretical speed fit (b)
Figure 1: Position (a) and speed (b) of Bolt in the 100 m sprint at the 12th IAAF WCA. Thedotted (blue) line corresponds to the experimental data while the solid (red) one correspondsto the theoretical fitting.The parameter values for both fittings are shown in table 1. We do not report errors, because thestandard error of the fitting on each parameter lies between the second and the third significantdigit, which is finer than the measurement error in the data.Table 1: Fitted values of the parameters A , B and k . Parameter Position fitting Velocity fitting A (m/s) 110.0 110.0 B (m/s) 12.2 12.1 k (1/s) 0.9 0.8The accuracy of the position and velocity fittings is R p = 0 . and R v = 0 . respectively, sowe decided to use from now on the results of the parameters A , B and k from the position fitting.The computed values of the magnitude of the constant force F , and the drag coefficients, γ and σ , are shown in table 2, taking Bolt’s mass as 86 kg [9].We also show in figure 2 the plot of the magnitude of the acceleration we obtained; no fittingwas made because there are no experimental data available.Table 2: Values of the physical parameters F , γ and σ . Constant Value F (N) 815.8 γ (kg/s) 59.7 σ (kg/m) 0.6 a ( m / s ) t (s) Figure 2: Theoretical acceleration of Bolt in the 100 m sprint at the 12th IAAF WCA.
4. Results
As any mechanical system subject to drag, the runner experiments a terminal velocity u T which is formally obtained when ˙ u = 0 in the equation of motion (1); that is, by solving theequation F = D ( u T ) (8)for u T . Nevertheless, the solution of the equation for the terminal velocity can also be foundwhen t → ∞ in Equation (5), and it turns out to be u T = B . Therefore, under this model,the runner acquires a terminal speed of u T = 12 . m/s, which is physically feasible (seefigure 1(b)). According to the data obtained from [7], the average speed in the second halfof the sprint, which is surprisingly equal to 99% of the maximum speed recorded [7], is . m/s. Moreover, the initial acceleration of Bolt is a (0) = 9 . m/s, which is of the order of theacceleration of gravity, g ; this value of the initial acceleration is fully reasonable, consideringthat the acceleration a man must exert in order to be able to jump half of his own height, shouldbe just slightly greater than g . Furthermore, the value of the constant force in table 2, F = 815 . N, is totally consistent with the fact that one expects that the maximum constant (horizontal)force he could exert should be of the order of his weight, i.e. w = 842 . N.Now, σ = 0 . ρC d A represents the hydrodynamic drag, where ρ is the density of air, C d the dragcoefficient of the runner and A his cross section area. The density of air at the time of the springcan be approximated as follows. Berlin has a mean altitude of 34 m over the sea level, andan average mean temperature for the month of August ‡ [10] of 18.8 ◦ C. Bearing in mind thatthe race took place at night, we consider an average temperature between the average meantemperature and the mean daily minimum temperature for August at Berlin, which is 14.3 ◦ C.Thus, the density of air is ρ = 1 . Kg/m and the drag coefficient of Bolt is C d = 2 σ / ρA = 1 . ,where the cross section area of Bolt § was estimated as A = 0 . m . This value of C d lays in thetypical range for human beings reported in the literature (between 1.0 and 1.3) [11, 12, 13].The instantaneous power that Bolt develops, considering the drag effect is simply P ( t ) = F u = m ˙ uu = mABk ( A + B ) (1 − e − kt ) e − kt ( A + Be − kt ) . (9)In figure 3 we plot the power of the sprint for Bolt and the drag. It is remarkable that themaximum power of P max = 2619 . W (3.5 HP) is reached at a time of t P max = 0 . s, whenthe speed u ( t P max ) = 6 . m/s was only about half of the maximum speed. The fact that themaximum instantaneous power arises in such a short time indicates the prompt influence ofthe drag terms in the dynamics of the runner.The effective work (considering the effect of the drag force) is then W Eff = Z τ P ( t ) dt = Z τ mdu = 12 mu ( τ ) , (10)where τ is the running time (the official time of the sprint minus the reaction time of the runner).The effective work is the area under the curve of figure 3, and it is W Eff = 6 . kJ. On the otherhand, as Bolt is assumed to develop an essentially constant force, his mechanical work is simply W B = F d = 81 . kJ, where d is the length of the sprint (100 m). This means that from the totalenergy that Bolt develops, only 7.79% is used to achieve the motion, while 92.21% is absorbedby the drag; that is, 75.22 kJ are dissipated by the drag, which is an incredible amount of lostenergy. ‡ The sprint took place on August 16, 2009. § To calculate such a cross section area, we used a similar procedure to the one used in [9], where instead of acircle we estimated the area of the head with an ellipse. We averaged several scaled measures from Bolt picturestaken from [14] P ( W ) t (s) Figure 3: Theoretical power of Bolt in the 100 m sprint at the 12th IAAF WCA.
5. Discussion
As mentioned in section 2, a central assumption in our model is that a 100 m sprinter (not onlyBolt) is able to develop a constant force during the race (except in the initial few tenths of asecond where he pushes himself against the starting block). In order to delimit how good isthis assumption, we use the experimental values of u , the calculated acceleration, and the fittedvalues of the constants γ and σ , to compute F . The result is shown in figure 4. It is interestingto note that the average value of the force obtained from this figure is 818.3 N, which is veryclose to the value obtained from the fitting of the data, 815.7 N. The high value of the force inthe first tenths of the race is due do the acceleration he obtains when he pushes himself fromthe starting block.At a first glance, observing the values of the drag coefficients in table 2, one is impelled toargue that, because σ ≪ γ , the hydrodynamic drag could have been neglected. However,one can calculate the drag terms in the equation of motion at the terminal speed u T , attaining γu T = 725 . N and σu T = 90 . N. Thus, from the total drag γu T + σu T , 11.05% correspondsto turbulent drag, which turns to be an important contribution.If we would like to make predictions considering different wind corrections, this can be doneas follows. Once a runner acquires the wind speed (which is almost instantly), the second termin the right side ( γu ) of equation (1) behaves as if the sprinter would be running in still air,because γ is proportional to the air viscosity, which is independent of air pressure. Howeverthat is not the case for the third term in (1), ( σu ), which arises from the collisions per unit timeof the air molecules against the sprinter and it is proportional to the speed of the runner respectto ground. In a simple model, the hydrodynamical drag force is D H = σ ( ρ )( v + v w ) , where v is t (s) F ( N ) Figure 4: Force exerted by the runner during the race. The red line is calculated with theexperimental data, the dash-dot-dot (green) line is the average force of 818.3 N, while theshort-dash-dot (black) line is the value of the force F obtained from the adjustment (815.7 N).the speed achieved by the runner without wind and v w is the speed of the wind. The value of σ depends on the number of molecules that impact on the runner per unit time and should bedifferent in still air conditions. Then, the equation of motion (1) can be rewritten as m ˙ u = m ˙ v = F − γv − σ ( v + v V ) , (11)and without wind as m ˙ v = F − γv − σ ′ v . (12)Subtracting (11) and (12), we obtain σ ( v + 2 vv w + v w ) = σ ′ v , (13)so then σ ′ = σ (cid:18) v w v + v w v (cid:19) ∼ σ (cid:18) v w v (cid:19) , (14)where the third term in the second expression has been neglected ( v w << v ). In order to estimatethe value of σ ′ , we consider v as the terminal speed of Bolt, u T . With these conditions, σ ′ = 0.69with still air ( v w = 0 m/s) and σ ′ = 0 . with a tailwind of v w = 2 m/s. It should be clearthat the present calculation is only a crude way to estimate the differences of running timewith and without wind. The results, which are close to the values reported in literature[15], aresummarized in table 3.Although this is a simple way to calculate a correction due to wind, it turns to be a goodproposal for it. A more realistic assumption would be to modify equation (14) to be σ ′ = σ (cid:18) αv w u T (cid:19) , (15)Table 3: Predictions of the running time for Bolt without tailwind, and with a tailwind of 2m/s. v w (m/s) Estimated running time (s) α lying between 1 and 2.The results we obtained, altogether with the facts pointed out in this discussion, shows theappropriateness and quality of the model developed in this paper. We look forward for thenext IAAF WCA, which will be held in Moscow, Russia, from August 10 to August 18, 2013,to test our model with the experimental data obtained from such sprints, as well as to waitexpectantly if the fastest man on earth is able to beat his own world record once again. Acknowledgments
This work was partially supported by PAPIIT-DGAPA-UNAM Project IN115612.
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