Physics of Sprinting

Physics of sprinting I. INTRODUCTION In the past few years there has been a growing interest in the application of physical models to problems of animal locomotion in generall ,2 and to human athletic performance in particular.3-5 A problem of special interest is to devise a model that represents with reasonable accuracy the propulsive and resistive forces acting on a human runner. One common model assumes that the sum total of re- sistive forces F, acting on a runner can be represented by Fr,= - oMu,lgor AlexandroP and Philip Lucht Department of Physics, ttniuenity of (Inh Salt Lake City, Utah 841!2 (Received I I February 1980; acepted 18 June 1980) Sprinting is described by a simple physical model. The model is used to predict the differences between the recorded times for raoes on a straight track and on a'curve. It is shown that the choice of the ninning lane makes a non-negligible difference. dzx(t) _du(t) _ r_dt2 -tr=f - ou(t)' (3) where M is the mass of the runner, u is the speed, and o is a parameter of the model, presumed constant or nearly constant for a given runner.6 Although such a form is sug- gested by analogy with viscous forces, it is far from clear that it can adequately represent human runners where the forces are rapidly varying and, in general, complex. Nonetheless such a form was used by Keller3 to model competitive running and draw certain conclusions regarding optimal running strategy and maximal future performances-. One significant conclusion reached by Keller is that a well-conditioned athlete is able to sustain a maximum and nearly constant muscular effort for races over distances of 290 m or less. We shall define all such races as sprints. A basic question is: Wh at are typical values of the pa- rameter o? It is to be expected that o will depend on the runner, his speed, the type race, and possibly on external conditions such as the track surface, altitude, etc. Keller3 uses a value of o = 0.44 sec-l while Whitt and Wilsons quote similar values. The latter, however, were measured at walking speeds. To our knowledge there exists no mea- surement of o at sprinting speeds. We propose that for a given sprinter a measurement of o and the propulsive force param eter f (defined below) can be performed in principle by measuring that sprinter's running times for two straight line races of different dis- tances. Our model will then allow us to use the values of these parameters to predict that runner's performance for a sprint run on the curve. II. MODEL We assume as does Keller3 that a sprinter of mass M is subject to two horizontal forces. One is the resistive force expressed by Eq. ( I ) and the other is a constant propulsive force F, which we write as Fp = fM,where x(t) is the distance measured from the start of the race and u(t) is the sprinter's instantaneous velocity. The solution to this equation subject to the initial conditions x(0): 0, D(0) = u6 is u(r) = Wo)(l - e-ot) + t)o€-ot, (4) x(r) = (f/o)t + (f/oz - uo/o)(e-", - 1). (5) These well-known solutions describe the motion of a particle sedimenting under the action of a constant force of "grav- ity" Mf and a viscous force - oMu; the particle (sprinter) approaches the terminal velocity u1 = f/ o asymptoti- cally. Let us suppose that the times / 1 and t 2 for an athlete's performances over two different straight-line sprinting distances x1 and x2,ole known. Then Eq. (5) can be used (with D0 = 0) to solve for f and o (see Appendix). We give in Table I the best running times over two dif- ferent distances for four world-class sprinters from the 1967 -69 period.T'8 (We note that two of them are still holders or coholders of three separate world records.) The computed values of o and f foi each runner are shown in columns 6 and 7, respectively. The last entry'in Table I ii Tommie Smith who has the distinction of having a recorded time for the straight-track 100-m race and also for the rarely run straight-track 220-yard race for which he holds the world's record. In addition he is the holder of the world's record for the 200 m on the curve which he established in Mexico City in I 969.7 The computed values of the parameters f and o differ widely from runner to runner. We have no way of knowing how reliable the data are but if we assume tfrat the com-- puted values of "f and o for Tommie Smith are accurate, then we can extend our theory to predict his performance in a 200-m race on the curve. The cioseness of our prediction to the actual recorded time will be a test of the validity of the basic assumptions of the theory embodied in Eqs. ( I ) and (2). III. RUNNING THE CURVE In a typical 200-m race the starts are staggered so that each runner runs the first 100 m of the race on the curve. However, the radius of curvature of the curved portion of each lane is diflerent. The lanes are I .22 m wide so that the inner radius of the nth lane is R(n) = 100/r * (n - lXr.22) m. Let the runner be running with speed u on a curve of(1) (2) where f is the force per unit mass of the athlete.6 The equation of motion for the sprinter is then : i:.'r 254 Am. J. Phys. 49(3), March l98l 0002-9s 0s I 8t I 0302s4-04$00. 50o l98l American Association of Physics Teachers 254 Table l. Computed.values of the Darameters f and o for various runners (*indicates a current world record holder). The times are the runners best times over the given distances.7.8 Runner X1 r 1 (sec) X2 /2(sec) o(sec-r ) /(N /ke) u, (m/sec) l6o I oll6"f l"fl l6c,lt:,1 John Carlos 60 yards Bill Gaines 60 yards Jim Hines* 100 yards Tommie Smithx 100 m100 yards I00 yards 100 m 220 yards (straight)6.0 5.9 9.2 r0. r9.0 9.3 9.9 r9.5r4.667 r.250 0.581 1.2528.13 r3.45 7.10 t3.4612.19 10.7 6 12.22 r0.75l3Vo 257o 6s% t9%lAVo 23% 5 lVo l87o1t-J"/a 1i') -,(' | 4ac lcc Equation (6) is to be solved with given parametersf o., and R subject to the initial conditions D(0) = 0, x(0) - 0 An approximate analytical solution can be obtained by noticing that for typical situations the second terrn under the square root is small, uolR2 <<f , and by expanding the square root we obtain du/dt* 6u =f - Gl2)u4lfR2constant radius R and subject to the forces ( I ) and (2). Now, however, the propulsive force must supply a compo- nent that gives rise to the centripetal acceleration of mag- nitude uzlR.The athlete's equation of motion then be- comes du/dt+6u=W-u4/Rz)t/zrameters the difference between ut and u.(-) is of the order of 0.3 msec-1" Thus the runner accelerates slightll,'upon entering the straightaway. lo Equations ( 10) and ( 1 I ) allow us to compute the total time ty for the race in a closed (approximate) form tr = /roo * l00f u, * el o - 2{ 100) lu, + \lo * e t(100lut) - t3 l02o)) + ez\(r}} luy) . 2s I 02o)l + o(€3). ( 1 2) Equation (12) then can be used to predict the running time, for example, for Tommie Smith running a 200-m race on the curve. We use the pararneters of Table I and assign him to lane 3, the lane in which he ran for his rvorld record. Then R - 34.27 ffi, € . 0.0314.,ur = 10.75. We obtain /roo : 10"35 sec, uloo : 10.41^lsec, time to run the last 100 m = 9.33 sec, total predicted time for the race = 19.68 sec. In the Mexico City Olympics on l6 October 1968 in the rnen's 200-m final, Tommie Smith running in lane 3 set the still-current world record of 19.83 sec. ln order to test the accuracy of our calculation we nu- merically integrated the exact Eq. (6) and computed the time necessary to cover the first 100 m of the course on the curve. The comparison of this calculation with the results of our approximate analytical solution is shown in Table II. The two procedures give results that differ at most by 0.3Vo. This surprisingly close agreemen'r lends support to the va- lidity of the approxirnations made in the calculations. The close agreement between our "predicted" and the measured value of Tommie Smith's performance in the 200-m race on the curve comes as somewhat of a surprise considering the meagerness of the data and the crudeness of the model. Finally, we apply our model to the following problem. Since the value of the radius ol curvature R enters Eqs. (8)-(12) we would expect that athletes running in the outside lane (lane 8) would enjoy an advant age over those in the inside lane. In fact, it is well known that sprinters dislike running in Table Il. Comparison of numerical solutions with approximate analytical solutions. Times are calculated lor the first I 00 m of the race on the curve; f - 13.46 N/kg, o = 1.252 sec-r. Lanet roo(sec) / 166(sec) numerical approximate dilference R(m) solution solution (Vr)(6) (7) ( 10) (11)As it stands, Eq. (7) is still not subject to a simple ana- lytical solution. We can, however, solve it by treating the term in u4 as a small perturbation (see Appendix). The solution so obtained subject to the initial conditions D. (0): 0, x.(0) - 0 gives the speed ur(r) and the distance trav- elled along the curve xr(t) as: t),(t) = (f/ o)(r - Z-r) - rl2(f/ o)(fR-to-2)zFlZ), (8) x,(t) = (f/ ,z)(lnZ - 1+ Z-t) - ( | I 2)(f/ o2)(fR- | o-2)2 F z(Z), (e) whereZ-e"tand F{Z) = 1 + (1013 - lnZ)z-r - 5Z-2 + 2Z-3 - | 13 Z-4, FzQ)- -37 I 12 * lnZ+ ( 213 + lnZ)z-l + 3Z-2 - 213 Z*3 + | I t2 7-+. The accuracy of our perturbation procedure depends on the smallness of the dimensionless parameter e = (fR-r o-2)2 12 which for the case of interest to us has a value of 0.031 39. For running times on the order of 10 sec, terms in Z - 1 and higher powers of Z - 1 are negligible and can be dropped from the expressions for F1 and F2. With these approximations we can write expressions for xr(t) and u.(-), the latter being the terminal speed of the runner on the curve. We note that with our typical value of o the sprinter attain s 98Vo of his terminal speed within ap- proximately 3 sec: u,(*) = (l - €)u,, ,xr(t) = u1(l - e)t - (ut/ o)(l :- 31 I 12 e). Equations ( 10) and ( 1 1) are useful in that they allow us to bring out the effect of curvature of the track represented by the parameter e . Physically, 2e is the square of the ratio of the centripetal acceleration u?/R calculated at terminal speed to the propulsive force per unit mass/ With our pa-I 3 8 straight track3 r.83 34.21 443't @10.36 r ta 321 t0.265 r0.r0010.389 r0.348 14.27 7 r0.r000.2170 4.20% 0.11% 255 Am. J. Phys., Vol. 49, No. 3, March l98l I. Alexandrov and P. Lucht 255 the inside lane complaining that the turn is "too tight." Our calculation bears this out. We find that, with the parameters we used for our test case, the total recorded time flor a runner in lane I would have been 19.72 sec whereas for lane 8 it would have been 19.60 sec. At finishing speeds of about I I m/sec that difference in time translates into a distance difference of over a meter. Since many sprint races are lost .or won by mere centimeters the above eflect is quite sig- nificant. Curiously enough sprinters do not consider lane 8 to be the most advantageous one. Because ofthe staggered start runners in lane 8 run for half the race ahead of the other competitors. This, they claim, puts them at the psycholog- ical disadvantage of not being able to see what their com- petition is doing. IV. ERRORS We wish to make a few brief comments about the "ex- perimental" errors. Since neither the records books nor the sport press include error bars with their reported results we will limit ourselves to a few crude estimates. We assume that most of the error is the result of inaccurate time mea- surements, Many of the times given in Table I were prob- ably the averages of several hand-recorded times. Other times might have been recorded electronically and then rounded offl to the nearest tenth of a second. For the purposes of this estimate, we assume that the accuracy in the recorded times is *0.05 sec. We notice that the resulting variation in the calculated values of f and o are quite large. This is due in a large part to the fact that the calculation involves a denominator which is a difference between two numbers numerically close to one another. One consequence of this effect is that data for races over similar distances such as 50 and 60 yards or 100 yards and 100 m are practically useless. However, as is apparent from Eq. (12), the total time t7 depends primarily on Dr, so to some degree the errors inrf and o tend to cancel one another. The uncertainties included in Table I were calculated according to the assumption At 1- -Ltz = *0.05 sec. Assuming this error in the measure- ment of time the resulting error in the predicted t7 for T. Smith is Ltr - *0.054 sec. A less conservative error estimate of At; - -Atz = t0.l I sec brings the predicted and observed t7 within the range of "experimental" error. V. CONCLUSIONS ln summary we have found that the forces acting on a sprinter at typical sprinting speeds can be reasonably well modeled by Eq. (3). The use of that equation over two dif- ferent distances can determine the sprinter's parameters f and, o . The values of the parameters that we calculate for a few sample cases are in some disagreement with those commonly found in the literature. Our model, extended to include centripetal effects, then allows us to predict an athlete's time for a race run on the curve. This in turn can be used to compute the difference in the times for two comparable athletes running in different lanes. This dif- ference is solely due to the different radii of curvature of 256 Am. J. Phys., Vol.49, No. 3, March l98ltheir respective lanes. We find its magnitude to be non- negiigible. tfJurly our results suggest a line of further investigation for those interested in the biomechanics of sports. To the extent that a sprinting human caR be characteri zed by pa- ramete rs f and o tt might be of interest to measure these parameters under carefully controlled conditions and cor- relate them with the physical characteristics of the sprinters, such as height, weight, type of build, reaction time, strength of leg muscle, etc" It is conceivable that these pararneters f and o might also depend on external conditions such as track surface, altitude, and other factors. Finally, we wish to comrnent that it has been our expe- rience that there is a fairly large audience among today's college students interested in learning about the biome- chanics of sports from the point of view of fairly sophisti- cated physics. Such a trend is certainly indicated by the amount of research done in many countries in the area of sports medicine and technique. \ APPENT}tX In order to determine the parameters f and 6 for a given sprinter, let us assume that the sprinter has recorded times / 1 and t 2 ovar two straightaway distances of lengths x1 and xz. Uling Eq.(5) twicewith r,0 = 0 and neglecting the exponential term we solve for f and o to obtain o: (rlt,)[(xz/x,) - l]ll@z/xr) - Qz/t,)), (13) f - 62x2/(otz- 1). ( 14) In solving Eq . (7) we procede as follows. First, we solve it with the ua term set equal to zero. This solution which we call w is the same as Eq. (a) with uo : 0, w(t) - (fl o)(l - e-"'). Next, we write the solution to Eq" (7) as u - w * u where we treat u as a small perturbation . u then satisfies du/dt * ou : -l 12 ua/(fR2). ( 1 5) Replacing u in Eq. (15) bV its approximation w, we inte- grate twice to obtain the solutions (8) and (9). For our test case (Tommie Smith) the parameters f and 6 are directly determined from Eqs.(13) and (14) using the data given in Table I Equations (8) and (9) can be further simplified if we notice that for our test case 6 ,- 1.25 sec-l and for running times of the order of 10 sec, Z- t ^., 10-6 so that Fl and F2 can be approximated by F1 t= l, F2 =. to - 37 l12. This gives Eqs. ( 10) -(12). These equations then predict the time to complete the curve portion of the race / 100, the speed at the end of 100 m, and the total time. For comparison we find that, using the same values of f and o, d 200-m race run entirely on the straightaway would have taken 19.40 sec as compared with our lane 3 calculated time 19.68 sec. Even for a runner in lane 8 the straightaway time is faster by 0.20 sec. "On leave of absence from Department of Physics-Astronomy, California State University at Long Beach, Long Beach, CA 90840. ' rK. Schmidt-Nielsen, Science 177,222 (1912) I. Alexandrov and P. Lucht 256 2Mechanics and Energetics of Animal Locomotion, edited by R. M. Alexander and G. Goldspink (Halsted, New york, lg77)3J. B. Keller, Phys. Today 2G(g),42 (tg73). 4H. Lin, Am. J. phys. 46(t), (197g). sF. whitt and D. wilson, Bicycling science (MIT, Cambridge, MA, te7 4) 5we use mks units throughout so that lld - kg, x = m, u = mlsec, F, \-257 Am. J. Phys., Vol. 49, No. 3, March l gg l i_ = N,.f= N/kg, and o = sec-I. 7N. McWhirter, Guinness Book of World Records (Sterling, New york, teTe). ssports lllustrared (New York, NY) and Track and Field News (Los Altos, cA) miscellanbous issues between 1967 and I 969.ePersonal observation (IA). l0We wish to thank an anonymous referee for pointing this out to us. f . Alexandrov and P. Lucht 257