Content uploaded by Vivianne Marquina

Author content

All content in this area was uploaded by Vivianne Marquina

Content may be subject to copyright.

IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 34 (2013) 1227–1233 doi:10.1088/0143-0807/34/5/1227

On the performance of Usain Bolt in

the 100 m sprint

JJHern

´

andez G ´

omez, V Marquina and R W G´

omez

Facultad de Ciencias, Universidad Nacional Aut´

onoma de M´

exico, Circuito Exterior CU,

M´

exico DF, 04510, Mexico

E-mail: jorge_hdz@ciencias.unam.mx,marquina@unam.mx and rgomez@unam.mx

Received 16 May 2013, in ﬁnal form 21 June 2013

Published 25 July 2013

Online at stacks.iop.org/EJP/34/1227

Abstract

Many university texts on mechanics consider the effect of air drag force, using

the slowing down of a parachute as an example. Very few discuss what happens

when the drag force is proportional to both uand u2. In this paper we deal

with a real problem to illustrate the effect of both terms on the speed of a

runner: a theoretical model of the world-record 100 m sprint of Usain Bolt

during the 2009 World Championships in Berlin is developed, assuming a drag

force proportional to uand to u2. The resulting equation of motion is solved

and ﬁtted to the experimental data obtained from the International Association

of Athletics Federations, which recorded Bolt’s position with a laser velocity

guard device. It is worth noting that our model works only for short sprints.

(Some ﬁgures may appear in colour only in the online journal)

1. Introduction

In Z¨

urich, Switzerland, 21 June 1960, the German Armin Harry astounded the sports world

by equalling what was considered an insuperable physiological and psychological barrier for

the 100 m sprint: the 10 s race. It was not until 20 June 1968, at Sacramento, USA, that Jim

Hines ran 100 m in 9.9 s, breaking through this barrier. Many sprinters have run this distance

in under 10 s subsequently, but 31 years were needed to lower Harry’s record by 0.14 s (Carl

Lewis, 25 August 1991, at Tokyo, Japan). The current world record of 9.58 s was established

by Usain Bolt (who also held the 200 m world record of 19.19 s up to 2012) in the 12th

International Association of Athletics Federations (IAAF) World Championships in Athletics

(WCA) at Berlin, Germany in 2009.

The performance of Usain Bolt in the 100 m sprint is of physical interest because he can

achieve speeds and accelerations that no other runner has been able to. Several mathematical

models to ﬁt the position and velocity (or both) of a sprinter have been proposed [1–6].

Recently, Helene et al [6] ﬁtted Bolt’s performance during both the summer Olympics in 2008

at Beijing and the world championships in 2009 at Berlin, using a simple exponential model

for the time dependence of the speed of the runner.

0143-0807/13/051227+07$33.00 c

2013 IOP Publishing Ltd Printed in the UK & the USA 1227

1228 JJHG

´

omez

et al

2. Theoretical model

The important forces acting during the race are the horizontal force that Bolt exerts and a drag

force that depends upon the horizontal velocity (speed). Other factors affecting the mechanics

of his motion, such as humidity, altitude above sea level (36 m), oxygen intake and his turning

his head to watch other runners, are not taken into account. Based on the fact that Bolt’s 200 m

time is almost twice that for 100 m, our main assumption is that in the 100 m sprint he is able

to develop a constant horizontal force F0during the whole race. The drag force, D(u),isa

function of Bolt’s horizontal speed with respect to the ground u(t), with or without wind. This

force causes a reduction of his acceleration, so his speed tends to a constant value (terminal

speed). Thus, the equation of motion is

m˙u=F0−D(u). (1)

This equation can readily be cast as a quadrature,

t−t0=mu

u0

du

F0−D(u).(2)

The integral above does not have an analytical solution for a general drag function; however,

the drag force can be expanded in Taylor series,

D(u)D(0)+dD(u)

du

0

u+1

2

d2D(u)

du2

0

u2+O(u3). (3)

The constant term of the expansion is zero, because the runner experiences no drag when at

rest. The second and third terms must be retained. While the term proportional to the speed

represents the basic effects of resistance, the term proportional to the square of the speed takes

into account hydrodynamic drag, obviously present due to the highly non-uniform geometry

of the runner. In general, for relatively small speeds it sufﬁces to take only the ﬁrst three terms

of the expansion.

Renaming the uand u2coefﬁcients as γand σ, respectively, the equation of motion (1)

takes the form

m˙u=F0−γu−σu2,(4)

whose solution follows straightforwardly from equation (2),

u(t)=AB 1−e−kt

A+Be−kt ,(5)

where the coefﬁcients are related by σ=km/(A+B),F0=kmAB/(2A+2B)and

γ=km(A−B)/(A+B).

The position can be obtained by integrating equation (5),

x(t)=A

kln A+Be−kt

A+B+B

kln Aekt +B

A+B,(6)

while the acceleration can also be calculated by deriving equation (5),

a(t)=ABk(A+B)e−kt

A+Be−kt 2.(7)

On the performance of Usain Bolt in the 100 m sprint 1229

0510

0

25

50

75

100

x (m)

t (s)

Measured data

Theoretical position fit

0510

0

4

8

12

v (m/s )

t (s)

Measured data

Theoretical speed fit

(a) (b)

Figure 1. Position (a) and speed (b) of Bolt in the 100 m sprint at the 12th IAAF WCA. The

dotted (blue) line corresponds to the experimental data while the solid (red) one corresponds to the

theoretical ﬁtting.

Tab le 1. Fitted values of the parameters A,Band k.

Parameter Position ﬁtting Velocity ﬁtting

A(m s−1) 110.0 110.0

B(m s−1) 12.2 12.1

k(1 s−1) 0.9 0.8

Tab le 2. Values of the physical parameters F0,γand σ.

Constant Value

F0(N) 815.8

γ(kg s−1) 59.7

σ(kg m−1) 0.6

3. Experimental data ﬁtting

The experimental data we used were from the 12th IAAF WCA, which were obtained from

[7], and consist of Bolt’s position and speed every 1/10 s. To corroborate the accuracy of the

data obtained from [7], we reproduced the velocity versus position plot given in [8] with them,

obtained by the IAAF using a laser velocity guard device. The parameters A,Band kwere ﬁtted

using a least-squares analysis via Origin 8.1 (both position and speed data sets) considering a

reaction time of 0.142 s [6]. In ﬁgures 1(a) and (b) we show such ﬁttings, together with the

experimental data.

The parameter values for both ﬁttings are shown in table 1. We do not report errors,

because the standard error of the ﬁtting on each parameter lies between the second and the

third signiﬁcant digit, which is ﬁner than the measurement error in the data.

The accuracy of the position and velocity ﬁttings is R2

p=0.999 and R2

v=0.993,

respectively, so we use the results of the parameters A,Band kfrom the position ﬁtting

henceforth. The computed values of the magnitude of the constant force, F0, and the drag

coefﬁcients, γand σ, are shown in table 2, taking Bolt’s mass as 86 kg [9].

1230 JJHG

´

omez

et al

0510

0

5

10

a (m/s2)

t (s)

Figure 2. Theoretical acceleration of Bolt in the 100 m sprint at the 12th IAAF WCA.

We also show in ﬁgure 2the plot of the magnitude of the acceleration we obtained; no

ﬁtting was made because there are no experimental data available.

4. Results

As for any mechanical system subject to drag, the runner experiments a terminal velocity uT

that is formally obtained when ˙u=0 in the equation of motion (1); that is, by solving the

equation

F0=D(uT)(8)

for uT. Nevertheless, the solution of the equation for the terminal velocity can also be found

when t→∞in equation (5), and it turns out to be uT=B. Therefore, in this model the runner

acquires a terminal speed of uT=12.2ms

−1, which is physically feasible (see ﬁgure 1(b)).

According to the data obtained from [7] the average speed in the second half of the sprint

(surprisingly equal to 99% of the maximum speed recorded [7]) is 12.15 m s−1. Moreover, the

initial acceleration of Bolt is a(0)=9.5ms

−1, which is of the order of the acceleration of

gravity, g; this value for the initial acceleration is reasonable, considering that the acceleration

a man must exert in order to be able to jump half of his own height should be slightly greater

than g. Furthermore, the value of the constant force in table 2,F0=815.8 N, is entirely

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=842.8N.

Now, σ=0.5ρCdArepresents the hydrodynamic drag, where ρis the density of air, Cd

the drag coefﬁcient of the runner and Ahis cross section area. The density of air at the time

of the spring can be approximated as follows. Berlin has a mean altitude of 34 m above sea

level, and an average mean temperature for the month of August1[10]of18.8◦C. Bearing

in mind that the race took place at night, we consider an average temperature between the

average mean temperature and the mean daily minimum temperature for August in Berlin,

which is 14.3 ◦C. Thus, the density of air is ρ=1.215 kg m−3and the drag coefﬁcient of Bolt

1The sprint took place 16 August 2009.

On the performance of Usain Bolt in the 100 m sprint 1231

0510

0

500

1000

1500

2000

2500

P (W)

t (s)

Figure 3. Theoretical power of Bolt in the 100 m sprint at the 12th IAAF WCA.

is Cd=2σ/ρA=1.2, where the cross section area of Bolt2was estimated as A=0.8m

2.

This value of Cdlies in the typical range for human beings reported in the literature (between

1.0 and 1.3) [11–13].

The instantaneous power that Bolt develops, considering the drag effect, is simply

P(t)=Fu =m˙uu =mABk(A+B)(1−e−kt )e−kt

A+Be−kt 3.(9)

In ﬁgure 3we plot the power of the sprint for Bolt and the drag. It is remarkable that the

maximum power of Pmax =2619.5 W (3.5 HP) was at time tPmax =0.89 s, when the speed

u(tPmax )=6.24 m s−1was only about half of the maximum speed. The fact that the maximum

instantaneous power arises in such a short time indicates the prompt inﬂuence of the drag

terms in the dynamics of the runner.

The effective work (considering the effect of the drag force) is then

WEff =τ

0

P(t)dt=τ

0

1

2mdu2=1

2mu2(τ ), (10)

where τis 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 WEff =6.36 kJ.

On the other hand, as Bolt is assumed to develop an essentially constant force, his mechanical

work is simply WB=F0d=81.58 kJ, where dis the length of the sprint (100 m). This

means that from the total energy that Bolt develops only 7.79% is used to achieve the motion,

while 92.21% is absorbed by the drag; that is, 75.22 kJ are dissipated by the drag, which is an

incredible amount of lost energy.

5. Discussion

As mentioned in section 2, a central assumption in our model is that a 100 m sprinter (not

only Bolt) is able to develop a constant force during a race (except in the initial few tenths

of a second, where he pushes himself against the starting block). To delimit how good this

2To calculate such a cross section area, we used a similar procedure to that used in [9], where instead of a circle

we estimated the area of the head with an ellipse. We averaged several scaled measures from Bolt pictures taken

from [14].

1232 JJHG

´

omez

et al

0246810

1000

800

t (s)

600

F0 (N)

Figure 4. Force exerted by the runner during the race. The red line is calculated with the

experimental data, the dash-dot-dot (green) line is the average force of 818.3 N, while the short-

dash-dot (black) line is the value of the force F0obtained from the adjustment (815.7 N).

assumption is we use the experimental values of u, the calculated acceleration and the ﬁtted

values of the constants γand σto compute F0. The result is shown in ﬁgure 4. It is interesting

to note that the average value of the force obtained from this ﬁgure is 818.3 N, which is very

close to the value obtained from the ﬁtting of the data, 815.7 N. The high value of the force in

the ﬁrst part of the race is due do the acceleration he obtains when he pushes himself from the

starting block.

At ﬁrst glance, observing the values of the drag coefﬁcients in table 2, one is impelled

to argue that because σγthe hydrodynamic drag could have been neglected. However,

one can calculate the drag terms in the equation of motion at the terminal speed uT, attaining

γuT=725.59 N and σuT

2=90.18 N. Thus, from the total drag γuT+σu2

T, 11.05%

corresponds to turbulent drag, which turns to be an important contribution.

If we would like to make predictions considering different wind corrections, this can be

done as follows. Once a runner acquires the wind speed (almost instantly), the second term in

the right side (γu) of equation (1) behaves as if the sprinter were running in still air, because γ

is proportional to the air viscosity, which is independent of air pressure. However, that is not

the case for the third term in (1), (σu2), which arises from the collisions per unit time of the air

molecules against the sprinter and it is proportional to the speed of the runner with respect to

the ground. In a simple model, the hydrodynamical drag force is DH=σ(ρ)(v+vw)2, where

vis the speed achieved by the runner without wind and vwis the speed of the wind. The value

of σdepends on the number of molecules that impact on the runner per unit time and should

be different in still air conditions. Then, the equation of motion (1) can be rewritten as

m˙u=m˙v=F0−γv−σ(v +vw)2,(11)

and without wind as

m˙v=F0−γv−σv2.(12)

Subtracting (11) and (12), we obtain

σv2+2vvw+v2

w=σv2,(13)

so then

σ=σ1+2vw

v+v2

w

v2∼σ1+2vw

v,(14)

where the third term in the second expression has been neglected (vwv). To estimate the

value of σ, we consider vas the terminal speed of Bolt, uT. With these conditions, σ=0.69

with still air (vw=0ms

−1) and σ=0.49 with a tailwind of vw=2ms

−1. It should be

clear that this calculation is only a crude way to estimate the differences of running time with

On the performance of Usain Bolt in the 100 m sprint 1233

Tab le 3. Predictions of the running time for Bolt without tailwind, and with a tailwind of 2 m s−1.

vw(m s−1) Estimated running time (s)

0 9.68

0.9 9.58

2 9.46

and without wind. The results, which are close to the values reported in the literature [15], are

summarized in table 3.

Although this is a simple way to calculate a correction due to wind, it turns out to be a

good proposal for it. A more realistic assumption would be to modify equation (14)tobe

σ=σ1+αvw

uT,(15)

with the parameter αlying between 1 and 2.

The results obtained, together with the facts indicated in this discussion, show the

appropriateness and quality of the model developed in this paper. We look forward to the

next IAAF WCA, which will be held in Moscow, Russia, 10–18 August 2013, to test our

model with the experimental data obtained from such sprints, as well as waiting to see if the

fastest man on earth is able to beat his own world record again.

Acknowledgment

This work was partially supported by PAPIIT-DGAPA-UNAM Project IN115612.

References

[1] Keller J 1973 A theory of competitive running Phys. Today 26 42–47 (Reprinted in [16])

[2] Alexandrov I and Lucht P 1981 Physics of sprinting Am.J.Phys.49 254–7 (Reprinted in [16])

[3] Tibshirani R 1997 Who is the fastest man in the world? Am. Stat. 51 106–11

[4] Wagner G 1998 The 100-meter dash: theory and experiment Phys. Teach. 36 144–6

[5] Heck A and Ellermeijer T 2009 Giving students the run of sprinting models Am.J.Phys.77 1028

[6] Helene O and Yamashita M 2010 The force, power and energy of the 100 meter sprint Am.J.Phys.78 307

[7] www.youtube.com/watch?v=SyY7RgNLCUk

[8] International Amateur Athletics Federation 2009 Berlin Biomechanics Project http://berlin.iaaf.org/news/kind=

101/newsid=53084.html

[9] Charles J D and Bejan A 2009 The evolution of speed, size and shape in modern athletics J. Exp.

Biol. 212 2419–25

[10] World Meteorological Organization World Weather Information Service (Berlin: World Meteorological

Organization) http://worldweather.wmo.int/016/c00059.htm

[11] Shanebrook J and Jaszczak R 1976 Aerodynamic drag analysis of runners Med. Sci. Sports 843–5

(PMID: 1272006)

[12] Zatsiorsky V 2002 Kinetics of Human Motion (Champaign, IL: Human Kinetics) pp 492–6

[13] Brownlie L 1982 Aerodynamic and thermal characteristics of running apparel MSc Thesis Simon Fraser

University

[14] Bolt U http://usainbolt.com/

[15] Mureika J R 2008 The legality of wind and altitude assisted performance in the sprints New Stud. Athletics

15 53–60

[16] Armenti A Jr 1992 The Physics of Sports (New York: American Institute of Physics)