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Optimal Force–Velocity Profile in Ballistic

Movements—Altius:Citius or Fortius?

PIERRE SAMOZINO

1

, ENRICO REJC

2

, PIETRO ENRICO DI PRAMPERO

2

, ALAIN BELLI

3

,

and JEAN-BENOI

ˆT MORIN

3

1

Laboratory of Exercise Physiology (EA4338), University of Savoy, Le Bourget du Lac, FRANCE;

2

Department of Biomedical Sciences and Technologies, University of Udine, Udine, ITALY; and

3

Laboratory of Exercise Physiology (EA4338), University of Lyon, Saint Etienne, FRANCE

ABSTRACT

SAMOZINO, P., E. REJC, P. E. DI PRAMPERO, A. BELLI, and J.-B. MORIN. Optimal Force–Velocity Profile in Ballistic Move-

ments—Altius:Citius or Fortius? Med. Sci. Sports Exerc., Vol. 44, No. 2, pp. 313–322, 2012. Purpose: The study’s purpose was to

determine the respective influences of the maximal power (P

;

max) and the force–velocity (F–v) mechanical profile of the lower limb

neuromuscular system on performance in ballistic movements. Methods: A theoretical integrative approach was proposed to express

ballistic performance as a mathematical function of P

;

max and F–vprofile. This equation was (i) validated from experimental data

obtained on 14 subjects during lower limb ballistic inclined push-offs and (ii) simulated to quantify the respective influence of P

;

max and

F–vprofile on performance. Results: The bias between performances predicted and obtained from experimental measurements was

4%–7%, confirming the validity of the proposed theoretical approach. Simulations showed that ballistic performance was mostly in-

fluenced not only by P

;

max but also by the balance between force and velocity capabilities as described by the F–vprofile. For each indi-

vidual, there is an optimal F–vprofile that maximizes performance, whereas unfavorable F–vbalances lead to differences in performance up

to 30% for a given P

;

max.ThisoptimalF–vprofile, which can be accurately determined, depends on some individual characteristics

(limb extension range, P

;

max)andontheafterloadinvolvedinthemovement(inertia,inclination). The lower the afterload, the more the opti-

mal F–vprofile is oriented toward velocity capabilities and the greater the limitation of performance imposed by the maximal velocity of

lower limb extension. Conclusions:Highballisticperformancesaredeterminedbybothmaximizationofthepoweroutputcapabilitiesand

optimization of the F–vmechanical profile of the lower limb neuromuscular system. Key Words: MAXIMAL POWER, JUMPING

PERFORMANCE, LOWER EXTREMITY EXTENSION, INCLINED PUSH-OFF, EXPLOSIVE STRENGTH, MUSCLE FUNCTION

Ballistic movements, notably jumping, have often

been investigated to better understand the mechani-

cal limits of skeletal muscle function in vivo, be it

in animals (19,23) or in humans (6,9,20). One of the main

questions scientists, coaches, or athletes ask when explor-

ing factors for optimizing ballistic performance is which

mechanical quality of the neuromuscular system is more

important: ‘‘force’’ or ‘‘velocity’’ mechanical capability?

Ballistic movements may be defined as maximal move-

ments aiming to accelerate a moving mass as much as pos-

sible, that is, to reach the highest possible velocity in the

shortest time during a push-off. From Newton’s second

law of motion, the velocity reached by the body center of

mass (CM) at the end of a push-off (or takeoff velocity, v

TO

)

directly depends on the mechanical impulse developed in

the movement direction (22,26,42). Because the ability to

develop a high impulse cannot be considered as a mechan-

ical property of the neuromuscular system, the issue is to

identify which mechanical capabilities of the lower limbs

determine the impulse. Developing a high impulse during a

lower limb push-off and, in turn, accelerating body mass as

much as possible have often been assumed to depend on

power capabilities of the neuromuscular system involved in

the movement (14,19,26,29,36,40,43). This explains the

wide interest of sports performance practitioners in im-

proving muscular power (9,10,12,14,27). On this basis,

maximal power output (P

;

max) may be improved by in-

creasing the ability to develop high levels of force at low

velocities (force capabilities or strength) and/or lower levels

of force at high velocities (velocity capabilities) (10,11,27).

The best strategy continues to be an everlasting source of

interest and debate (5,10–12,14,29).

The overall dynamic mechanical capabilities of the lower

limb neuromuscular system have been well described by

inverse linear force–velocity (F–v) and parabolic power–

velocity (P–v)relationshipsduringvarious types of multijoint

Address for correspondence: Pierre Samozino, Ph.D., Laboratoire de Physi-

ologie de l’Exercice, Universite

´de Saint-E

´tienne, Me

´decine du sport et

Myologie - Centre Hospitalier Universitaire Bellevue, 42055 Saint-E

´tienne

Cedex 02, France; E-mail: pierre.samozino@univ-savoie.fr.

Submitted for publication November 2010.

Accepted for publication July 2011.

Supplemental digital content is available for this article. Direct URL

citations appear in the printed text and are provided in the HTML and PDF

versions of this article on the journal’s Web site (www.acsm-msse.org).

0195-9131/12/4402-0313/0

MEDICINE & SCIENCE IN SPORTS & EXERCISE

!

Copyright "2012 by the American College of Sports Medicine

DOI: 10.1249/MSS.0b013e31822d757a

313

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Copyright © 2012 by the American College of Sports Medicine. Unauthorized reproduction of this article is prohibited.

concentric extension movements (3,33,35,40,43). These re-

lationships describe the changes in external force generation

and power output with increasing movement velocity and

may be summarized through three typical variables: the the-

oretical maximal force at null velocity (F

;

0), the P

;

max the

lower limbs can produce over one extension, and the theo-

retical maximal velocity at which lower limbs can extend

during one extension under zero load (v

Y

0). These three pa-

rameters represent the maximal mechanical capabilities of

lower limbs to generate external force, power output, and

extension velocity, respectively. Because they characterize

the mechanical limits of the entire neuromuscular function,

they encompass individual muscle mechanical properties

(e.g., intrinsic F–vand length–tension relationships, rate of

force development), some morphological factors (e.g., cross-

sectional area, fascicle length, pennation angle, tendon prop-

erties), and neural mechanisms (e.g., motor unit recruitment,

firing frequency, motor unit synchronization, intermuscular

coordination) (9). Graphically, F

;

0and v

Y

0correspond to the

force axis and velocity axis intercepts of the linear F–vcurve,

respectively, and P

;

max corresponds to the apex of the para-

bolic P–vrelationship. Under these conditions, the relation-

ship among these three parameters can be described by the

following mathematical equation (41):

!

Pmax ¼

!

F0v

!

0

4½1$

Consequently, two athletes with similar P

;

max could theo-

retically present different F–vmechanical profiles, i.e., dif-

ferent combinations of F

;

0and v

Y

0. The issue is therefore to

determine whether the F–vprofile may influence ballistic

performances independently of P

;

max. In other words, is it

preferable to be ‘‘strong’’ or ‘‘fast’’ to reach the highest per-

formance in ballistic movements? Such an analysis might

provide greater insight into the relationship between me-

chanical properties of the neuromuscular system and func-

tional performance, either to further explore animal motor

behaviors (19,20) or to program athletic training in humans,

as underlined in recent reviews (10,12,14).

The effects of the F–vmechanical profile on ballistic per-

formance have been experimentally approached only through

studies led in athletes with different training backgrounds

(40,43), through different training protocols (5,7,11,16,27),

or both (4,8). However, in these studies, the various F–v

profiles of athletes were also associated with various P

;

max

values among subjects, making it impossible to identify

the sole effect of the F–vprofile. The influence of force

and velocity capabilities on jumping performance has been

recently addressed through a theoretical integrative approach

mathematically expressing the maximal jump height an indi-

vidual can reach as a function of F

;

0and v

Y

0(37). However,

the observed positive effects of F

;

0and v

Y

0on performance

were not independent from possible effects of P

;

max, the latter

being overlooked.

On the basis of this theoretical approach, the main aim

of this study was to determine the respective influences of

P

;

max and F–vprofile on performance in ballistic lower limb

movements. Moreover, force and velocity contributions to

power output depend on the load involved (10,14). Con-

sequently, the secondary aim of this study was to investigate

whether the effects of the F–vprofile on ballistic per-

formances (if any) depend on the afterloads (additional loads

and/or push-off orientation against gravity) involved in

the movement. To achieve these aims, the aforementioned

theoretical analysis was compared with experimental mea-

surements during jumping.

THEORETICAL BACKGROUND

This section is devoted to an analysis of ballistic per-

formance through maximal jumps at different push-off

angles. The entire lower limb neuromuscular system is

considered as a force generator characterized by an inverse

linear F–vrelationship and a given range of motion. The

maximal jumping performance can be well represented by

the maximal v

TO

(vTOmax)ofthebodyCM.Asdetailedinthe

recent theoretical integrative approach, jumping perfor-

mance can be expressed as a function of some mechanical

characteristics of lower limbs.Inthisapproachmentioned

above (see Samozino et al. [37]), vTOmax can be expressed

as follows:

vTOmax ¼hPO ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Y

F2

0

4Y

v2

0

þ2

hPO

ð

Y

F0jgsin>Þ

sj

Y

F0

2Y

v0

0

@1

A½2$

where gis the gravitational acceleration (9.81 mIs

j2

), >is

the push-off angle with respect to the horizontal (-), h

PO

is the distance covered by the CM during push-off cor-

responding to the extension range of lower limbs (m), and

F

;

0(NIkg

j1

of moving mass) and v

Y

0(mIs

j1

)arethemaxi-

mal force at theoretical null velocity and the theoretical

maximal unloaded velocity of lower limbs, respectively.

The push-off angle >,assumedtobethesameastheaxisof

the force developed, is considered constant over the entire

push-off. In equation 2, the afterload opposing in motion

is taken into account through inertia (i.e., the moving

mass present here in the normalization of F

;

0)andgravity

(g(sin >,i.e.,thecomponentofthegravityopposedtothe

movement).

The F–vmechanical profile of lower limbs can be repre-

sented by the ratio between F

;

0and v

Y

0, i.e., by the slope of

the linear F–vrelationship (S

Fv

) given by the following

equation:

SFv ¼j

Y

F0

Y

v0

½3$

(with the force graphically represented on the vertical axis of

the F–vrelationship).

Thus, the lower the S

Fv

, the steeper the F–vrelation-

ship and the higher the force capabilities compared with

velocity ones (7). Note that S

Fv

and P

;

max are theorized to be

independent.

http://www.acsm-msse.org314 Official Journal of the American College of Sports Medicine

APPLIED SCIENCES

Copyright © 2012 by the American College of Sports Medicine. Unauthorized reproduction of this article is prohibited.

Substituting equation 3 in equation 2 gives the following:

vTOmax ¼hPO ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

S2

Fv

4þ2

hPO

ð

Y

F0jgsin>Þ

sþSFv

2

0

@1

A½4$

On the other hand, from equations 1 and 3, F

;

0can be

expressed as a function of P

;

max and S

Fv

:

Y

F0¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

j4

Y

PmaxSFv

p½5$

Substituting equation 5 in equation 4 gives the following:

vTOmax ¼hPO ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

S2

Fv

4þ2

hPO

ð2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

j

Y

PmaxSFv

pjgsin>Þ

sþSFv

2

0

@1

A½6$

Consequently, vTOmax can also be expressed as a func-

tion of P

;

max,S

Fv

, and h

PO

. Equation 6 is true for

h

PO

90,

Y

Pmax 9gsin>2=j4SFv, and SFv Gjgsin>2=4

Y

Pmax

(see appendices, Supplemental Digital Content 1a,

http://links.lww.com/MSS/A114, for details on the com-

putations of these values). In the present study, equation 6

was (i) validated from experimental measurements and (ii)

simulated to analyze the respective influences of P

;

max and

S

Fv

on jumping performance.

METHODS USED IN THE EXPERIMENTAL

VALIDATION

Subjects and experimental protocol. Fourteen sub-

jects (age = 26.3 T4.5 yr, body mass = 83.9 T18.3 kg, stature

= 1.81 T0.07 m) gave their written informed consent to

participate in this study, which was approved by the local

ethical committee and in agreement with the Declaration of

Helsinki. All subjects practiced physical activities including

explosive efforts (e.g., basketball, rugby, soccer); eight of

them were rugby players (four played in the Italian first

league). After a 10-min warm-up and a brief familiarization

with the laboratory equipment, each subject performed two

series of maximal lower limb push-offs: (i) horizontal ex-

tensions with different resistive forces allowing us to deter-

mine F–vrelationships of the lower limbs and (ii) inclined

jumps used to compare experimental performances with

theoretical predictions.

Tests were realized on the Explosive Ergometer

(EXER, see Figure, Supplemental Digital Content 2,

http://links.lww.com/MSS/A115, for a schematic view of

the EXER) consisting of a metal frame supporting one rail

on which a seat, fixed on a carriage, was free to move

(for more details, see Rejc et al. [34]). The total moving

mass (seat + carriage) was 31.6 kg. The main frame could be

inclined up to a maximum angle of 30-with respect to the

horizontal. The subject could therefore accelerate himself or

herself and the carriage seat backward by pushing on two

force plates (LAUMAS PA 300; Parma, Italy) positioned

perpendicular to the rail, the output of which was independent

of the point of application of the force within a wide area. The

velocity of the carriage seat along the direction of motion was

continuously recorded by a wire tachometer (LIKA SGI,

Vicenza, Italy) mounted on the back of the main frame. Force

and velocity analog outputs were sampled at a frequency of

1000 Hz using a data acquisition system (MP100; BIOPAC

Systems, Inc., Goleta, CA). The instantaneous power was cal-

culated from the product of instantaneous force and velocity

values. Data were processed using the AcqKnowledge soft-

ware (BIOPAC Systems, Inc.). An electric motor, positioned

in front of the carriage seat, allowed us to impose known

braking forces, acting along the direction of motion. The mo-

tor, controlled by a personal computer, was linked to the seat

by a chain, its braking action initiating immediately at the

onset of the subject’s push. The braking force of the motor,

ranging from about 200 to 2300 N, was set using a custom-

built LabVIEW program (National Instruments, Austin, TX).

For each test, the subject was seated on the carriage seat,

secured by a safety belt tightened around the shoulders and

abdomen, with the arms on handlebars. The starting posi-

tion, set with feet on the force plates and knees flexed at 90-,

was fixed thanks to adjustable blocks positioned on the rail

of the EXER to prevent the downward movement of the

carriage seat and, in turn, any countermovement.

F–vrelationships of lower limb neuromuscular

system. To determine individual F–vrelationships, each

subject performed horizontal maximal lower limb extension

against seven randomized motor braking forces: 0%, 40%,

80%, 120%, 160%, 200%, and 240% of the subject’s body

weight. The condition without braking force (0% of body

weight) was performed with the motor chain disconnected

from the carriage seat. For each trial, subjects were asked

to extend their lower limbs as fast as possible. Two trials,

separated by 2 min of recovery, were completed at each

braking force. Mean force (F

Y

), velocity (v

Y), and power (P

Y

)

for the best trial of each condition were determined from the

averages of instantaneous values over the entire push-off

phase. The push-off began when the velocity signal in-

creased and ended when the force signal (if takeoff) or the

velocity signal (if no takeoff) fell to zero. As previously

suggested (3,33,43), F–vrelationships were determined by

least squares linear regressions. Because P–vrelationships

are derived from the product of force and velocity, they were

logically described by second-degree polynomial functions.

F–vcurves were extrapolated to obtain F

;

0(then normalized

to total moving mass, i.e., body + carriage seat mass) and v

Y

0,

which correspond to the intercepts of the F–vcurve with the

force and velocity axis, respectively. According to equation

3, S

Fv

was then computed from F

;

0and v

Y

0. Values of P

;

max

(normalized to body + carriage seat mass) were determined

from the first mathematical derivation of P–vregression

equations. Moreover, to test the validity of equation 1, P

;

max

was also computed from this equation (PmaxTH ).

Inclined push-off performance. To validate equa-

tion 6, each subject then performed two inclined maximal

push-offs at three sled angles (>) (10-, 20-, and 30-above

the horizontal) with the motor chain disconnected from

the carriage seat, following the same procedures described

above. v

TO

was determined for each trial as the instantaneous

OPTIMAL FORCE–VELOCITY PROFILE Medicine & Science in Sports & Exercise

d

315

APPLIED SCIENCES

Copyright © 2012 by the American College of Sports Medicine. Unauthorized reproduction of this article is prohibited.

velocity value when the force signal fell to zero. Push-off

distance (h

PO

) was determined for each subject by integrating

the velocity signal over time during the push-off phase.

Statistical analyses. All data are presented as mean T

SD. For each subject and each sled angle condition, the

highest v

TO

reached in the two trials was compared with

vTOmax computed according to equation 6, from P

;

max,h

PO

,

and S

Fv

.Aftercheckingdistributionsnormalitywiththe

Shapiro–Wilk test, the difference between v

TO

and vTOmax

(bias) was computed and tested using a t-test for paired

samples. To complete this comparison, the absolute dif-

ference between v

TO

and vTOmax (absolute bias) was also

calculated as jðvTOmax jvTOÞvj1

TOj100 (36). Using the same

comparison method, experimental values of P

;

max were

compared with theoretical values (PmaxTH). After checking

the homogeneity of variances, the effect of sled angle was

tested with a one-way ANOVA for repeated measures on

v

TO

and vTOmax .Whenasignificanteffectwasdetected,a

post hoc Newman–Keuls comparison was used to locate

the significant differences. For all statistical analyses, a

Pvalue of 0.05 was accepted as the level of significance.

METHODS USED IN THE SIMULATION STUDY

The relative influences of P

;

max and S

Fv

on vTOmax were

analyzed via equation 6. First, vTOmax changes with S

Fv

were

determined for different P

;

max values at different push-off

angles (>). The range of P

;

max and S

Fv

values used in the

simulations was obtained from data (P

;

max,F

;

0,v

Y

0) previously

reported for human maximal lower limb extensions: P

;

max

from 10 to 40 WIkg

j1

and S

Fv

until to j40 NIsIm

j1

Ikg

j1

(32,33,36,43). The effect of h

PO

on performance, previously

studied and discussed (see Samozino et al. [37]), was not

specifically treated here; h

PO

was set at 0.4 m, which is a

typical value for humans. Then, sensitivity analyses were

performed to assess the respective weight of each variable

plotting relative variations in vTOmax against relative variations

in P

;

max and S

Fv

at different push-off angles (>), each variable

being studied separately.

RESULTS

Validation of the theoretical approach. Individual

F–vand P–vrelationships were well fitted by linear

(r

2

= 0.75–0.99, Pe0.012) and second-degree polynomial

(r

2

= 0.70–1.00, Pe0.024) regressions, respectively. Figure 1

shows these relationships for two typical subjects with dif-

ferent F–vprofiles (i.e., different F

;

0,v

Y

0, and S

Fv

) and dif-

ferent P

;

max capabilities. Mean TSD values of h

PO

,v

Y

0,F

;

0,

P

;

max, and S

Fv

were 0.39 T0.04 m, 2.78 T0.63 mIs

j1

, 24.2 T

2.97 NIkg

j1

(or 17.3 T1.60 NIkg

j1

when normalized to

body + carriage seat mass), 16.34 T2.26 WIkg

j1

(or 11.78 T

1.80 WIkg

j1

when normalized to body + carriage seat

mass), and j9.33 T3.31 NIsIm

j1

Ikg

j1

(or j6.64 T

2.12 NIsIm

j1

Ikg

j1

when normalized to body + carriage seat

mass), respectively. The difference between P

;

max and PmaxTH

was not significant and very low (absolute bias = 1.81% T

0.76%), which shows the validity of equation 1. Mean TSD

values of v

TO

and vTOmax , as well as mean values of absolute

bias, are presented in Table 1. For each push-off angle, v

TO

and vTOmax were not significantly different, and bias was

j0.05 T0.17 mIs

j1

(see Figure, Supplemental Digital

Content 3, http://links.lww.com/MSS/A116, which shows

bias and limits of agreement in a Bland–Altman plot). On

the other hand, the effect of push-off angle was significant

on both v

TO

and vTOmax , with differences between every

condition (Table 1).

Theoretical simulations. As expected, P

;

max positively

affects vTOmax , which is clearly shown in Figure 2 for both

vertical (>= 90-) and horizontal (>=0-) push-offs. The

main original result was the curvilinear changes in vTOmax

with S

Fv

for a given P

;

max (Fig. 2). Such variations highlight

the existence of an optimal S

Fv

(SFvopt ) maximizing vTOmax for

given P

;

max and h

PO

. Moreover, SFvopt values seem to change

slightly as a function of both P

;

max and >values, ranging

FIGURE 1—Typical F–v(left panel) and P–v(right panel) relationships for two subjects with different F–vprofiles (S

Fv

=jF

–

0

/v

–

0

) and P

–

max

values

(gray cross). Subject 1 (open circles) presents a lower P

–

max

and an F–vprofile more oriented toward force capabilities than subject 2 ( filled circles), who

presents an F–vprofile more oriented toward velocity capabilities.

http://www.acsm-msse.org316 Official Journal of the American College of Sports Medicine

APPLIED SCIENCES

from j18 to j6NIsIm

j1

Ikg

j1

for the conditions simulated

in Figure 2. The dependence of SFvopt on P

;

max,>, and h

PO

can be mathematically analyzed: the expression of SFvopt as a

function of these three variables is a real solution canceling

out the first mathematical derivative of vTOmax with respect

to S

Fv

(see appendices, Supplemental Digital Content 1b,

http://links.lww.com/MSS/A114, for detailed computations

of SFvopt ). Whatever the value of P

;

max,SFvopt decreases when

>increases (Fig. 3). For both vertical and horizontal push-

offs, the sensitivity analysis showed that vTOmax is more in-

fluenced by P

;

max than by S

Fv

, at least when the S

Fv

reference

value is equal to SFvopt (Fig. 4). Moreover, the respective

effects of P

;

max and S

Fv

on vTOmax seem to decrease with de-

creasing >(Fig. 4).

DISCUSSION

The original and main findings of this study are that bal-

listic performance of the lower limbs depends on both P

;

max

capabilities and the F–vprofile, with the existence of an

individual optimal F–vprofile corresponding to the best

balance between force and velocity capabilities. This opti-

mal F–vprofile, which can be accurately determined,

depends on some individual characteristics (limb extension

range, P

;

max) and on the afterload involved in the movement

(inertia, inclination). The concept of optimal F–vprofile and

the proposed approach make it possible to clarify some

scientific issues previously discussed about the mechanical

capabilities of lower limbs that determine ballistic perfor-

mance and about the relationships between lower limb

neuromuscular system structure and function. The following

discussion is devoted to detailing these different points.

Validity of the theoretical approach. These findings

were obtained using a theoretical integrative approach based

on fundamental principles of dynamics and on the F–vlinear

model characterizing the dynamic mechanical capabilities of

the neuromuscular system during a lower limb extension.

This linear model, as well as the parabolic P–vrelationship,

has been well supported and experimentally described for

FIGURE 2—Changes in maximal CM v

TO

(v

TOmax

) reached at the end of a lower limb push-off, as a function of the changes in the F–vprofile (S

Fv

) for

different P

–

max

values and at two push-off angles (>). The h

PO

is fixed here at 0.4 m. For the vertical push-off (>= 90-), the corresponding jump height

(obtained from basic ballistic equations) is presented on the additional yaxis. Open circles represent the v

TOmax

reached for an optimal F–v profile

(S

Fv

opt

).

FIGURE 3—Changes in optimal F–v profile (S

Fvopt

) as a function of

the push-off angle (>) for different P

–

max

values. The h

PO

is fixed here at

0.4 m.

TABLE 1. Mean TSD of v

TO

obtained with experimental and theoretical approaches,

absolute bias between these two approaches, and t-test comparison results.

>(-)

Experimental Values

(v

TO

(mIs

j1

))

Theoretical Values

(v

TO

max

(mIs

j1

)) t-Test

Absolute

Bias (%)

10 2.45 T0.22 2.43 T0.18 ns 4.40 T4.94

20 2.32 T0.25

a

2.25 T0.16

a

ns 6.56 T5.46

30 2.14 T0.23

ab

2.07 T0.15

ab

ns 5.73 T3.89

a

Significantly different from >= 10-.

b

Significantly different from >= 20-.

ns, nonsignificative difference between experimental and theoretical values.

OPTIMAL FORCE–VELOCITY PROFILE Medicine & Science in Sports & Exercise

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317

APPLIED SCIENCES

multijoint movements (3,33,43,44). The linearity of the F–v

relationship, usually presented as hyperbolic for isolated

muscles (17), is explained by the integrative feature of the

model. The force generator and, in turn, its maximal force

(F

;

0), unloaded velocity (v

Y

0), and power (P

;

max) refer here to

the entire in vivo neuromuscular system involving several

muscles with different mixed fiber composition, architec-

tural characteristics, anatomical joint configuration, level

of neural activation, and specific coordination strategies

(7–9,44). The limits of this theoretical approach have been

previously discussed (37), but the significance and accu-

racy of its predictions have not been quantified yet. Besides

validating equation 1 (P

;

max and PmaxTH are very close), the

present results showed no differences between predicted

(vTOmax)andmeasured(v

TO

) values, associated to a low

absolute bias from 4% to 6.6%. This is within the range of

reproducibility indices previously reported for different

variables (performance, velocity, force, or power) mea-

sured during lower limb maximal extensions (3,18). These

results support the validity of the proposed theoretical ap-

proach, which was strengthened by the sensibility of both

predicted and experimental values to changes in push-off

angles. Obviously, the accuracy of equation 6 is enhanced

when muscular properties (P

;

max and S

Fv

)areassessedin

the same conditions (e.g., joints and muscle groups in-

volved, range of motion) under which the actual perfor-

mance is studied, as it was done here on the EXER.

Muscular capabilities determining jumping per-

formance. Among the muscular characteristics determin-

ing jumping performance, P

;

max has the greatest weight.

Although expected, the importance of P

;

max in setting bal-

listic performance needed to be established, as concluded by

Cronin and Sleivert (12) in their recent review: ‘‘power is

only one aspect that affects performance and it is quite likely

that other strength measures may be equally if not more

important for determining the success of certain tasks.’’ The

present results clearly demonstrate this idea. On the other

hand, the dependence of ballistic performances on muscular

power capability brings new insights into the recurrent de-

bate about the role of ‘‘power’’ in impulsive performance,

such as jumping (22,26,42). On the basis of Newton’s sec-

ond law of motion, some authors stated that jumping per-

formance does not depend on the muscular capability to

develop power but rather on the capability to develop a high

impulse (26,42). Even if fundamental principles of dynamics

directly relate mechanical impulse to v

TO

(and in turn jump-

ing performance), the capability to generate impulse does not

represent an intrinsic mechanical property of the lower limb

neuromuscular system, contrary to P

;

max. It is important to

differentiate mechanical outputs (e.g., external force, move-

ment velocity, power output, impulse, mechanical work) from

mechanical capabilities of lower limbs (P

;

max,v

Y

0,F

;

0). On

the one hand, mechanical outputs represent the mechanical

entities that can be externally measured during a movement

and are often used to characterize movement dynamics from

a mechanical point of view. On the other hand, mechanical

capabilities of lower limbs characterize the mechanical lim-

its of the neuromuscular function and refer to the theoretical

maximal values of some mechanical outputs that could be

reached by an individual. The proposed theoretical approach

demonstrates that the ability to develop a high impulse

against the ground and, in turn, the ability to reach maximal

CM velocity at the end of a push-off are highly related to

the P

;

max the lower limbs can produce (over a given exten-

sion range).

That said, the present results show that P

;

max is not the

only muscular property involved in jumping performance.

Indeed, two individuals with the same P

;

max (and the same

h

PO

) may achieve different performances, be it during a

vertical jump or a horizontal push-off (Fig. 2). These dif-

ferences are due to their respective F–vprofiles (S

Fv

), i.e., to

their respective ratios between maximal force (F

;

0) and ve-

locity (v

Y

0) capabilities. For each individual (given his/her

P

;

max and h

PO

), there is an optimal F–vprofile that max-

imizes performance. The more this F–vprofile differs from

the optimal one, the lower the performance in comparison

FIGURE 4—Sensitivity analyses: relative changes in maximal CM v

TO

(v

TOmax

) as a function of the relative variations of P

–

max

and F–vprofile (S

Fv

) for

two >values. The reference value for P

–

max

is 25 WIkg

j1

and corresponds to the optimal F–vprofile value for S

Fv

(j14.0 for >=90-and j8.20 for >=0-).

The h

PO

is fixed here at 0.4 m. For S

Fv

, the higher the normalized variation, the lower the value because S

Fv

values are only negative and the more

the F–vprofile tends toward force capabilities.

http://www.acsm-msse.org318 Official Journal of the American College of Sports Medicine

APPLIED SCIENCES

with the one that could be reached with the same power

capabilities (Fig. 2). The values of S

Fv

observed here (from

j16.8 to j4.9 NIsIm

j1

Ikg

j1

) are consistent with F

;

0and

v

Y

0values previously reported (3,32,33,43). P

;

max and F

;

0

values were slightly lower than those reported during vertical

push-offs (3,32,33), which is likely due to the specific sitting

position imposed by the EXER compared with the totally

extended hip configuration usually tested. Individuals, nota-

bly rugby players, as most of our subjects were, may present

very different F–vprofiles, as shown by coefficients of vari-

ation for S

Fv

of beyond 30% compared with coefficients of

variation below 20% for P

;

max or F

;

0. Most of these different

individual F–vprofiles differ from the optimal ones, thus

characterizing unfavorable balances between force and ve-

locity capabilities. Indeed, individual F–vprofiles observed in

this study ranged from 36% to 104% of the optimal ones

maximizing vertical jumping performance. Simulations of

equation 6 showed that such unfavorable F–vbalances may

be related to differences up to 30% in jump height between

two individuals with similar power capabilities (Figs. 2 and

4). Consequently, we think that the F–vprofile represents a

muscular quality that has to be considered attentively not only

by scientists working on muscle function during maximal

efforts but also by coaches for training purposes.

Effect of afterloads on optimal F–vprofile. The

optimal F–vprofile depends on some individual char-

acteristics (h

PO

,P

;

max) and on the afterload opposing in

motion (inertia, inclination). On the one hand, the F–vpro-

file does affect jumping performances when S

Fv

is expressed

through values normalized to the total moving mass

(NIsIm

j1

Ikg

j1

), which may be body mass, body mass plus

additional loads, or projectile mass. Thus, the interpretation

of F–vprofiles is dependent on the movement considered.

On the other hand, the computation of the optimal F–v

profile also takes account of the total moving mass: SFvopt is a

function of P

;

max, itself expressed relative to moving mass.

Consequently, for a given athlete, the optimal F–vprofile is

not the same for a javelin throw (high P

;

max relative to

moving mass) and for a shot put (low relative P

;

max, see the

different curves in Fig. 3). The optimal F–vprofile also

depends on the push-off angle and more generally on the

magnitude of the gravity component opposing motion (the

lower the push-off angle, the more the optimal F–vprofile is

oriented toward velocity capabilities). Thus, the optimal F–v

profile is not the same when seeking to maximize perfor-

mance during the first push of a sprint or during a vertical

jump; velocity capabilities are more important in the former

case; force capabilities, in the latter. This is in line with the

theoretical framework proposed by Minetti (28) showing

that power output developed during maximal efforts is less

dependent on muscle strength when the exercise does not

involve gravity, as in horizontal extensions. Such horizontal

(or very horizontally inclined) push-offs are thus especially

limited by the velocity capabilities of lower limbs. The

originality of the present theoretical approach is to allow

the accurate determination of the optimal balance between

force and velocity capabilities (through SFvopt ) according to

movement specificities. The subjects tested here presented

an overall unfavorable balance toward velocity capabilities

for vertical jumps (S

Fv

from 36% to 104% of their respective

SFvopt ) and toward force capabilities for horizontal push-offs

(S

Fv

from 66% to 227% of SFvopt ).

F–vprofile and athletic training. Assessing F–v

profiles when seeking to identify the optimal balance be-

tween force and velocity capabilities may be of interest to set

training loads and regimens, as previously proposed using

power–load relationships (10,20,27,38). Values of S

Fv

allow

comparisons among athletes independently from their power

capabilities (which is not possible from only F

;

0and v

Y

0

values) and, thus, to know whether an athlete, as compared

with another one, is characterized by a ‘‘force’’ or a

‘‘velocity’’ profile (Fig. 1). To the best of our knowledge,

only Bosco (2) proposed an index to compare athletes’ F–v

profiles dividing jump height reached with an additional

load (100% of body mass) by unloaded jump height: the

higher this index, the higher the force capabilities compared

with the velocity ones. However, Bosco’s index does not

allow the orientation of training loads for a given athlete

according to his/her own strengths and weaknesses and to

movement specificities. Therefore, we propose the individ-

ual value of S

Fv

, expressed relatively to SFvopt , as a good and

practical index to characterize the F–vprofile and to design

appropriate training programs. The present results showed

that improving ballistic performance may be achieved

through increasing power capabilities (i.e., shifting F–v

relationships upward and/or to the right [21]) and moving

the F–vprofile as close to the optimal one as possible. Such

changes in the F–vrelationship, notably in its slope, may be

achieved by specific strength training (7,8,21). An athlete

presenting an unfavorable F–vbalance in favor of force

(relatively to his/her optimal profile corresponding to target

movement specificities) should improve his/her velocity

capabilities as a priority by training with maximal efforts

and light (e.g., G30% of one repetition maximum, the latter

being close to F

;

0) or negative loading, which is often called

‘‘ballistic’’ or ‘‘power’’ training (7,8,11,25,27). On the con-

trary, an athlete with an imbalanced F–vprofile oriented

toward velocity should follow a strength training with heavy

loads (975%–80% of one repetition maximum) to increase

his/her force capabilities as a priority (7,8,27). In both cases,

it is likely that (i) P

;

max will increase and (ii) the F–vprofile

will be optimized (i.e., change toward the optimal one),

partly or totally correcting unfavorable F–vbalances. As

shown in the present study, these two changes would

both result in a higher performance. The mechanisms un-

derlying these changes in F–vrelationships, specific to the

kind of training, include changes in mixed fiber composi-

tion, muscle architecture (hypertrophy, pennation angle),

and neural activation (voluntary activation level, firing

frequency, rate of EMG rise, intermuscular coordina-

tion strategies) (1,7,15,27). These theoretical findings

support previous experimental results about the velocity (or

OPTIMAL FORCE–VELOCITY PROFILE Medicine & Science in Sports & Exercise

d

319

APPLIED SCIENCES

load)-specific changes in performance after training with

light or heavy loads (10,25,27), with the additional origi-

nality of controlling the respective effects of F–vqualities

and P

;

max capabilities.

F–vprofile and optimal load. The proposed approach

brings new insight into the understanding of the relation-

ships between structure and mechanical function of the

lower limb neuromuscular system and, notably, the effect of

specific changes in the F–vrelationship on athletic perfor-

mance. The concept of the F–vprofile could be related to the

maximum dynamic output hypothesis proposed and dis-

cussed by Jaric and Markovic (20) and supported by recent

studies (6,13,30). Their hypothesis states that the optimal

load-maximizing power output in ballistic movements for

physically active individuals corresponds to their own body

weight and inertia (20). They argued that this optimal load

would be related to the particular design of the muscular

system (notably its mechanical properties), itself influenced

by the actual load individuals regularly overcome during

their daily activities. They pointed out, however, that the

different evidences provided needed to be supported by

theoretical frameworks describing the general aspects of the

neuromuscular system’s ability to provide the P

;

max output

against a particular load. This may be done using the theo-

retical approach proposed here. Indeed, the slope of the F–v

relationship and, thus, the ratio between F

;

0and v

Y

0are di-

rectly related to the optimal velocity and force-maximizing

power output and so to the corresponding optimal load.

From F–vand P–vrelationships (Fig. 1), the higher the v

Y

0,

the higher the optimal velocity and the lower the optimal

load. Conversely, high F

;

0values are associated with high

optimal loads. Consequently, the optimal load corresponds

to the mass and inertia of the body only for individuals de-

veloping their individual P

;

max during an unloaded vertical

jump. Because (i) maximal jump height changes with F–v

profile for a given P

;

max (Fig. 2, left panel) and (ii) jump

height and power output (relative to body mass) developed

during a vertical jump are positively related (36), the power

output developed during an unloaded maximal jump de-

pends on the F–vprofile. Consequently, jumping perfor-

mance depends directly on the mean power output developed

during push-off (for a given h

PO

), and the latter can be

maximized by both maximizing P

;

max and optimizing the

F–vprofile. This is illustrated in Figure 5, which shows the

power output developed during a vertical jump (expressed

relatively to P

;

max) according to the F–vprofile expressed

relatively to the optimal one (power output was computed

from equations 6 and 9 of Samozino et al. (36), see appen-

dices for more details, Supplemental Digital Content 1c,

http://links.lww.com/MSS/A114). An optimal F–vprofile,

i.e., an optimal balance between F

;

0and v

Y

0, allows the de-

velopment of P

;

max during an unloaded jump (Fig. 5, left

panel) and thus maximization of jumping performance

(Fig. 5, right panel). Consequently, the body mass represents

the optimal load for individuals with optimal F–vprofiles.

An athlete with an unfavorable F–vbalance develops a

power output lower than P

;

max during an unloaded jump.

Such an athlete would produce P

;

max against a load lower

than body mass if he/she presents a velocity profile and

higher than body mass in the case of a force profile.

The present theoretical framework may help to explain and

understand the possible interindividual differences in opti-

mal load previously observed, discussed, and debated

(6,20,24,30,31,39). The influence of training history re-

cently proposed supports our findings because training

background specificities directly affect the F–vprofile

(8,10,27), which influences the optimal load (39). This is in

line with the maximum dynamic output hypothesis stating

that strength-trained athletes (with high force capabilities)

present optimal loads higher than their body mass (20,39).

In animals and humans, the lower limbs’ neuromuscular

system is likely designed to work optimally against loads

usually supported and mobilized (20,23). Consequently,

animals would naturally present F–vprofiles optimizing

ballistic performance such as horizontal jumps, when these

latter represent their main survival behavior.

FIGURE 5—Left panel: changes in power output developed during a vertical jump (expressed in %P

–

max

) with changes in F–vprofile (S

Fv

, expressed in

%S

Fvopt

). Right panel: effect of the power output developed during a vertical jump (expressed in %P

–

max

) on the jump height reached (expressed

relatively to the jump height that could be reached, should the F–vprofile be optimal). Values of P

–

max

and h

PO

were fixed here at 25 WIkg

j1

and 0.4 m,

respectively.

http://www.acsm-msse.org320 Official Journal of the American College of Sports Medicine

APPLIED SCIENCES

CONCLUSIONS

Ballistic performance is mostly determined not only

by the P

;

max lower limbs can generate but also by the F–v

mechanical profile characterizing the ratio between maximal

force capabilities and maximal unloaded extension velocity.

This F–vprofile of lower limbs, independent from power

capabilities, may be optimized to maximize performance.

Altius is neither citius nor fortius but an optimal balance

between the two. This optimal F–vprofile depends on in-

dividual and movement specificities, notably on the after-

load involved (inertia and gravity): the lower the afterload,

the more the optimal F–vprofile will be oriented toward

velocity capabilities. Considering F–vprofile may help better

understand the relationships between neuromuscular system

mechanical properties and functional performance, notably

to optimize sport performance and training. This original me-

chanical quality was put forward by a theoretical integrative

approach and validated here from comparisons between

theoretically predicted performances and experimental mea-

surements during jumping. This approach was discussed here

for lower limb extensions, but the results may be also ap-

plied to other multijoint muscular efforts, such as upper limb

ballistic movements, or more complex movements such as

sprint running.

The authors thank Alberto Botter (Udine Rugby Football Club) for

his help in recruitment of the subjects tested and the subjects for

their ‘‘explosive’’ implication in the protocol.

No funding for this study from the National Institutes of Health,

Wellcome Trust, Howard Hughes Medical Institute, or others was

received.

The authors declare that they have no conflict of interest.

The results of the present study do not constitute endorsement by

the American College of Sports Medicine.

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NOTATION

CM body center of mass

mbody mass or moving mass (kg)

ggravitational acceleration on Earth (9.81 mIs

j2

)

F

Y

mean external force developed over push-off along the push-off axis (relative to moving mass (NIkg

j1

))

v

Ymean CM velocity over push-off along the push-off axis (mIs

j1

)

P

Y

mean power output developed over push-off (relative to moving mass (WIkg

j1

))

F

;

0theoretical maximal value of F

Y

that lower limbs can produce during one extension at a theoretical null v

Y(relative to moving mass (NIkg

j1

))

v

Y

0theoretical maximal value of v

Yat which lower limbs can extend during one extension under the influence of muscle action in a theoretical unloaded condition (mIs

j1

)

P

;

max maximal P

Y

that lower limbs can produce during a push-off (WIkg

j1

)

PmaxTH theoretical value of P

;

max estimated from equation 1 (WIkg

j1

)

h

PO

push-off distance determined by lower limb extension range (m)

v

TO

CM velocity at takeoff (mIs

j1

)

vTOmax maximal v

TO

an individual can reach (mIs

j1

)

>push-off angle with respect to the horizontal (-)

S

Fv

slope of linear F–vrelationship (NIsIm

j1

Ikg

j1

)

SFvopt optimal value of S

Fv

maximizing vTOmax for given values of P

;

max and h

PO

(NIsIm

j1

Ikg

j1

)

http://www.acsm-msse.org322 Official Journal of the American College of Sports Medicine

APPLIED SCIENCES

S

UPPLEMENTAL

D

IGITAL

C

ONTENT

1

APPENDICES

(a) variables values for which the presented equations are true

The variables presented in the equations (

0

F

,

0

v

,

max

P

,

Fv

S

and

PO

h

) have to be consistent

with push-off dynamics.

• Being a distance,

PO

h

has to be a real positive value:

0

PO

h

>

[A1]

• In order to takeoff, the mean vertical force developed during push-off (in N) has to be

higher than the body weight component along the axis of movement direction. Hence,

when expressed relative to body mass (in N.kg

-1

), the mean vertical force, and in turn

0

F

, have to be a real positive value higher than the gravitational acceleration

component along the axis of movement direction:

0

sin

α

>Fg [A2]

• In the same manner, in order to takeoff, the mean vertical velocity of the CM during

push-off has to be a positive value. Hence,

0

0

v

>

[A3]

• From equations [1] and [3],

0

v

can be expressed as a function of

max

P

and

Fv

S

:

max

0

2

Fv

P

v

S

=− [A4]

with

0

Fv

S

<

, since

0

0

>

F and

0

0

v

>

.

Since

0

sin

α

>Fg , and according to equation [1]:

0

max

sin

4

α

>vg

P

[A5]

Substituting equation [A4] in equation [A5] gives:

max

max

sin

2

α

>−

Fv

P

g

P

S

[A6]

then,

max

(sin)²

4

α

>−

Fv

g

PS [A7]

•

Since

0

sin

α

>Fg , and according to equation [3]:

0

sin

α

−>

Fv

g

Sv [A8]

Substituting equation [A4] in equation [A8], and after reduction, gives:

max

(sin)²

4

α

<−

Fv

g

SP [A9]

(b) mathematical expression of

Fv

Sopt

as a function of

max

P

and

PO

h

The optimal slope of F-v relationship (

Fv

Sopt

) is the

Fv

S

value maximizing

max

TO

v. The

mathematical expression of

Fv

Sopt

as a function of

max

P

and

PO

h

is a real solution of:

max

0

=

TO

Fv

d v

d S [A10]

The first mathematical derivative of

max max

(,,)

TO Fv PO

vPSh

with respect to

Fv

S

is:

max

max 2

max

max

2

max

4

.

²11 2 1

(2 )

24 2

12

4(2)

4

−

−

=++−−+

+−−

Fv

PO Fv

TO PO

Fv Fv Fv

Fv PO

Fv Fv

PO

P

S

hPS

d v h

SPSgS

d S g h

SPSg

h

[A11]

Equation [A10] has four solutions, of which only one corresponds to real values of

Fv

S

among values for which equation [6] is true:

max

max

,

,

44 32

2

max

2 2

max max max

()

()

(( ) 12 )

33 3

PO

PO

PO PO

Fv

PO PO

P

P

h

h

ggPZ

g

Sopt

PPZ P

hh

hh

+

−−

=− − [A12]

with

(

)

max

1/3

66 352 44 396 88

max max max max

,

()

() 18 54 632 27=− − − + +

PO

PO PO PO PO PO

P

h

ZggPPgPP

hh h h h

[A13]

(c) power output developed during a vertical jump as a function of

Fv

S

According to basic ballistic principles, the height reached during a vertical jump (h in m) can

be expressed as a function of the CM vertical take-off velocity (

TO

v

):

2

²

=

TO

v

h

g

[A14]

From equations [6] and [A14], and substituting

TO

v

by

max

TO

v, the maximal jump height an

individual can reach can be expressed as a function of

max

P

,

Fv

S

and

PO

h

.

On the other hand, the mean power output (

P

in W.kg

-1

) developed during a vertical jump

was expressed as a function of

h

and push-off distance (

PO

h

in m) (Samozino et al., 2008):

2

1()=+

PO

gh

h

Pg

h

[A15]

Consequently, from equations [6], [A14] and [A15],

P

developed during the push-off of a

maximal vertical jump can be expressed as a function of

max

P

,

Fv

S

and

PO

h

. This allows to

analyze changes in

P

according to changes in

Fv

S

(for a given

max

P

and

PO

h

, Fig. 5).

SUPPLEMENTAL DIGITAL CONTENT 2

Schematic view of the Explosive Ergometer (EXER). WT wire tachometer, CS carriage seat,

FP force platforms, M electric motor. Rail system (R) and lower frame (LF) are hinged (Hi)

so that R can be tilted upward (see text for further details). To allow the motor to act on the

seat without delays, a constant tension (We = 196 N) is applied to the steel chain (dotted line).

S

UPPLEMENTAL

D

IGITAL

C

ONTENT

3

Averaged v

TO

by the two approaches (m.s

-1

)

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Differences in v

TO

between the two approaches (m.s

-1

)

-0.4

-0.2

0.0

0.2

0.4

Bias

Bias + 1.96 SD

Bias - 1.96 SD

Bland and Altman plot of differences between values of centre of mass take-off velocity (

TO

v

)

predicted from the proposed theoretical approach and those measured during the experimental

tests for push-off angles of 10° (black filled circles), 20° (grey filled circles) and 30° (open

circles). The solid horizontal line corresponds to the bias (mean differences). Upper and lower

horizontal dotted lines represent the limits of agreement (bias ± 1.96 SD of the differences).