Jour nal of Sports Sciences, 1997, 15, 517 - 526
A mathem atical analysis of the bioenergetics of
hurdling
A .J. WA R D -SM IT H
Departm ent of Sport Sciences, B runel U niversity, Oster ley Campus, B orough Road, Isleworth, M iddlesex
TW 7 5DU, U K
Accepted 11 Februar y 1997
A m athematical model of the bioenergetics of running has been extended to apply to hurdling. This has been
achieved by two principal modi® cations. First, a new term has been added to the energy equation to account
for the potential energy required to negotiate each hurdle. Secondly, the term describing the degradation of
m echanical energy to thermal energy has been modi® ed to account for the adjustments in stride pattern
necessary to negotiate the hurdles. A comparison of predicted and actual running times of elite athletes was
m ade and good levels of agreem ent were achieved.
K eywords : Athletics, bioenergetics, biomechanics, hurdling.
Introduction
H urdling can be regard ed as a specialized form of
sprinting in which athletes negotiate a hurdle by exaggerating their norm al stride pattern. In the men’s
110 m and wom en’s 100 m hurdles, about one stride in
every four is exaggerated, whereas in the m en’s and
wom en’s 400 m hurdles - depending on the natural
stride length of the athlete - one stride in anything from
13 to 19 is exagge rated.
The starting point for the present paper is the m athem atical analysis of the running perform ance of m ale
athletes (Ward-Sm ith, 1985a). Recently, the m ethod
has been reviewed, been found to be robust and been
app lied to fem ale runners by Ward-Sm ith and M obey
(1995). T he theoretical app roach is based on a consideration of the dominant energy transfor m ations that
take place during running, as shown in Fig. 1. Secondorder effects, such as the small-sc ale cyclical variations
of energy level associated w ith the stride pattern, are
ignored on order-of-m agnitude grounds.
Q ualitative aspects of hurdling
To m aintain a good horizontal speed in hurdling, the
obstacles m ust not only be cleared, but be cleared in
such a way that the clearance process has only a m ini0264 - 0414/97
© 1997 E & FN Spon
m al effect on the horizontal running speed. T he takeoff distance in front of the hurdles depends on the
height, leg length, speed, ¯ exibility and technique of
the athlete. Exp ert hurdlers instinctively m odify the
take-off distance in accordance with their horizontal
velocity over the ground. Thus, as an athlete’s speed
increases over the ® rst few hurdles, the take-off distance increases accordingly. Equally, adjustments have
to be m ade over the ® nal few hurdles whenever fatigue
reduces the athlete’s speed. As the athlete prepares to
clear the hurdle, the lead leg and the torso m ove forwards and towards each other so that the athlete’s
centre of m ass is raised above the ground by the smallest amount consistent w ith clearing the hurdle. T his
effect is enhanced by som e athletes who, when clearing
the hurdle, lower their head as far as possible. Ideally,
to m aintain the fastest horizontal speed, the athlete
skim s over the hurdle with the centre of m ass as low as
possible and consistent with a balanced landing on the
track. As soon as the lead leg passes the obstacle, the
hurdler begins to bring it back into contact with the
ground, ensuring that the aerial distance after the
hurdle is as short as possible. Speed is only gained or
m aintained through propulsive action when the hurdler
is in contact with the ground; m inim izing the hurdling
distance and duration therefore leads to a m axim ization
of the proportion of time devoted to spr inting. The
518
action of the trailing leg during hurdling is rather different from that in norm al sprinting. W hereas in sprinting
it passes vertically beneath the body, in hurdling the
trailing leg is abducted away from the running plane to
avoid contact w ith the hurdle. As a consequence of this
action, the centre of m ass is raised by a sm aller amount
than would otherw ise be the case. By minim izing the
vertical m ovem ent of the centre of m ass, the amount of
potential energy that has to be generated at each hurdle
is m inim ized.
It is known from steady-state treadm ill tests that
there is an essentially linear relationship between running speed and oxygen consum ption (M argaria, 1976;
D avies, 1980). It follows (Ward-Sm ith, 1984) that the
rate of increase of energy dissip ated as heat in running
is directly proportional to velocity under steady-state
conditions. Ward-S m ith (1984, 1985a) has shown, by
extrapolation, that this relationship can also be app lied
to the acceleration phase at the start of spr inting. In
hurdling, the norm al sprinting action is com promised,
adjustments being m ade by the athlete to both the
stride length and vertical displacement of the centre of
m ass to accom m odate the height and spacing of the
hurdles. Because the stride pattern is com prom ised, it
follows that the rate of degradation of m echanical
energy into therm al energy, or, expressed another way,
the rate of heat production, is greater in hurdling than
in spr inting.
Sum m arizing the above discussion, from an energy
standpoint there are two principal differences between
hurdling and normal sprinting. In hurdling, potential
energy has to be supplied to the centre of m ass at each
hurdle over and above the am ount associated with norm al running. Also, the stride pattern is distorted relative to that in running. U ltimately, both of these factors
contribute to an increase in the energy dissip ated as
heat.
M athematical analysis
T he preceding discussion has shown that hurdling can
be regarded as a specialized form of sprinting adapted
to the clearance of obstacles. Here we begin the analysis of hurdling by using the m athem atical m odel of
running, based on energy considerations, established
by Ward-Sm ith (1985a). M odi® cations appropriate to
hurdling will then be introduced. The analysis considers energy changes asso ciated with the m otion of the
centre of m ass of the runner; for distances greater than
a few stride lengths, the cyclical variations asso ciated
with the stride pattern are ignored on order-ofm agnitude grounds. T he energy exchanges are sum m arized in Fig. 1.
Ward-Smith
Figu re 1 Representation of energy exchanges occurring during running.
Energ y balance dur ing running
The chemical energy released, C , is equal to the sum of
the external work done on the centre of m ass of the
hurdler, W, plus the m echanical energy converted into
therm al energy, H. The overall energy balance is given
by:
C5
H
1
W
(1)
D ifferentiation of the energy balance with respect to
time, t, results in the corresponding power equation,
given by:
dC
dt
5
dH
dt
1
dW
dt
(2)
The individual contributions to the power equation are
as follow s.
519
A mathematical analysis of the bioenergetics of hurdling
The rate of chem ical energy conversion is, follow ing
Ward-Sm ith (1985a):
dC
dt
5
t) 1
l
P m ax exp( 2
R[1 2
exp ( -
l
t)]
dt
5
(4)
Av
where v is velocity and A is a param eter governing the
rate of dissipation of m echanical energy.
The work done by adding kinetic energy to the
centre of m ass will be denoted by W 1 and the work
done against aerodynam ic drag by W 2 . T hen, the rate at
which the horizontal component of kinetic energy is
added to the runner of m ass m is given by:
dW 1
dt
5
mv
dv
(5)
dt
T he rate of working again st aerodynam ic drag, for still
air conditions, is:
dW 2
dt
5
Dv 5
1
2
r
v 3 SC D
1
5
2
r
v3AD
(6)
where D is the aerodynam ic drag experienced by the
body, r is the air density, S is the projected frontal area
of the body, C D is the drag coef® cient and A D ( 5 SC D )
is the drag area.
Substitution of the individual components into equation (2) yields the full form of the power equation for
running, w hich is:
P m ax exp( 2
l
t) 1
1
2
r
R[1 2
v 3 SC D
1
exp( 2
mv
l
dv
dt
t)] 5
Av
l
(3)
where P m ax represents the m axim um power availab le
from the anaerobic m echanism , R is the m axim um
aerobic power and l is a param eter governing the rate
of release of chemical energy.
The sum of the four contributions H 1 , H 2 , H 3 and
H 4 shown in Fig. 1 is denoted by H. The overall rate of
degradation of m echanical energy into heat is given
by:
dH
P m ax
1
(7)
In this form ulation of the power equation, the relatively
small effects of the vertical m ovem ent of the runner’s
centre of m ass in the earth’s gravitational ® eld are
absorbed into the dissipation term, Av, as explain ed in
Fig. 1.
The energy equation is derived by integration of
equation (7) w ith respect to t between the lim its t 5 0,
x 5 0 and t 5 T, x 5 X, and is
[1 2
exp( 2
AX
T)] 1
l
1
K
1
R[T 2
E
T
0
l
(1 2
1
v 3 dt 1
2
exp ( 2
mV 2
l
T))] 5
(8)
where
r
K5
SC D
2
(9)
M odi® cations for hurdling
For hurdling, the analysis developed for running needs
to be m odi® ed in two ways. First, it is useful to consider explicitly an additional com ponent in the power
and energy equations which accounts for the changed
vertical m ovem ent of the body’s centre of mass associated with the negotiation of the hurdles. Just as in running, over the race as a whole, the net vertical
displacem ent of the centre of mass is alm ost zero; this
term is therefore ultimately manifested as a contribution to the degradation term . However, it is convenient
to represent the term explicitly in the analysis of hurdling. Secondly, it m ust be recognized that, to negotiate
the hurdles, athletes are forced to depart from their
optimal stride pattern adopted in running, and as a
consequence there is an increased degradation of
m echanical energy into heat asso ciated with this
effect.
By conservation of energy, the vertical com ponent of
kinetic energy when taking off to clear the hurdle is
converted to potential energy because of the gain in
height of the centre of m ass of the athlete. T hus:
1
2
mV V 2 5
mgh
(10)
where h is the vertical displacem ent of the centre of
m ass relative to its nom inal horizontal level, and V V is
the vertical velocity com ponent at take-off. If the num ber of hurdles is denoted by N H , then the additional
energy transferred to potential energy in clearing the
hurdles is N H mgh. W hereas the term s in equation (7) the power equation for running - are continuous, the
contributions to the power and energy equations from
the hurdling action are discontinuous. To establish the
power equation for hurdling, we average the hurdling
term s over the entire race. A power term associated
with the potential energy used to clear the hurdles may
be de® ned by dW 3 /dt and this is given by:
dW 3
dt
5
NH
mgh
T
(11)
520
Ward-Smith
The dissip ation term for hurdling will be different
from that for running and can be derived in the following way. It is inevitable that, to a greater or lesser
extent, a hurdler has to modify the stride pattern not
only in those strides used to clear the hurdles but in
others in the approach and landing as well. As a basis
for developing the theoretical work, we assu m e that a
hurdler prog resses through a race: (1) by advancing
using a norm al running action, and (2) by m odifying
the norm al running action in a certain num ber of
strides that is proportional to the num ber of hurdles
cleared. Let A and A H represent the rate of degradation
of mechanical energy into therm al energy for norm al
running strides and hurdling strides, respectively. We
also de® ne N T and N , respectively, as the total num ber
of strides and the num ber of norm al running strides in
a hurdle race. Hence:
NT 5
N
1
P m ax exp( 2
t) 1
l
R[1 2
mv
NH
(13)
NT
dv
dt
5
1
v
dv
dt
l
A e ffv 1
t)] 5
N H mgh
1
Kv 3 1
(18)
T
R* 2
[(P*m ax 2
K*v 3 2
t) 1
l
R*)exp ( 2
A*e ff v 2
N H gh
T
]
(19)
where values shown with an asterisk (*) are values norm alized with respect to body m ass. T hus R* 5 R/m, etc.
Furtherm ore, x, v and t are related by:
dx
dt
5
(20)
v
T he total energy contributions from the anaerobic
and aerobic mechanism s, C an and C aer respectively, can
be obtained by integration of the corresponding power
term s w ith respect to time. T hus the total chem ical
energy converted at time t from rest is given by:
C5
T hen:
exp( 2
Equation (18) can be rewritten as:
(12)
NH
We introduce the symbol a to represent the proportion of the total num ber of strides used for negotiating the hurdles: Thus:
a 5
The corresponding power equation is:
C an 1
(21)
C ae r
where
N
NT
NT 2
5
NH
NT
5
a
(1 2
)
An average effective rate of energy degradation for
the entire race A e ff can then be de® ned to satisfy the
relation:
A e ff 5
a
(1 2
1 a
)A
C an 5
(14)
(15)
AH
P m ax
l
[1 2
exp( 2
l
t)]
(22)
and
C aer 5
R[t 2
l
1
(1 2
exp( 2
l
t))]
(23)
We can relate A and A H by the expression:
AH
5 b
A
w here
b 5
f( a ,h)
(16)
T he param eter b will have a m agnitude greater than 1
and in principle depends, to a greater or lesser extent,
upon the vertical displacem ent of the centre of m ass of
the athlete and upon the stride pattern required to
negotiate the hurdles.
Hence for hurdling the energy equation becom es:
P m ax 2 R
l
K
[1 2
E
T
0
exp( 2
v 3 dt 1
l
1
2
T)] 1
mv 2
1
RT 5
A e ffX
N H mgh
1
(17)
Application of analysis to hurdling
D ata relevant to the solution of the derived equations
are assem bled in this section. Attention will be focused
on the perform ance of elite athletes.
Position of the hurdles on the track
Over the past century, there have been a number of
different race distances. Currently, the recognized outdoor hurdle events are the m en’s 110 m and 400 m and
the wom en’s 100 m and 400 m . The m en’s and
wom en’s indoor 50 m and 60 m hurdles events have
recently been recognized by the International Am ateur
Athletic Federation (IAAF). Although the 3000 m
521
A mathematical analysis of the bioenergetics of hurdling
steeplechase requires hurdles to be cleared, this event
has additional features and, because it is radically different from the other events considered here, it has
been excluded from the analysis. The num ber of
hurdles in an event, and the positions of the hurdles on
the track, are given in Table 1.
Typical stride patter ns of elite athletes
T he quantitative data on stride patterns reported here
were obtained by observing videos of elite hurdlers in
action; additional unpublished data were provided by
Paul Grim shaw.
In the m en’s 110 m hurdles, most elite hurdlers
adopt a stride pattern of eight strides to the ® rst hurdle,
although som e exceptional athletes use seven; three
strides are used between hurdles, with the fourth used
to negotiate the hurdle, and the race is com pleted with
six strides to the ® nish. T he race therefore consists of
about 51 strides, 10 of w hich are exagge rated by hurdle
clearance, yielding a representative value for a of 10/51.
In the m en’s 400 m hurdles, most hurdlers use about
22 strides to the ® rst hurdle, 13 or 15 strides between
hurdles - although som e athletes are able to alternate
their leading leg and thereby use 14 strides - and about
25 strides from the last hurdle to the ® nish. As fatigu e
sets in, the natural stride shortens and so the stride
pattern in the early part of the race m ay be different
from that adopted later in the race. For a total of 192
strides, a is evaluated as 10/192. In the m en’s 60 m
hurdles, athletes usually adopt the sam e stride pattern
as in the 110 m hurdles, with eight strides to the ® rst
hurdle and, subsequently, three strides between each
hurdle. For this event, a is about 5/28. In the 50 m
hurdles, the stride pattern is the sam e as in the 60 m
event, with the exception of the num ber of strides used
in the ® nal distance. An app roxim ate estimate for a is
4/24.
In the wom en’s 100 m hurdles, the stride pattern is
the sam e as that in the m en’s 110 m hurdles, w ith eight
strides 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 representative
value for a of 10/51. In the wom en’s 400 m event, the
stride pattern usually consists of about 28 strides to the
® rst hurdle, either 17 or 19 strides between the hurdles,
depending on the physiqu e and natural stride of the
athlete, followed by about 26 strides to the ® nishing
line. As in the men’s event, the num ber of strides m ay
increase towards the end of the race due to fatigu e. A
representative value for a is 10/221. In the wom en’s
60 m hurdles, about eight strides are used to the ® rst
hurdle and three strides in between. A typical value for
a is 5/28. O ver 50 m , the stride pattern adopted is the
sam e as for the 60 m hurdles, with the exception of the
num ber of strides at the ® nish. A representative value
for a is 4/25.
B iophysical data
Representative biophysical data describing an elite m ale
athlete capable of running the 100 m in 10.23 s have
been established by Ward-Smith (1985a). M arar and
Grim shaw (1993) have reported that the m ain factor
which determ ines performance at hurdling is sprinting
speed. T herefore, the representative biophysical data
previously derived for a m ale spr inter are directly
applicable to the case of a m ale hurdler. C orresponding
values for an elite fem ale runner capable of sprinting
the 100 m in 10.93 s have been evaluated by WardSm ith and M obey (1995). Again , it is appropriate to
use biophysical data for elite sprinters to evaluate
hurdling perform ance.
Tanner (1964) showed that the average height of
m ale hurdlers in the 1960 O lympics was 183 cm. A
Table 1 Positions of hurdles
Event
M en’s
M en’s
M en’s
M en’s
50 m
60 m
110 m
400 m
Women’s
Women’s
Women’s
Women’s
50 m
60 m
100 m
400 m
Number
of
hurdles
Distance
from start
to ® rst
hurdle
(m)
Distance
between
hurdles
(m)
Distance
from last
hurdle
to ® nish
(m )
4
5
10
10
13.72
13.72
13.72
45.00
9.14
9.14
9.14
35.00
8.86
9.72
14.02
40.00
4
5
10
10
13.00
13.00
13.00
45.00
8.50
8.50
8.50
35.00
11.50
13.00
10.50
40.00
522
Ward-Smith
Table 2 Magnitudes of parameters for male and
female hurdlers
Parameter
M ale hurdler
Female hurdler
l
0.03 s - 1
50.5 W kg- 1
23.5 W kg- 1
3.9 J kg- 1 m - 1
3.3 3 10 - 3 m - 1
1.83 m
0.385 m 2
70 kg
0.03 s - 1
47.2 W kg- 1
21.8 W kg- 1
3.98 J kg - 1 m - 1
3.6 3 10 - 3 m - 1
1.73 m
0.358 m 2
60 kg
P *m ax
R*
A*
K*
H
AD
m
representative value of 173 cm for the height of worldclass fem ale hurdlers has been obtained from unpublished data provided by Paul G rim shaw. A sum m ary of
the representative biophysical and other data for substitution in the com puter calculations is given in
Table 2.
C entre of mass
M easurements of the vertical displacem ent of the
centre of m ass of several athletes have been m ade from
G rim shaw ’s data. Inform ation was obtained by m easurem ent from three-dim ensional digital im ages created
from videos taken at various athletics m eetings. T he
whole body centre of m ass positions were obtained by
the usual segm entation m ethod. D ata on the clearance
height of a num ber of athletes were availab le for the
m en’s 110 m and the women’s 110 m hurdle events
only, and these are presented in Table 3. The representative clearance height used in the present calculations is
the average of the data in Table 3, and these data were
used for all events except the 400 m hurdles.
Table 3 Perform ance data for athletes in
competition (from Marar and Grimshaw, 1993)
Event
M en’s 60 m
M en’s 60 m
M en’s 60 m
M en’s
M en’s
M en’s
M en’s
110
110
110
110
Women’s
Women’s
Women’s
Women’s
Women’s
Women’s
m
m
m
m
100
100
100
100
100
100
m
m
m
m
m
m
Time (s)
Clearance (cm)
7.77
8.40
8.20
27
28
24
13.64
13.86
13.33
13.23
29
28
25
25
13.24
13.33
13.72
13.08
13.15
13.41
40
36
34
35
37
37
As no measurem ents had been m ade for either the
m en’s or wom en’s 400 m events, an estim ate of the
clearance was m ade; the estim ated value of the clearance was in close accord with data reported by Kaufm ann and Piotrowski (1976) for hurdlers of
interm ediate standard. The vertical displacement of the
centre of m ass of the hurdler was then calculated by
adding the clearance height to the height of the hurdle
and subtracting the height of the centre of m ass above
the ground during spr inting. Page (1978) quoted the
results of earlier work he had undertaken which showed
that the centre of mass of an adult m ale in a norm al
upright stance lies about 2.5 cm below his navel, or
approximately 57% of his full height from the ground.
A fem ale’s weigh t is distributed differently; she has a
wider pelvis and, usually, narrower and lighter shoulders. Her centre of mass is nearer the ground and is
about 55% of her full height from the ground (Page,
1978).
T he heigh t of the hurdles varies according to the
event. It is therefore necessar y to consider each event
separately in evaluating the mathem atical m odel. T he
height of the hurdles, the average clearance height and
the vertical displacement of the centre of m ass for each
event are sum m arized in Table 4.
Effective rate of energy degradation
At this stage, the one unknown quantity required to
solve equations (19) and (20) is a m easure of the
degradation of m echanical energy to therm al energy
asso ciated with hurdling. In principle, the m odi® ed
stride pattern, the num ber of hurdles and the vertical
m ovem ent of the centre of mass are factors that affe ct
A*e ff . H ere it will be assum ed that the effect of the vertical m ovem ent is m uch sm aller than the effect associated w ith the changed stride pattern, and can be
ignored on order-of-m agnitude grounds. This assu m ption can be tested when predicted and actual results are
Table 4 Estimated vertical displacement of centre of mass
Event
M en’ s
M en’ s
M en’ s
M en’ s
50 m
60 m
110 m
400 m
Women’s
Women’s
Women’s
Women’s
50 m
60 m
100 m
400 m
Height of
hurdle (m)
Clearance
(m )
Displacement
of centre of
m ass (m )
1.067
1.067
1.067
0.914
0.268
0.268
0.268
0.318
0.291
0.291
0.291
0.188
0.84
0.84
0.84
0.762
0.365
0.365
0.365
0.365
0.254
0.254
0.254
0.176
523
A mathematical analysis of the bioenergetics of hurdling
com pared. It is evident that A*e ff m ust exceed A* and
the assu m ption w ill be m ade that the excess is directly
proportional to a . T hus A*e ff is relatred to A* by the
exp ression:
A *e ff 5
(1
1 a
)A*
(24)
T his relationship corresponds to a value for b of 2;
exp ressed another way, for the strides directly affe cted
by the hurdling process, the ef® ciency of locom otion is
half that for norm al running.
Results and discussion
Equations (19) and (20) were solved to obtain v 5 v(t)
and x 5 x(t) using a numerical schem e based on the
fourth-order Runge-K utta m ethod. A time-step of
0.01 s was adopted, and program s previously written
for m ale spr inting (Ward-Sm ith, 1985a) and fem ale
sprinting (Ward-Sm ith and M obey, 1995) were used as
a basis for the prog ram developm ent.
The program m ade use of representative values of
the biophysical data contained in Table 2; data for h
and a derived above and sum m arized in Table 5 were
also used. Z ero w ind and a nom inal sea level density for
air of 1.21 kg m - 3 were assum ed. For all events, current
Table 5 Input data used in the program to
calculate hurdling perform ance
Event
M en’s
M en’s
M en’s
M en’s
50 m
60 m
110 m
400 m
Women’s
Women’s
Women’s
Women’s
50 m
60 m
100 m
400 m
a
h (m)
NH
0.29
0.29
0.29
0.19
4
5
10
10
0.17
0.18
0.20
0.052
0.25
0.25
0.25
0.18
4
5
10
10
0.16
0.18
0.20
0.045
world record times were used as a basis of com parison
with the m athem atical m odel.
C om pariso ns of predicted and actual running times
are shown in Table 6, for both outdoor and indoor
events. It is perhaps helpful to m ention that several factors can contribute to the differences between the predicted and actual times. First, there is the basic
variab ility in perform ance between one individual athlete and another, w ith the consequence that world
records do not ® t som e precise correlating equation.
Secondly, the physiolo gical param eters of individual
athletes m ay depart to som e extent from the representative data incorporated into the present calculations.
Thirdly, how well the m athem atical m odel works as a
whole depends on how well the individual contributions to the energy balance are modelled. For example,
there is evidence that a tail w ind enhanced the world
records in the m en’s 110 m event (w ind 1 0.5 m s - 1 )
and the wom en’ s 100 m event (wind 1 0.7 m s - 1 ). The
data in Table 6 are not adjusted for this effect and we
shall return to this m atter later. O verall, the differences
in Table 6 between predicted and actual times were
generally less than 2.5% , w ith the exception of the
wom en’s 100 m event, where the predicted time was
som e 6% greater than the world record. Bearing in
m ind the three broad sources of possible discrepancies
discussed above, the general level of correlation for the
other seven events is considered to be ver y good. We
note at this stage that, in deriving an exp ression for
A*e ff , the effect of the vertical movem ent of the centre of
m ass was ignored compared to the effect of the m odi® ed stride pattern on order-of-m agnitude grounds.
This assu m ption is seen to be justi® ed, as there is no
substantial disparity between the actual and predicted
times which can be system atically correlated w ith this
factor.
A num ber of supplementary calculations has been
m ade. The energy contributions from the anaerobic
Table 6 Comparison of predicted hurdling times with world record times
Event
M en’s
M en’s
M en’s
M en’s
50 m
60 m
110 m
400 m
Women’s
Women’s
Women’s
Women’s
50 m
60 m
100 m
400 m
Actual time,
t A (s)
Predicted time,
t P (s)
6.25
7.30
12.91
46.78
6.10
7.24
13.19
47.01
6.58
7.69
12.21
52.61
6.44
7.70
12.94
51.63
D t
(t A 2 t P )
1
0.15
0.06
2 0.28
2 0.23
1
1
0.14
( D t /t A ) %
1
2.4
0.8
2 2.2
2 0.5
1
1
2.2
2 0.01
2 0.2
1
1
2 0.73
0.98
2 6.0
1.9
524
Ward-Smith
Table 7 C ontributions from the aerobic and anaerobic
mechanisms
Actual
time (s)
Event
Men’s
Men’s
Men’s
Men’s
50 m
60 m
110 m
400 m
Women’s
Women’s
Women’s
Women’s
50 m
60 m
100 m
400 m
Anaerobic
(%)
Aerobic
(%)
6.25
7.30
12.91
46.78
95.7
95.0
91.2
71.4
4.3
5.0
8.8
28.6
6.58
7.69
12.21
52.61
95.5
94.8
91.8
68.6
4.5
5.2
8.2
31.4
Table 8 Predicted in¯ uence of head and tail winds on
hurdling performance (a following wind is positive; a
head wind is negative)
Event
Men’s 110 m
Wind speed
(m s - 1 )
Predicted time
(s)
2 2
13.46
13.32
13.19
13.07
12.97
2 1
0
1
2
Women’s 100 m
2 2
2 1
0
1
2
and aerobic m echanism s were evaluated by substituting
the world record times in equations (21) and (23). T he
results are given in Table 7. T hese show that the power
exerted through out a hurdling event is predom inantly
from the anaerobic source, a result which is consistent
with previous results for running (Ward-Smith, 1985a).
T hese calculations are also consistent w ith data
reported by  strand and Rodahl (1986), who have
tabulated the app roxim ate contributions from the aerobic and anaerobic processes under general conditions
of m axim al effort in exercise.  strand and Rodahl give
the anaerobic-to-aerobic ratio as 85% at 10 s and
65 - 70% after 60 s, dim inishing to 1% after 2 h.
Indoor hurdling events are held under conditions
where air m ovements are sm all; in the 400 m outdoor
events, the effects of wind tend to be self-cancelling and
are quite sm all. However, perform ance in the m en’s
outdoor 110 m hurdles and the women’s outdoor
100 m hurdles is signi® cantly in¯ uenced by w ind conditions. To take account of wind effects, equation (6)
was m odi® ed. T he rate of working again st aerodynam ic
drag is then given by:
dW 2
dt
5
Dv 5
1
1
v(v 2
2
r
2
r
v(v 2
V W ) 2 SC D
V W )2 A D
5
(25)
where V W represents the wind velocity, positive for a
following w ind.
Calculations of the effects of both head and following
winds were m ade, and the results are given in Table 8.
T hese calculations indicate that a following w ind of
2 m s - 1 confers an advantage of just over 0.2 s for both
the m en’s 110 m and the wom en’s 100 m events. T hese
® gures are very sim ilar to those predicted for sprinting;
13.22
13.07
12.94
12.83
12.73
Table 9 Predicted in¯ uence of vertical movement
of centre of mass h on hurdling time
Event
h (m )
Predicted time (s)
Men’s 50 m
0.24
0.29
0.34
6.07
6.10
6.13
Men’s 60 m
0.24
0.29
0.34
7.20
7.24
7.28
Men’s 110 m
0.24
0.29
0.34
13.09
13.19
13.29
Men’s 400 m
0.14
0.19
0.24
46.87
47.01
47.15
Women’s 50 m
0.20
0.25
0.30
6.40
6.44
6.47
Women’s 60 m
0.20
0.25
0.30
7.66
7.70
7.75
Women’s 100 m
0.20
0.25
0.30
12.83
12.94
13.06
Women’s 400 m
0.13
0.18
0.23
51.48
51.63
51.79
for exam ple, an advantage of 0.18 s has been computed
for the m en’s 100 m sprint (Ward-Smith, 1985b).
525
A mathematical analysis of the bioenergetics of hurdling
Returning again to the conditions under which the
world records were set and using the method on which
Table 8 is based, the m athem atical m odel for the m en’s
110 m hurdles, with V W 5 0.5 m s - 1 , predicts a time of
13.13 s; for the wom en’s 100 m hurdles, with
V W 5 0.7 m s - 1 , a time of 12.86 s is predicted. The
corresponding percentage differences between the predicted and actual world record times (see Table 6) are
thereby reduced to 2 1.7% and 2 5.2% , respectively.
T hese calculations do suggest that the world record of
12.21 s for the wom en’s 100 m hurdles, established by
Yordanka D onkova on 20 August 1988, is quite exceptional w hen com pared again st all the other world
record hurdling perform ances, even after the effects of
wind assistance are taken into account.
The results of the com putation depend to som e
extent upon the representative values of the vertical displacem ent of the centre of m ass, h. T he values used are
shown in Table 4, and are based on the assum ed position of the centre of m ass of the runner as well as the
average results for clearance height set out in Table 3.
Although in Table 3 there is no direct correspondence
between clearance height and running times, the times
cover a range of perform ances, som e of which fall signi® cantly short of world standard. As it is useful to have
som e insight into how variations in h affect the predicted time, calculations at 1 cm intervals in h were
m ade; a selection of the results is set out in Table 9. A
detailed inspection of the calculations showed that,
over the range investigated, the predicted time varied
linearly with h; sm all but signi® cant bene® ts arise from
controlling this effect during hurdling.
The effects on hurdling performance of a num ber of
variables, including body m ass (m), projected frontal
area of the athlete (S), air density ( r ) and drag coef® cient (C D ), are taken into account by the param eter K*.
Rather than investigate the separate effects of these are
variables, particularly bearing in m ind that m and S are
interrelated, calculations were m ade to investigate the
overall effect of K* on hurdling time. T he results are
shown in Table 10. The effect of changing K* on predicted time is small; typically, a 20% change in K* produced a change in predicted time of 1% or less.
C onclusion
It has been shown that a m athem atical m odel of running can be successfully adapted to the analysis of
hurdling. This has been achieved by two principal
m odi® cations. First, a new term has been added to the
power (or energy) equation to account for the vertical
displacem ent of the centre of m ass required to negotiate the hurdles. Secondly, the term expressing the rate
of degradation of m echanical energy to therm al energy
Table 10 Predicted in¯ uence of K * on hurdling
performance times (s)
K * (m - 1 )
Event (m)
Men’s
Men’s
Men’s
Men’s
50 m
60 m
110 m
400 m
0.0030
0.0033
0.0036
6.08
7.22
13.14
46.75
6.10
7.24
13.19
47.01
6.11
7.26
13.24
47.27
K * (m - 1 )
Women’s
Women’s
Women’s
Women’s
50 m
60 m
100 m
400 m
0.0033
0.0036
0.0039
6.42
7.68
12.90
51.39
6.44
7.70
12.94
51.63
6.45
7.72
12.98
51.88
has been increased in m agnitude to account for the
effects of the adjustments in stride pattern to negotiate
the hurdles. G ood correlations of actual and predicted
times for outdoor and indoor hurdles events were
achieved.
Acknowledgem ents
The author wishes to thank D r P.N. Grim shaw for
releasing unpublished inform ation. Contributions to
the early stages of the work were undertaken by A.
M obey as a ® nal-year project.
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