MECHANICS AND ENERGETICS IN RUNNING SPECIAL REFERENCE TO EFFICIENCY MASAHIRO

WITH

KANEKO*

Department of Exercise Physiology and Biomechanics. Osaka College of Physical Education, Kumatori. Sennan-gun. Osaka. Japan

Abstract-In running, the musclesexert forces acting internally on the body and externally on the ground to move the body’s center of mass forward. The energctics of running can be studied by calculating efficiency: mechanical work done/net energy consumed. Variations in efficiency, however.can be attributed not only to the human machinery but also to methodological differences. Mechanical work has been calculated primarilbby two procedures: one is from energy changes taking place on and around the body’s center of mass, and the other is from segmental analysis. Using the former and assuming the energy cost to be I kcal k -t km-‘. we have shown that efficiency increases with running speed from45 to 70 ?% with elastic enerav 4, il alavinu a role fCavaana and Kaneko. 1. Phvsiol. 26% 467-481. 1977). However. the efficiency v&d&as& &h-speed iithe oiygen debt is included in the calculation ofthe energy cost. With the same method ~wchave also shown that the efficiency of distance runners is higher than that of sprinters if the runningispeed is less than 25 km h- *, and that there is a most economical step rate for maximizing the efficiency et a given constant running speed.

INTRODUCTION

total mechanical work falls to zero if the same amount of positive and negative work are simply added toIn contrast to walking, running can be characterized gether. Fenn (1930a. b) approached this problem by as a speedy form of locomotion that has no doubleusing absolute values of changes in potential and support time. Any type of running, from the shortest kinetic energy to calculate mechanical work during sprint to the longest distance run, is performed by level running at a constant speed, and he applied this relatively quick cyclic movements of the limbs. The to the maximum speed to obtain the efficiency ratio. skeletal muscles exert forces through the locomotor Use of a segmental analysis to calculate muscle mosystem, acting internally on the body and externally ments and power was lirst described by Elftman on the ground; to move the body’s center of gravity (1940). He examined efficiency oftop speed running on forward. Runntng may also, by mechanical modelling, the same film used by Fenn, and stated that ‘the rate be likened to a bouncing ball (Cavagna et al., 1964): (power) would be at least 2.89 h.p. This confirms the part of the elasiic energy stored in muscle and tendon value of 2.95 h.p. calculated by Fenn from the changes when the foot istrikes the ground can be unleashed in potential and kinetic energy’ (p. 684). Following immediately dGring the positive work phase. these pioneering studies, research and discussion on The energet@s of muscle function during running mechanical work and its efficiency have become can be studied, by measuring both mechanical work popular. and energy cost and then calculating ‘mechanical Efficiency values vary with running speed (Cavagna efficiency’, defined by Hill (1927) as mechanical work and Kaneko, 1977) and stride rate (Hogberg 1952) as done per unit dnergy expended. In fact, the ‘bouncing well as among individuals. Many of the disparitiesseen ball model’ ref&red to above has been derived from an in eliicicncy, however, for dimerenttypes and speedsof efficiency study on running (Cavagna et al., 1964). locomotion can be attributed not only to variables of Despite the tremendous number of publications on encrgetics in the human machinery, but also to methenergy cost of locomotion, relatively few studies have odological digerences in determining mechanical been conducted on mechanical work in human power and energy expenditure. In considering methlocomotion. The first well known study on efficiency odological problems, Williams and Cavanagh (1983) of running waslpublished by Furusawa et al. (1927). In pointed out the wide range of mechanical power that study, mechanical work in sprinting was estim- values in the literature. In the present paper the author ated by the propelling force (= 0.7 x body mass) x dis- describes mainly his own investigations on running tance, while energy cost was estimated by directly efficiency, with a short review to identify other studies measuring oxygen uptake during the recovery period. that have a bearing on this area of research. Using this method they reported efficiency values of 0.35-0.45. In level running at constant speed, however, CALCULATING

To calculate mechanical work (or power) in running, two procedures have primarily been used:one is based on energy changes taking place on and/or

*Address coqcspondcna to: M. Kaneko, Ph.D., Osaka College of Physijal Education, 1558-l Noda, Kumatori-cho, Scnnan-gun, Osaka 590-04. Japan. M

2%suw1-c

MECHANICAL WORK

57

58

M. KANEKO

around the body’s center of mass. and the other is based on segmental energy analysis. Using a force platform held by springs, Fenn (193Oa) measured the mechanical work used to accelerate the body’s center of mass-so-called ‘external’ mechanical work (We,,). This can be expressed simply as ul,,, = A(tngh + +nY’)

(1)

where m is body mass, g is gravity, and V is linear velocity of the total body center of mass (c.o.m.). A modern electrical force platform was first used by Cavagna et al. (1964) to measure external work in running. Subsequent studies have shown that external power increases linearly with increases in running speed (Cavagna et al.. 1976; Fukunaga et al., 1980). Fenn (1930b) also described a method for determining the mechanical work used to accelerate limb segments around the total body c.o.m.-later to be known as’intemal’ mechanical work (IV,,,). This work can be measured through cinematographic analysis of the movements of limb segments according to the following equation. WI”, = ZA(lm, I’; + fm,k;w;)

301

(2)

where m, is massof the ith limb segment, V, is the linear velocity of that segment, k, is the radius of gyration of the segment, and a), is the angular velocity around that segment’s center of mass. Total mechanical work ( W,,) can be obtained by summing all positive magnitudes of change in external and internal work as defined above, i.e.

O1

. c

5

6 7 6 Rumnp soccd (In/S)

9

Fig. 1. Mechanical power values in relation to running speed. (1) Total power by Fcnn’smethod (Fcnn. 193Ob) in which energy transfers between adjacent limb segments were allowed for (Cavagna and Kaneko. 1977). (2) Half of the total power by ‘Joint Power Method’(Aleshinsky. 1986). The data were supplied by courtesy of Dr M. Ae (University of Tsukuba). (3) Total oower bv ‘Seament Enernv Method (Winter. 1979). which assum& cokplete ener;; transfers among all body segments. (4) External power by Fenn’s method (Fenn. 1930a).

where /, is moment of inertia of the ith segment. To obtain total mechanical work, some studies have summed the absolute energy changes in each segment, i.e. changes in potential energy and in translational and rotational kinetic energies (Norman et al., 1976; w,Ol = I W.,,I + I %,I* (3) Grcgor and Kirkcndall. 1978; Luhtanen and Komi. 1978). This assumes no energy transfer between body In Fenn’s method, the external and internal work were thought to be independent of each other, i.e. no segments or between dilTerent forms of energy taking energy was transferred between the two, and were place within the same segment (called ‘pseudo-mcchcalled ‘maximum work’. Some amount of energy, anical work’). Subsequently, the idea was put forward however, can be transferred between limb segments. to assume complete energy transfers among all body For example, if arm movement is checked suddenly at segments. e.g. between right and left limbs, and bea point above the elbow, the kinetic energy of the tween upper and lower limbs (Winter. 1979, Pierrynupper arm will transfer to the adjoining forearm to owsky et al., 1980). The question arises, however, accelerate the latter without the need for muscular whether in practice mechanical energy can really be activity. Likewise, a transfer can take place in the completely transferred. Consider, for example, upper opposite direction, from a more distal to a more body movement in the transverse plane (twist) checked by lower body movement to the opposite proximal segment. For this reason, Cavagna and Kancko (1977) modi- direction to stabilize the body. Winter’s method fied Fenn’s method to allow for transfer of energy should give a much smaller W,, value than values between adjacent limb segments in determining W,,,,. obtained using Fenn’s method. In this particular Total mechanical work was then obtained by adding example, Sakurai and Miyashita (1985) showed that this W,,, to w.,,, as in equation (3) (hereafter referred just the external work determined by Fenn’s method to as ‘modified Fenn’s method’). The internal power exceeded the total work calculated using Winter’s values calculated this way arc about 15-20 % lessthan method. According to our calculations, Winter’s value those values that allow for no energy transfer, and of total power (‘Segment Energy Method’ in Fig. I) was 20-30% less than the values calculated using the increase exponentially with the speed of locomotion. In the segmental analyses, mechanical power can be modification of Fenn’s method, and about 15% greater than I@_,. obtained by calculating work according to the followIn addition to energy transfer. factors regarding ing equation: elastic energy storage and metabolic differences in W= ZlAEJ (4) positive and negative work also afTcct the efficiency value. Taking all of these variables into consideration, Eb = XA(m& + fm, V? + $20:)

59

Energetin in running

Williams lation

and Cavanagh

method

(1983) proposed a new calcu-

and calculated

an efficiency

value of

is externally

loaded. This may prompt an investigator

to set the

base-line

value

at

the

level

0.44 for posititie work, and of 1.32 for negative work,

expended during exercise on an unloaded

based on their assumption of negative work being three times more efficient than positive work.

is known as ‘work efficiency’. Whipp

Aleshinsky presented method

(1986) criticized

methods,

and

all of these previously

proposed

using ‘muscle moment

the

following

and power,

which

is

Efficiency determined

In a method

of Cappoao

is subtracted

ZIIIMioildt]

v&city

mass. Although

around

(5) at the ith joint,

method. the d!ta

that segment’s center of

give the same value as Winter’s

processing, e.g. smoothing,

can give

rise to discrepaincies. Total power calculated method

wi is

the method using the above equation

should theoret/ically

has pioduced

for calculating

‘delta

efficiency’

(or

efficiency’), energy cost at a lower work rate from that of a higher work

rate. Using

this method, relatively high efficiency values have been

where M, is the muscle moment the angular

and Wasserman

be 0.298. ‘apparent

W=

on the basis of such a base-line

show that the work efficiency in their example would

based on an idka similar to that of Elftman (1940) and dt al. (1976):

of energy ergometer.

using this

values about twice as large as

reported for running: 0.538 by Asmussen and BondePetersen (1974) in running with horizontally forces, and 0.69 by Pugh (1971) wind

resistance. Such methodological

determining

against

differences

in

a base-line can lead to an endless ‘philos-

ophic’ discussion, since isolating penditure

impeding

in running

the ‘net’ energy ex-

used only for muscular work appears to be

an elusive task. The base-line can be set on the basis of

Fenn’s total p+wer. because both positive and negat-

such factors

ive powers are included.

For this reason, half of k,,,

preceding conditions,

derived

method

priate method depending on the purpose of the study.

by Aleshinsky’s

Power Methd’

of Dr M. Ae. University

by Mann (1981). Chapman regarding

ments to running

as ‘Joint

data courtesy

of Tsukuba).

As exemplified

et al. (1984). and Ae et ul.

(1985). this type of procedure information

is shown

in Fig. 1 (unpublished

may

provide

contributions

useful

of muscle mo-

For

example,

might

bc determined

cult problem cpntributing cncy values. gne

to large variations

in cfici-

issues is the so-called

‘base-line’ i.e. which part of oxygen consumption be used for calculating Wasserman

(1969)

is to

all

rcportcd

the cncrgy

a value of I kcal

cost for standing

was suhtractcd.

in a fully

For

aerobic situation.

an oxygen

presented

by Whipp

uses bicycle pcdalling

and

as the cx-

this thuy

debt was contracted.

to a An

example of such a case would be high speed running. Another

problem

is estimating

the energy cost of

n al. (1952) compared the costs of positive work (En + ) vs negative work (En- ) on two negative work. Abbott

bicycle ergometers coupled in opposition.

efficiency.

A theoretic@1 analysis

food

suggested that the same value would be applicable

energy cost is also a dilii-

of the major

is to consider

load, gross efficiency

by including

FI ul. (1963)

cast in which PI metabolic

or

1974).

mcasurc. obtained

AND EFFICIENCY

Evaluation

of work.

’ km - ’ for net cncrgy cost in running. in which the

cncrgy CAI.CUt.ATING ENERGY EXPENDlTlJRE

purpose

at a given work

consumed (Wilkie, Morgaria

duration

so one must choose an appro-

if the

consumption

kg-

performance.

as rate of work.

En-/En’

was l/3.7

increasing

work

The ratio of

at 35 rpm. and incrcascd

rate.

Negative

work

efficiency

with re-

ternal load. Resting oxygen uptake is usually assumed

ported by Abbott and Bigland (1953) was in the range

to be a base-line in determining

of 0.47-l

poz level, however, value quickly

net energy cost. The

does not return

following

to the base-line

the termination

Because of thi$ prolonged

of activity.

recovery phase, Hill (1965)

.06. Margaria

of I.2 in downhill (1983) postulated for

which

et al. (1963) determined a value

walking.

Williams

and Cavanagh

negative work etliciency to be 1.32.

the efficiency

ratio

of negative/positive

suggested usink the base-line that follows mild exercise

work was assumed to be I :3 (Asmusscn and Bonde-

for

Petersen, 1974).

the

subsequent

mcasuremcnt

during

heavy

Using a value of 1.2 for negative work efficiency, Ito

exercise. ‘Whipp and Wasserman value

would

be 0.14

estimate that the efficiency

if the work

performed

were

et ol. (1983)estimated work

(W-)

divided by theienergy represented by all of the oxygen

calculated

consumed.

(Km-

consumed investigator

Sotne of that oxysen,

however,

would

even if no exercise were performed, n+ight want

oxygen consu@ption only on the e/lergy

to use the resting

as a base-line. expended

value is called Inet efficiency’. Whipp

level of

ERiciency

beyond

be

so an based

this base-line

and Wasserman

E-). Although

expected to have taken place in a more sudden manner than

it would

prompted

have

in walking

work efficiency in a ballistic movement Variations

of negative

only insofar as the ergometer

or

running.

This

Kaneko et al. (1984) to examine the negative

would be 0.2Ob. be considered important

then

= W’/

the negative work phase could

where

might

= W-/1.2.

not be isolated during running, that phase should be

ergometer,

the bicycle ergometer

to be E-

positive work efficiency as W’/E+

estimate that the net efficiency in their bicycle example The work 01 pedalling

) for negative

the energy cost (E-

in running

negative

work

work

greatly exceeded the variability ency (0.15-0.26)

using a sledge-

could

efficiency

bc isolated. (0.46190)

of positive work effici-

in the latter experiment.

suggesting

60

M. KANEKO

‘Ior

70 -

so -

2

50-

,F .- LO5 30 20 -

I

0

.r

3

L

5 6 7 Runnln9So*,8 (m/s)

8

9

rg

Fig. 2. The efficiency values determined for running at variousspeeds.Seedetailsin the text. that an unstable combination of factors contribute to negative work efficiency in ballistic movements. We see from these investigations that one cannot simply assign a set value to negative work efficiency and expect that value to be valid. EFFlClENCY IN RELATION TO RUNNING Sl’):):D

How does the efficiency value change with speed of locomotion? Cavagna and Kaneko (1977) studied this relationship in walking and running on a level surface, since they could find no such data prior to 1977. In that study they assumed that. if overall efficiency of positive work exceeds the elliciency of purely positive work, some of the energy to perform positive work must apparently be derived from elastic energy stored in the preceding negative work phase (Cavagna er D/., 1964). The procedures used to calculate mechanical work and efliciency were essentially the same as those reported by Fenn (1930a. b). The exception was as that energy transfer between adjacent limb segments was assumed as mentioned above. Internal mechanical work was measured by a cinematographic technique for speeds of 8.5-25 km h-’ in a running experiment. From these data and previously determined external mechanical power (Cavagna er al., 1976). total positive power ( fi,,) was obtained. The relation

between

p,,

(cal kg-’

min-

‘) and

MOST ECONOMICAL STEP RATE

A ‘most economical speed’ to minimize energy expenditure has been well established for walking (e.g. Zarrugh et al., 1974), but not for running. An ‘optimum’ energetic situation, however, can prevail in running if the speed is constant. In running at a given speed, an optimum stride length lo minimize Vo2 has

run-

V (km h-l) was I&=9.42 f4.73 V +0.266 V1.99’. The efficiency, k,,,,/net energy cost, shown in Fig. 2, rose from about 0.45 to 0.70 as running speed increased. This level of efficiency is clearly greater than Fenn’s value of 0.227. This discrepancy can be attributed entirely to diKerent energy cost levels, since total mechanical work values were of the same order as Fenn’s (1930b). An appreciable contribution by elastic energy was cited as the basis for the efficiency value being over 0.25. ning

Using the same procedure for determining IV,. Kaneko et al. (1983.1985) reported later that efficiency values actually decrease as running speed increases. This discrepancy, shown in Fig. 2, was not due to physiological factors but rather to an artifactual factor related to net energy cost. Cavagna and Kaneko (1977) used the value of I kcal kg-’ km-’ (Margaria et al., 1963) for all speeds,assuming that this value was applicable not only to aerobic but also to anaerobic exercise such as sprinting. On the other hand, Kaneko et 01. (above) directly measured net energy cost in which oxygen debt was included (Hill, 1927). The oxygen debt would lead to an overestimation of net energy cost (Margaria et al., 1963; Di Prampero, personal communication). There is no way, however, of directly determining net energy cost during anaerobic exercise. Gregor and Kirkendall(l978) have also shown that efficiency decreases with running speed (3-4.4 ms-I), but the efficiency level reported by Ito et al. (1983) remained virtually constant over a speed range of about 2.5-5.5 m s-l.

speeds

2 3 Step Frequency (slepsls) Fig. 3. Total mechanical power

( ?,:,A net energy cxpend-

iturc (k,,,). and effkiency ( W,,/&,,,) in relation to step frequency(Kancko et aI.. 1987). Running speeds are indicated by different lines. Note that the most efficient step frqucncy is virtually the same as the freely chosen frequency indicated by the arrows.

Energetics in running been reported by Hbgberg (1952) and by Cavanagh and Williams (1982). but a step rate to maximize mechanical efficiency has only been determined in walking with ,the measurement of mechanical work (Zarrugh, 1981; Cavagna and Ftanzetti, 1982). Under these circumstances, Kaneko et al, (1987) examined whether or no! a most efficient step rate (or stride rate) could be detetmined in level running performed at a given speed. Under variqus combinations of step rate and stride length, total tnechanical power was determined by summing extqrnal power (WY,.,)and internal power ( &,,) as descriibedpreviously and oxygen uptake was measured, ri;,, was found to decrease and a,“, to increase with increasing step rate, while the resultant total mechanical power ( d&, = tie,, + ri;,,) showed a minimum val+e at an intermediate step rate of about 3 stepss- * (Fig, 3). The minimum net energy cost and maximum efljciency values appeared at approximately the same step rate. Furthermore, the most efficient step ralie was virtually the same as the freely chosen step rate. The reason for this is not known. Neuromusculnr functions appear to ‘choose’ an optimal step rate (or stride Icngth) at the unconscious level so as to minimize energy cost and power output and thereby maximize efficiency. EFFICIENCY

IN RUNNERS

Efficient muscular encrgetics should be of prime importance to distance runners. Gregor and Kirkendall(l978) examined ‘performance eliicicncy’ of fcmalc

61

marathon runners, and reported that the average efficiency value across various speeds was highest in the fastest runner. A similar observation was made on young distance runners by Kaneko et al. (1981). in which the fastest runner demonstrated the highest efficiency at a race pace. As shown in Fig, 4, the efficiency values were compared between distance runners and sprinters at various constant speeds (3.9-9.4 m s- * ). Distance runners had greater efficiency than sprinters at relatively low speeds ( c 7 ms-’ ). but this relation tended to be reversed at higher speeds.This ditTerencebetween the two groups of runners was not due to total mechanical power but rather to differences in net energy cost (Kaneko et al., 1985). From a study using bicycle work, a similar trend in efficiency was shown by Stuart et 01. (1981). In that study, ‘instantaneous efficiency’ was higher in distance runners than in sprinters when the work rate was relatively low (< 90 W). At higher work rates. however, the sprinters had greater efficiency. Differences in efficiency between sprinters and distance runners may be accounted for by differences in muscle fiber composition. Distance runners’ muscles are characterized by a high per cent of slow-twitch (ST) fibers, whereas sprinters have a predominance of fast-twitch (FT) fibers (Gollnick et al., 1972; Costill a al., 1976). Such differences in muscle fiber type have been shown to affect athletic performance (Komi, 1979: Bosco et al., 1980). In relation to speed. Awan and Goldspink (1972) have shown in mammals that etliciency of ST iibcrs is high at a relatively low contraction speed and that of FT fibers at high speed. This may, in part, be explained by the ‘size principle’ (Henneman et al.. 1965). ST fibers work more efiiciently than FT fibers in both isometric and isotonic contractions (Gibbs and Gibson, 1972, Wendt and Gibbs, 1973). These findings provide a tentative basis for explaining why distance runners can run more economically than sprinters at relatively low speeds.

CONCLUDING 160’ 160 a f

1f.O~

‘,

I

--‘-‘

6 6 Rumirq

1 Speed

6 (mhrc)

,

10

II

Fig. 4. Etki+y of dislana runners ( 0 ) and sprinters (0) in relation to ruhning speed(Kancko et ni., 19115).Note that cfTiciency valjcs arc greater in distance runners than sprinteds at relatively low speeds ( < 7 ms- ’ ).

in

REMARKS

Determining a ‘true’ efficiency value for running is difficult since this involves many elusive factors such as mechanical energy transfer, elastic energy utilization, and energy cost related to ‘base-line’ activity. For this reason, several theories concerning methods to determine efficiency have been presented in the literature. Fenn’s method, which we have followed, has met with strong criticism in the literature. Although such theoretical analysis may be indispensable to the development of science, theory alone can sometimes be dangerous, because misplaced confidence in a theory may prevent us from doing experiments. An appropriate combination of experiments with assumptions would be a more fruitful approach, since the growth of knowledge progresses primarily through trial and error. In this context, the present author has stayed with one method and therewith has compared the

M. KANEKO

62

efficiency values of different runners at various run-

ning speeds. Although such data should eventually be superseded as some of the above difliculties are resolved, the data we obtained could contribute during the interim to considerations on the energetics of running in the field of sport science. Acknowledgemenrs-I am grateful to Dr Paul D. Andrew for his kind criticisms in correcting this manuscript and also to Dr M. Ae for his help and valuable suggestions. REFERENCES Abbott, B. C.. Bigland. B. and Ritchie. J. M. (1952) The physiological cost of negative work. 1. Physiol. 117, 380-390. Abbott, B. C. and Bigland. B. (1953) The effect of force and speed changes on the rate of oxygen consumption during negative work. 1. Physiol. It. 319-325. Ae, M.. Miyashita. M.. Shibukawa, K.. Yokoi. T. and Hashiham. Y. (I9851 Body segment contributions during the support phase while running at different velocities. In Biamvchunics 1X-B (Edited by Winter, D. A., Norman. R. W.. Wells. R. P., Heyes. K. C. and Patla. A. E.). pp. 343-349. Human Kinetics Publishers, Champaign. Aleshinsky, S. Y. (1986) An energy ‘sources’ and fracIions’ approach to the mechanical energy expenditure problem: II. Movement of the multi-link chain model. J. Biomrrhunics 19, 295-300. Asmusscn. E. and Bondc-Petersen, F. (1974) Apparent efficiency and storage ofclasticcnergy in human musclesduring exercise. Actu Physiol. Scund. 92, 537-545. Awan, M. Z. and Goldspink. G. (1972) EnergcIics of Ihe dcvelopmcnt and maintenance of isometric tension by mammalian fast and slow muscles. J. Mrchunnchcm. Cv// MoIilify 1, 97-108. Bosco, C.. Koni. P. V. and Sinkkonen. K. (1980) Mcuhanical power. net efliciency and muscle structure in male and female middle-distance runners. Scund. J. Spurrs Sci. 2. 47-51. Cappoxao, A.. Figura. F.. Marchetti. M. and Pedotti, A. (1976) The interplay of muscular and external forces in human ambulation. J. t)iomechunics9, 3&43. C%vagna. G. A. and Franrctti. P. (1982) Step frequency in walking. ARCS Med. Sci. Anur. Num. Biol.. Biul. Technul.. Curdiuvusc. Syrt. IO, 28 f -282. Cavagna. G. A. and Kaneko, M. (1977) Mechanical work and efficiency in level walking and running. J. Physiul. 268, 467-481. Cavagna. G. A., Komarek. L. and Maxzolcni. S. (197 I) The mechanics of sprint running. J. Physinl. 217, 709-721. Cavagna. G. A.. Saibene. F. P. and Margaria. R. (1964) Mechanical work in running. J. appl. Phjsiol. 18, l-9. Cavaana. G. A.. Thvs. H. and Zamboni. A. i 1976)The sources of cxlernal work-in level walking and running. J. Physiul. 262.639-657. Cavanagh. P. R. and Williams, K. R. (1982) The ctfect of stride length variation on oxygen uptake during d&lance running. Med. Sci. Sp0fI.s Exert. 14. 30-35. Chapman, A. E.. Lonergan. R. and Caldwell. G. E. (1984) Kinetic sources of lower-limb angular displacement in the recovery phase of sprinting. Med. Sci. Sports Exert. 18. 382-388. Costill. D. L.. Daniels. J.. Evans, W., Fink, W.. Krahenbohl. G. and SalIin, B. (1976) Skeletal muscle enzymes and kinetic factors in male and fcmalc track athlctcs. J. uppl. Physiol. 40. 149-l 54. Elftman. H. (1940) The work done by musclesin running. Am. J. Physiol. 129. 672-684. Fenn. W. 0. (19301) Frictional and kinetic factors in the work of sprint

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