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A simple method for measuring power, force, velocity properties,
and mechanical effectiveness in sprint running
P. Samozino1, G. Rabita2, S. Dorel3, J. Slawinski4,N.Peyrot
5, E. Saez de Villarreal6, J.-B. Morin7
1Laboratory of Exercise Physiology (EA4338), University Savoie Mont Blanc, Le Bourget du Lac, France, 2Research Department,
Laboratory Sport, Expertise and Performance, French Institute of Sport (INSEP), Paris, France, 3Laboratory “Motricité, Interactions,
Performance” (EA 4334), University of Nantes, Nantes, France, 4CeSERM – EA 2931, UFRSTAPS, Université de Paris Ouest
Nanterre la Défense, Paris, France, 5Laboratory IRISSE (EA4075), University of La Réunion, Le Tampon, La Réunion, France,
6Laboratory of Human Performance, Department of Sports, University Pablo de Olavide, Seville, Spain, 7Laboratory of Human
Motricity, Education Sport and Health (EA6312), University of Nice Sophia Antipolis, Nice, France
Corresponding author: Pierre Samozino, PhD, Laboratoire de Physiologie de l’Exercice, Université de Savoie Mont-Blanc, UFR
CISM – Technolac, 73376 Le Bourget du Lac, France. Tel: +33 4 79 75 81 77, Fax: +33 4 79 75 81 48, E-mail:
pierre.samozino@univ-savoie.fr
Accepted for publication 7 April 2015
This study aimed to validate a simple field method for
determining force– and power–velocity relationships and
mechanical effectiveness of force application during
sprint running. The proposed method, based on an
inverse dynamic approach applied to the body center of
mass, estimates the step-averaged ground reaction forces
in runner’s sagittal plane of motion during overground
sprint acceleration from only anthropometric and spatio-
temporal data. Force– and power–velocity relationships,
the associated variables, and mechanical effectiveness
were determined (a) on nine sprinters using both the
proposed method and force plate measurements and (b)
on six other sprinters using the proposed method during
several consecutive trials to assess the inter-trial reliabil-
ity. The low bias (<5%) and narrow limits of agreement
between both methods for maximal horizontal force
(638 ±84 N), velocity (10.5 ±0.74 m/s), and power output
(1680 ±280 W); for the slope of the force–velocity rela-
tionships; and for the mechanical effectiveness of force
application showed high concurrent validity of the pro-
posed method. The low standard errors of measurements
between trials (<5%) highlighted the high reliability of
the method. These findings support the validity of the
proposed simple method, convenient for field use, to
determine power, force, velocity properties, and mechani-
cal effectiveness in sprint running.
Sprint running is a key factor of performance in many
sport activities, not only to reach the highest top velocity
but also, and most importantly, to cover a given distance
in the shortest time possible, be it in track-and-field
events or in team sports. This ability implies large
forward acceleration, which has been related to the
capacity to produce and apply high amounts of power
output in the horizontal direction onto the ground, i.e.,
high amounts of horizontal external force at various
velocities over sprint acceleration (Jaskolska et al.,
1999b; Morin et al., 2011a, 2012; Rabita et al., 2015).
The overall mechanical capability to produce horizon-
tal external force during sprint running is well described
by the inverse linear force–velocity (F–v) and the para-
bolic power–velocity (P–v) relationships (Jaskolska
et al., 1999a, b; Morin et al., 2010, 2011a, 2012; Rabita
et al., 2015). Indeed, although the F–v relationships
obtained on isolated muscles or mono-articular move-
ments are described by a hyperbolic equation (Hill,
1938; Thorstensson et al., 1976), linear relationships
were consistently obtained for multijoint lower limb
movements such as pedaling, squat, leg press, or sprint
running movements (e.g., Yamauchi & Ishii, 2007; Dorel
et al., 2010; Morin et al., 2010; Bobbert, 2012;
Samozino et al., 2012; Rabita et al., 2015). These rela-
tionships characterize the external mechanical limits of
the entire neuromuscular system during specific
multijoint movements and are well summarized through
the theoretical maximal force (F0) and velocity (v0) this
system can develop, and the associated maximal power
output (Pmax). Moreover, the slope of the F–v relationship
determines the F–v mechanical profile (SFV), i.e., the
individual ratio between force and velocity qualities.
These mechanical properties obtained from multijoint
F–v and P–v relationships are a complex integration of
different mechanisms involved in the total external force
produced during one (for acyclic movements) or several
consecutive (for cyclic movements) limb extensions.
They encompass individual muscle mechanical proper-
ties, morphological factors, neural mechanisms, and
segmental dynamics (Cormie et al., 2010a, b, 2011;
Bobbert, 2012). Furthermore, because sprint running is a
dynamic movement mainly requiring force production
in two dimensions (in contrast to squat or leg press
Scand J Med Sci Sports 2015: ••: ••–••
doi: 10.1111/sms.12490
© 2015 John Wiley & Sons A/S.
Published by John Wiley & Sons Ltd
1
exercises), the F–v and P–v relationships during running
propulsion also integrate the ability to apply the external
force effectively (i.e., horizontally in the antero-posterior
direction) onto the ground (Morin et al., 2011a, b, 2012;
Rabita et al., 2015). This concept of mechanical effec-
tiveness of force application has been previously pro-
posed in sprint pedaling considering tangential and
normal force components of the force applied onto the
pedals (Dorel et al., 2010). The technical ability of force
application (or mechanical effectiveness) during sprint
running has been quantified at each step by the ratio (RF)
of the net horizontal and resultant ground reaction forces
(GRF), and over the entire acceleration phase by the rate
(DRF) of linear decrease in RF as velocity increases. DRF,
which is independent from the amount of total force
applied (i.e., physical capabilities), describes the run-
ner’s ability to maintain a forward horizontal orientation
of the resultant GRF vector despite increasing speed
(Morin et al., 2011a, b; Morin et al., 2012; Rabita et al.,
2015; see Methods section). So, F–v and P–v relation-
ships provide a macroscopic and integrative view of the
P–F–v mechanical profile of an athlete in the specific
sprint running task.
Therefore, determining individual F–v and P–v rela-
tionships and mechanical effectiveness during sprint
propulsion is of great interest for coaches, sport practi-
tioners, or physiotherapists. Indeed, sprint performance
is highly correlated to Pmax, be it quantified during sprint
running or other movements such as vertical jump or
sprint cycling (e.g., Cronin & Sleivert, 2005; Morin
et al., 2012; Rabita et al., 2015). In addition to this power
output capability, the F–v mechanical profile (character-
ized by the slope of the F–v relationship) has recently
been shown to influence maximal jumping perfor-
mances, independently from the effect of Pmax, with the
existence of an individual optimal F–v profile character-
izing the best balance, for a given subject, between force
and velocity qualities to maximize performances
(Samozino et al., 2012, 2014). These results suggest that
the F–v mechanical profile in sprint running, which
shows high inter-individual differences (Jaskolska et al.,
1999b; Morin et al., 2010), can also be interesting to
consider and adjust by individualized training loads and
exercises. Finally, recent studies showed that sprint per-
formances (6-s sprints, 100-m events, repeated sprints)
are related more to the effectiveness of force application
to the ground than to the total force developed by lower
limbs (Morin et al., 2011a, 2012; Rabita et al., 2015). So,
quantifying individually the mechanical effectiveness
can help distinguish the physical and technical origins of
inter- or intra-individual differences in both P–F–v
mechanical profiles and sprint performances, which can
be useful to more appropriately orient the training
process toward the specific mechanical qualities to
develop.
To date, such evaluations require to measure horizon-
tal antero-posterior and vertical GRF components and
forward horizontal velocity during an entire sprint accel-
eration (∼30–60 m). Sprint F–v and P–v relationships
have hitherto been obtained using specific instrumented
treadmills on which subjects accelerate the belt them-
selves by the action of their lower limbs, while their
waist is tethered backward to a fixed point (e.g.,
Jaskolska et al., 1999a, b; Chelly & Denis, 2001; Morin
et al., 2010). Despite the high accuracy of these methods,
their main limitation is the treadmill condition that does
not exactly reproduce natural overground sprint running
movement due to waist attachment, a belt narrower than
a typical track lane, the impossibility to use starting
block, and the need to set a default torque to, only partly,
compensate for the friction of the treadmill belt bed
(Morin et al., 2010; Morin & Seve, 2011). Because, to
date, no 30- to 60-m long force plate systems exist,
GRFs over an entire sprint acceleration phase were very
recently determined in elite sprinters using data from
several sprints measured on a 6.60-m long force plate
system, which allowed, for the first time, to provide
the data to entirely characterize the mechanics of
overground sprint acceleration (Rabita et al., 2015).
Sport practitioners do not have easy access to such rare
and expensive devices, and often do not have the tech-
nical expertise to process the raw force data measured. In
the best cases, this forces athletes to report to a labora-
tory, which explains that, although very accurate and
potentially useful for training purposes, this kind of
evaluation is almost never performed. In addition, sport
scientists investigating sprint mechanics and perfor-
mance usually assess, at best, only very few steps of a
sprint because of this technical limitation (e.g., Lockie
et al., 2013; Kawamori et al., 2014). A simple method for
determining F–v and P–v relationships and force appli-
cation effectiveness during sprint running in overground
realistic conditions could therefore be very interesting to
generalize such evaluations for training or scientific
purposes.
Mechanics and energetics of sprint running have been
approached by various kinds of mathematical models
that aimed at describing sprint performance from the
balance between the mechanical energy demand of
sprint running acceleration and the energy release capac-
ity of the aerobic and anaerobic metabolism (e.g.,
Furusawa et al., 1927; van Ingen Schenau et al., 1991;
Arsac & Locatelli, 2002; Helene & Yamashita, 2010).
Based on the mechanical analyses used in these models,
an inverse dynamic approach applied to the runner body
center of mass (CM) could give valid estimation of GRF
during sprint running acceleration from simple kine-
matic data, as recently proposed by di Prampero et al.
(2015), but never compared with force plate measure-
ments. This could then be used to obtain the aforemen-
tioned sprint mechanical properties without force
platform system in typical field conditions of practice.
The aim of this study was to propose and validate a
simple field method based on an inverse dynamic
Samozino et al.
2
approach applied to the runner body CM during sprint
running acceleration for (a) estimating the GRFs in the
sagittal plane of motion from only anthropometric and
spatiotemporal data and (b) determining the associated
F–v and P–v relationships and effectiveness of force
application. The concurrent validity of this method
was tested by comparison to reference force plate
measurements.
Biomechanical model used in the proposed method
This section, devoted to present the biomechanical
model on which the proposed simple method is based, is
an analysis of kinematics and kinetics of the runner’s
body CM during sprint acceleration using a macroscopic
inverse dynamic approach aiming to be the simplest
possible (Furusawa et al., 1927; Helene & Yamashita,
2010). All variables presented in this section are
modeled over time, without considering intra-step
changes, and thus correspond to step-averaged values
(contact plus aerial times).
During a running maximal acceleration, horizontal
velocity (vH)–time (t) curve has long been shown to
systematically follow a mono-exponential function for
recreational to highly trained sprinters (e.g., Furusawa
et al., 1927; Chelly & Denis, 2001; di Prampero et al.,
2005; Morin et al., 2006):
vt v e
HH
t
()
=⋅−
()
−
max 1τ[1]
where vHmax is the maximal velocity reached at the end
of the acceleration and τis the acceleration time con-
stant. The horizontal position (xH) and acceleration
(aH) of the body CM as a function of time during the
acceleration phase can be expressed after integration
and derivation of vH(t) over time, respectively, as
follows:
x t v t dt v e dt
t
HH H
()
=
()
=⋅−
⎛
⎝
⎜⎞
⎠
⎟
∫∫ −
max 1τ[2]
xt v t e v
t
()
=⋅+
⎛
⎝
⎜⎞
⎠
⎟−⋅
−
HH
max max
.ττ
τ[3]
at dv t
dt
ve
dt
t
H
H
Hmax
()
=
()
=
⋅−
⎛
⎝
⎜⎞
⎠
⎟
−
1τ
[4]
at ve
t
H
()
=⎛
⎝
⎜⎞
⎠
⎟⋅−
max
ττ[5]
Applying the fundamental laws of dynamics in the
horizontal direction, the net horizontal antero-posterior
GRF (FH) applied to the body CM can be modeled over
time as:
Ft ma t F t
H H aero
()
=⋅
()
+
()
[6]
where mis the runner’s body mass (in kg) and Faero(t)is
the aerodynamic drag to overcome during sprint running
that is proportional to the square of the velocity of air
relative to the runner:
Ftkvtv
aero
()
=⋅
()
−
()
Hw
2[7]
where vwis the wind velocity (if any) and kis the run-
ner’s aerodynamic friction coefficient, which can be esti-
mated as proposed by Arsac and Locatelli (2002) from
values of air density (ρ, in kg/m3), frontal area of the
runner (Af,inm
2), and drag coefficient (Cd =0.9; van
Ingen Schenau et al., 1991):
kAfCd=⋅⋅⋅05.ρ[8]
with
ρρ=⋅ ⋅ +°
0760
273
273
Pb
T[9]
Af h m=⋅⋅
()
⋅0 2025 0 266
0 725 0 425
..
.. [10]
where ρ0=1.293 kg/m is the ρat 760 Torr and 273 °K,
Pb is the barometric pressure (in Torr), T° is the air
temperature (in °C), and his the runner’s stature
(in m). The mean net horizontal antero-posterior
power output applied to the body CM (PHin W) can
then be modeled at each instant as the product of FH
and vH.
In the vertical direction, the runner’s body CM during
the acceleration phase of a sprint goes up from the start-
ing crouched position (be it with or without using start-
ing blocks) to the standing running position, and then
does not change from one complete step to another.
Because the initial upward movement of the CM is
overall smoothed through a relative long time/distance
(∼30–40 m; Cavagna et al., 1971; Slawinski et al.,
2010), we can consider that it does not require any large
vertical acceleration, and so that the mean net vertical
acceleration of the CM over each step is quasi-null
throughout the sprint acceleration phase. Consequently,
applying the fundamental laws of dynamics in the verti-
cal direction, the mean net vertical GRF (FV) applied to
the body CM over each complete step can be modeled
over time as equal to body weight (di Prampero et al.,
2015):
Ft mg
V
()
=⋅ [11]
where gis the gravitational acceleration (9.81 m/s2).
Morin et al. (2011a) proposed that the mechanical
effectiveness of force application during running could
be quantified over each support phase or step by the
ratio (RF in %, eqn. [12]) of FHto the corresponding
total resultant GRF (FRes, in N), and over the entire
Simple method to compute sprint mechanics
3
acceleration phase by the slope of the linear decrease in
RF when velocity increases (DRF, in %.s/m):
RF F
F
F
FF
=⋅= +⋅
H
Res
H
HV
100 100
22 [12]
To date, RF values have been computed from the
second step (i.e., the first complete step) to the step at
maximal speed (Morin et al., 2011a, b, 2012; Rabita
et al., 2015). Because the starting block phase (push-off
and following aerial time) lasts between 0.5 and 0.6 s
(Slawinski et al., 2010; Rabita et al., 2015) and so occurs
at an averaged time of ∼0.3 s, RF and DRF can be rea-
sonably computed from FHand FVvalues modeled for
t>0.3 s.
The above-described biomechanical model makes
possible to estimate GRFs in the sagittal plane of motion
during one single sprint running acceleration from
simple inputs: anthropometric (body mass and stature)
and spatiotemporal (split times or instantaneous running
velocity) data. This model can then be used as a simple
method to determine the F–v and P–v relationships and
the associated variables, as well as the mechanical effec-
tiveness of force application parameters (see Methods
section for details about the practical methodology asso-
ciated to this computation method). Two different
experimental protocols were conducted (a) to test the
concurrent validity of the proposed computation method
by comparison to force plate measurements and (b) to
test its inter-trial reliability.
Methods
First protocol: concurrent validity compared with force
plate measurements
Subjects and protocol
Nine elite or sub-elite sprinters (age: 23.9 ±3.4 years; body mass:
76.4 ±7.1 kg; height: 1.82 ±0.69 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. Their personal 100-m official best times were
10.37 ±0.27 s (range: 9.95–10.63 s). After a standardized 45-min
warm-up, subjects performed seven maximal sprints in an indoor
stadium (2 ×10, 2 ×15, 20, 30 and 40 m with 4-min rest between
each trial) in order to collect GRF data over an entire 40-m
distance. From these sprints, antero-posterior and vertical GRF
components; F–v, P–v, and RF–v relationships; and associated
variables (F0,v0,Pmax,SFV,DRF) were obtained from both force
plate measurements and above-described computation method.
Force plate method: materials and data processing
The experimental protocol used here to determine F–v and P–v
relationships from force plate measurements has recently been
proposed and detailed by Rabita et al. (2015). Briefly, during seven
sprints, vertical and antero-posterior GRF components were mea-
sured by a 6.60-m long force platform system (natural fre-
quency ≥500 Hz). This system consisted of six individual force
plates (1.2 ×0.6 m) connected in series, time synchronized, and
covered with a tartan mat leveled with the stadium track. Each
force plate was equipped with piezoelectric sensors (KI 9067;
Kistler, Winterthur, Switzerland). The force signals were digitized
at a 1000-Hz sampling rate.
The protocol was designed in order to virtually reconstruct for
each athlete the GRF signal of an entire single 40-m sprint by
setting differently for each sprint the position of the starting blocks
relatively to the 6.60-m long force platform system. The starting
blocks were placed over the first platform for the 10-m sprints and
were placed remotely for the other trials (15–40 m) so that 17
different steps (18 foot contacts) from the block to the 40-m mark
could be measured (cf. Fig. 1 in Rabita et al., 2015). Force plat-
form signal was low-pass filtered (200-Hz cutoff, third-order zero-
phase Butterworth) and instantaneous data of vertical (FV,inN)
and horizontal antero-posterior (FH, in N) GRF components as
well as the resultant (FRes) were averaged for each step consisting
of a contact and an aerial phase (determined using 10-N threshold
on FVsignal).
Over each measurement areas, the instantaneous horizontal
velocity of the CM (vH, in m/s) was computed as the first integra-
tion over time of the antero-posterior horizontal acceleration (aH,
in m/s2) obtained at each instant dividing FHby body mass, with
initial vHvalues as integration constants obtained as follows. For
the 10-m sprints, the initial vHwas set to 0 as the starting blocks
were placed over the force plate area. For the other sprints, the
initial vHvalues were measured by high-speed video using a 300
frames per second digital camera (Exilim EX-F1, Casio, Tokyo,
Japan) placed perpendicularly to the sagittal plane of motion of the
athletes in a fixed position focusing on the entrance of the force
plate area (for more details, see Rabita et al., 2015). The instanta-
neous power output in the horizontal direction (PH) was computed
as the product at each instant of FHand vH. Instantaneous data of
vHand PHwere averaged over each step. For the following analy-
ses, the data of the seven sprints were pooled to reconstruct a
complete 40-m dataset for each subject. For each step, RF was
obtained from averaged step values of FHand FRes using eqn. [12].
Individual DRF values were determined as the slope of the linear
RF–vrelationships.
Proposed computation method: material and specific
data processing
Sprint times were measured with a pair of photocells (Microgate,
Bolzano, Italy) located at the finish line of the seven sprints. For
Fig. 1. Correlation between force values measured at each step
using the force plate method and values computed by the pro-
posed method. Horizontal and vertical components and resultant
ground reaction force values are represented for all subjects. The
identity line is represented by the continuous black line.
Samozino et al.
4
each sprint, the timer was triggered when subjects’ right thumb left
the ground. To remove all possible bias due to this kind of trig-
gering procedure on variables obtained with this computation
method, the time between the beginning of the force production on
the starting blocks (which represents the actual start of the sprint)
and the trigger of the timer was determined and added to photocell
times using sagittal high-speed video recording of sprint starts
(Exilim EX-F1, Casio, Tokyo, Japan) synchronized with force
plate data.
For the two 10- and 15-m trials, only the best times were
considered at each distance for data analysis. For each subject, the
five split times at 10, 15, 20, 30, and 40 m were then used to
determine vHmax and τusing eqn. [3] and least-square regression
method. From these two parameters, vH(t) and aH(t) were modeled
over time using eqns. [1] and [5], respectively. From aH(t), FH(t)
was modeled over time using eqn. [6] and estimation of Faero(t)
from eqns. [7] to [10]. For individual computations of k, both the
subject’s body mass and stature were measured before tests, Pb
was 760 Torr, and T° was 20 °C. No wind was present during this
indoor testing session and the very limited effect of air humidity
on ρwas not considered. PHcan then be modeled at each instant as
the product of FHand vH.FV(t) and RF(t) were obtained using
eqns. [11] and [12], respectively, and individual DRF values were
determined as described in the theoretical background section. All
these variables were computed every 0.1 s (from eqns. [1], [6],
[11], and [12]) over each individual acceleration phase.
Common data analysis for both methods
Individual F–v and P–v relationships were determined for both
methods from step averaged (for force plate method) and modeled
(for proposed method) FH,PH, and vHvalues using least-square
linear and second-order polynomial regressions, respectively
(Jaskolska et al., 1999b; Morin et al., 2010, 2012; Rabita et al.,
2015). F–v relationships were extrapolated to obtain F0and v0as
the intercepts of the F–v curve with the force and velocity axis,
respectively. SFV value was determined for each subject as the
slope of the F–v relationship, and Pmax was determined as the apex
of the P–v relationship using the first mathematical derivation of
the associated quadratic equation. Pmax values were also computed
as previously proposed and validated (Vandewalle et al., 1987;
Samozino et al., 2012, 2014) as follows:
PFv
max =⋅
00
4[13]
Statistical analysis
All data are presented as mean ±standard deviation (SD). For each
subject, the standard errors of estimate (SEE) of FH,FV,FRes, and
RF were computed between values obtained from force plate data
at each step and values estimated from model computations for
corresponding vHvalues:
SEE FF
N
=−
()
−
∑Force Plate Model
steps
2
2[14]
The correlation between force values (FH,FV,FRes) obtained by
both methods was analyzed. F–v relationships, power output capa-
bilities, and mechanical effectiveness obtained with both methods
were compared through F0,v0,SFV,Pmax, and DRF values using bias
and limits of agreements (Bland & Altman, 1986). To complete
this quantification of inter-method differences, absolute bias was
also calculated for each subject as follows: absolute bias =|(com-
putation method–force plate method)/force plate method|.100. For
all statistical analyses, a P-value of 0.05 was accepted as the level
of significance.
Second experimental protocol: inter-trial reliability
Six high-level sprinters (age: 22.5 ±3.9 years; body mass:
81.8 ±5.1 kg; height: 1.86 ±0.04 m) performed three maximal
50-m sprints using starting blocks with 10 min of rest between
each trial. Instantaneous vHwas measured at a sampling rate of
46.875 Hz with a radar system (Stalker ATS System, Radar Sales,
Minneapolis, Minnesota, USA) placed on a tripod 10 m behind the
subjects at a height of 1 m corresponding approximately to the
height of subjects’ CM (di Prampero et al., 2005; Morin et al.,
2012). F0,v0,SFV,Pmax, and DRF values were obtained with the
same data processing as presented before for the proposed com-
putation method, except that vHmax and τwere determined from
vH(t) using eqn. [1] and least-square regression method. Because
only the best of several trials is commonly used to be considered
during explosive performance testing, the inter-trial reliability of
each variable was quantified by the coefficient of variation (CV in
%), the change in the mean, and the standard error of measurement
(SEM, expressed in percentage of mean values) between the two
best trials (Hopkins, 2000). These data were used to determine the
smallest worthwhile change (SWC) for intra-individual (SWCintra)
and for inter-individual (SWCinter) comparisons for each variable as
0.3 of the SEM (expressed in the variable unity) and 0.2 of the
between-subject SDs, respectively (Hopkins, 2004).
Results
For the validation protocol, the split times at 10, 15, 20,
30, and 40 m were 1.84 ±0.10, 2.49 ±0.11, 3.05 ±0.13,
4.08 ±0.18, and 5.10 ±0.25 s, respectively. For each
subject, the change in horizontal position with time
given by these split times was well fitted by the expo-
nential model described by eqn. [3] (r2>0.999;
P<0.0001). The associated vHmax and τwere
10.05 ±0.66 m/s and 1.24 ±0.14 s, respectively. The
modeled values of FH,FV,FRes, and PHwell fitted the
experimental values measured at each step using force
plates, which is shown by SEE of 39.8 ±13.3 N,
49.5 ±17.0 N, 52.8 ±16.5 N, and 234.4 ±69.9 W,
respectively. Force values measured by force plate
method at each step and force values computed for the
corresponding step using the proposed method were
highly correlated (P<0.001; Fig. 1). Figure 2 shows the
changes in step-averaged (force plate method) and
modeled (computation method) values of force, power
output, velocity, and RF over the acceleration phase
obtained for a typical subject (subject #5). The SEE of
RF over the acceleration phase was 3.72% ±0.76%.
For all subjects and considering both methods, F–v
relationships were well fitted by linear regressions
(median r2=0.953 from 0.920 to 0.987 for the force
plate method and r2>0.999 for the proposed simple
method, P<0.001), P–v relationships were well fitted
by second-order polynomial regressions (median
r2=0.886 from 0.857 to 0.961 for the force plate method
and r2=1.000 for the proposed simple method,
P<0.001), and RF–vrelationships were well fitted by
linear regressions (median r2=0.965 from 0.955 to
Simple method to compute sprint mechanics
5
0.976 for the force plate method and r2>0.996 for the
proposed simple method, P<0.001). Typical examples
of these relationships obtained by both methods are
shown in Fig. 3 for subject #5. Values of F0,v0,Pmax (both
computed as the apex of the P–v relationship and using
eqn. [13]), SFV, and DRF are presented in Table 1, as well
as the associated bias, 95% limits of agreement, and
absolute bias between both methods. Table 2 presents
the inter-trial reliability of different mechanical variables
through CV, change in the mean and SEM between the
two best trials of the reliability protocol, as well as
values of SWCintra and SWCinter.
Discussion
Although running mechanics over the entire sprint accel-
eration phase have very recently been described for the
first time in overground conditions using in-serie force
plates (Rabita et al., 2015), the present study showed
valid estimates of the main sprinting mechanical
variables from only basic anthropometric and spatiotem-
poral data (i.e., distance–time or speed–time measure-
ments). The proposed simple computation method
makes possible to determine force– and power–velocity
relationships and effectiveness of force application
during sprint running acceleration in real-practice con-
ditions. This new method shows very strong agreement
with the gold standard force plate measurements and a
Fig. 2. Changes over the acceleration phase in horizontal veloc-
ity (vH, black diamonds), horizontal antero-posterior (FH, black
filled circles) and vertical (FV, open triangles) force components,
horizontal antero-posterior power output (PH, open circles), and
ratio of force (RF, open diamonds) for a typical subject (subject
#5). Points represent averaged values over each step obtained
from force plate method (from five sprints) and lines represent
modeled values computed by the proposed simple method.
Fig. 3. Force–velocity (a), power–velocity (b), and RF–velocity
(c) relationships obtained by both methods for a typical subject
(subject #5). Open circles represent averaged values over each
step obtained from force plate method, dashed lines the associ-
ated regressions, and thin lines the modeled values computed by
the proposed simple method confounded with the associated
regressions (dotted lines).
Samozino et al.
6
high test–retest reliability, which supports its interest for
practitioners in numerous sports that involve sprint
running accelerations.
Modeled GRFs during sprint acceleration
The proposed method is based on a macroscopic biome-
chanical model using an inverse dynamic approach
applied to the runner body CM during sprint running
acceleration. This approach models the horizontal
antero-posterior and vertical GRF components applied
to the runner’s CM, and in turn the force developed
by the runner onto the ground in the sagittal plane,
during the entire acceleration phase of maximal intensity
sprint. The main simplifying assumptions admitted in
this model are those inherent to the application of fun-
damental laws of dynamics to the whole human body
considered as a system represented by its CM (e.g.,
Cavagna et al., 1971; van Ingen Schenau et al., 1991;
Samozino et al., 2008; Helene & Yamashita, 2010;
Samozino et al., 2010; Samozino et al., 2012; Rabita
et al., 2015), the estimation of the horizontal aerody-
namic drag from only stature, body mass and a fixed
drag coefficient (Arsac & Locatelli, 2002), and the
assumption of a quasi-null CM vertical acceleration over
the acceleration phase of the sprint. Note that our com-
putations lead to modeled values over complete steps,
i.e., contact plus aerial times, in contrast with previous
kinetic measurements during sprint running that aver-
aged mechanical variables over each support phase
(Morin et al., 2010, 2011a, 2012; Lockie et al., 2013;
Kawamori et al., 2014; Rabita et al., 2015). This induces
lower values of force or power output and a different
averaging period than support phase-averaged values:
step-averaged variables characterize more the mechanics
of the overall sprint running propulsion than specifically
the mechanical capabilities of lower limb neuromuscular
system during each contact phase. This does not affect
RF (and in turn DRF) values as it is a ratio between two
force components averaged over the same duration.
Despite the above-described simplifying assumptions,
present results showed that the modeled force values (FH,
FV,FRes) were very close to values measured by force
plates at each step with low SEE values of ∼30–50 N.
Furthermore, the values obtained by both methods were
highly correlated and closely distributed around the
identity line, even if the correlation was slightly lower
for FVthan for FHdenoting a better accuracy of estima-
tion in the horizontal than in the vertical direction. It is
worth noting that the SEE quantified here are mainly due
to the relatively high inter-step variability measured by
force plates rather than an inaccuracy of the proposed
method. Indeed, as shown in Fig. 2 for a typical subject,
force-modeled values well fitted the force plate data with
a scattering of the latter due to a possible asymmetry
between right and left legs, an inter-step variability
inherent to the complexity of this multijoint free move-
ment characterized by a great muscle coordination com-
plexity, and the fact that force plate data were obtained
from five different sprints. The inter-step variability is
not detectable by the proposed method as the model
gives the average tendency of change in GRF compo-
nents with time. The inter-step variability in force plate
data is more obvious in power output values combining
the variability of both force and velocity values, which
lead to a relatively higher, but still very acceptable,
Table 1. Mean ±SD values of variables attesting the concurrent validity of the proposed method
Reference
method
Proposed
method
Bias 95% agreement
limits
Absolute
bias (%)
F0(N) 654 ±80 638 ±84 −15.9 ±25.7 (−66.3; 34.5) 3.74 ±2.69
v0(m/s) 10.20 ±0.36 10.51 ±0.74 0.32 ±0.52 (−0.7; 1.3) 4.77 ±3.26
Pmax (W)apex of the P–v relationship 1546 ±195 1661 ±277 115 ±107 (−94.7; 324.7) 8.04 ±5.01
Pmax (W)from F0and v0(eqn. [13]) 1669 ±253 1680 ±280 10.56 ±45.01 (−77.7; 98.8) 1.88 ±1.88
SFV (N/s/m) −64.06 ±6.30 −60.8 ±7.71 3.26 ±5.22 (−6.97; 13.49) 7.93 ±5.32
DRF (%/s/m) −6.80 ±0.28 −6.80 ±0.74 −0.002 ±0.58 (−1.139; 1.135) 6.04 ±5.70
SD, standard deviation.
Table 2. Mean ±SD of the main variables attesting the reliability of the proposed method
CV (%) Change in
the mean
Standard error of
measurement (%)
SWCintra SWCinter
F0(N) 2.93 ±2.00 −1.53 ±32.2 3.57 6.76 12.06
v0(m/s) 1.11 ±0.86 −0.171 ±0.776 1.40 0.166 0.268
Pmax (W)apex of the P–v relationship 1.90 ±1.40 −0.164 ±0.669 2.37 0.147 0.441
Pmax (W)from F0and v0(eqn. [13])
1.87 ±1.36
1.87 ±1.36 −0.167 ±0.66 2.33 0.144 0.446
SFV (N/s/m) 4.04 ±2.72 −0.20 ±4.18 4.94 0.888 0.963
DRF (%/s/m) 3.99 ±2.80 −0.110 ±0.45 4.86 0.096 0.080
SD, standard deviation; SWC, smallest worthwhile change.
Simple method to compute sprint mechanics
7
power output SEE (∼234 W). For all subjects, FVvalues
measured with force plates over the first 20–30 m were
not particularly higher than body weight (as shown for a
typical subject in Fig. 2) and were very close to body
weight when averaged over the entire 40 m (grand aver-
aged difference between mean FVand body weight of
∼2.40%). This supports the assumption of a quasi-null
vertical acceleration of the CM over this phase due to a
very smoothed upward movement, and in turn supports
the validity of step averaged FVmodeled values as equal
to body weight. Collectively, these results support a very
good agreement in the model determination of GRF in
the sagittal plane of motion (horizontal, vertical, and
resultant) during sprint running acceleration.
Validity of force– and power–velocity
relationships determination
Both methods presented individual F–v relationships
strongly described by linear regression (r2>0.920) as
reported in our recent paper (Rabita et al., 2015), during
treadmill sprint running protocols (Jaskolska et al.,
1999b; Morin et al., 2010, 2012) or more generally
during multijoint lower limb movements such as pedal-
ing, squat, or leg press movements (e.g., Yamauchi &
Ishii, 2007; Dorel et al., 2010; Bobbert, 2012; Samozino
et al., 2012). The adjustment quality of the linear regres-
sions was logically better for the proposed method
(mean r2=1.00) based on modeled values than for the
reference method (mean r2=0.956) using experimental
force plate data that were inevitably associated to a
higher measurement and inter-step variability. The low
bias associated to narrow 95% agreement limits crossing
0 in the determination of F0,v0, and SFV showed that the
difference between both methods is very low (−2.43%,
3.14%, and 5.09% when expressed relatively to refer-
ence values, respectively) and can be attributed to mea-
surements variability. The very low absolute bias
(<5%), representing exactly the mean absolute error
value between both methods at each F–v variable deter-
mination, clearly supports the validity of the proposed
method to determine F–v relationship in sprint running.
Note that the absolute bias for SFV was slightly higher
than for F0and v0as it represents a regression slope
(often associated to higher variability than other vari-
ables) computed from F0and v0, thus including twice
variability.
Individual P–v relationships presented well-fitted qua-
dratic regressions for both methods, as expected from
previous sprint running protocols (Jaskolska et al.,
1999b; Morin et al., 2010, 2012; Rabita et al., 2015).
However, for force plate method, the adjustment quality
of P–v regressions was lower (mean r2=0.894) than that
of F–v relationships, and lower than for the proposed
method (mean r2=1.00). This can be explained by the
above-discussed higher inter-step and inter-sprint vari-
ability measured by force plate and by the very few
number of steps in the ascending part of the P–v rela-
tionship inherent to maximal power production occur-
ring after only approximately five to six steps (Rabita
et al., 2015). The lower adjustment quality of the qua-
dratic model obtained from force plate data induced
noise in the determination of Pmax from the apex of P–v
relationships, which explains the higher, but still very
acceptable, bias and absolute bias (∼8%) observed here
between both methods for this variable compared with
bias observed for F0and v0. Consequently, differences in
Pmax are mainly due to the inter-step variability detected
by force plates, which was supported by the very similar
Pmax values obtained using the proposed method from the
apex of the P–v relationships and using eqn. [13] (∼1%),
while they were quite different for the reference method
(∼8%). From a purely mathematical point of view, if the
F–v relationship is perfectly linear, the apex of the P–v
relationships should be equal to Pmax given by eqn. [13]
(Vandewalle et al., 1987; Samozino et al., 2012). More-
over, when computed from eqn. [13], Pmax values
obtained by both methods were very close with very low
bias values (absolute bias <2%) and narrowed limits of
agreement. These findings support the validity of the
proposed method in determining F–v and P–v relation-
ships and their associated mechanical variables (Pmax,F0,
v0,SFV) in sprint running.
Validity of the effectiveness of force
application determination
The very good agreement between both methods in GRF
components resulted in modeled RF values similar to
those computed at each step from force plate measure-
ments, as shown by the low SEE. RF values measured
here from both methods (RF values between 0% and
60%) were in line with values previously reported from
dynamometric treadmill measurements (between 10%
and 40%; Morin et al., 2011a, b; Morin et al., 2012), but
through a larger range of values as treadmill running
made impossible to measure RF for the first step (due to
the starting crouch position) and forced subjects to over-
come treadmill belt bed horizontal friction force at peak
running velocity, the latter being ∼20% lower than the
overground one (Morin & Seve, 2011). Individual
RF–velocity relationships determined by both methods
were well fitted by inverse linear regressions (r2>0.95),
as originally shown on treadmill (Morin et al., 2011a,
2012) and more recently during overground sprint
running (Rabita et al., 2015). Moreover, present results
show that the rate of decrease of these regressions (DRF)
were very similar between both methods (absolute
bias ∼6%). This supports the accuracy of the proposed
method to determine the effectiveness of force applica-
tion throughout the acceleration phase of a sprint and,
in turn, its rate of decrease when running velocity
increases.
Samozino et al.
8
Validation protocol
The validation of the proposed method was performed
using a recent validated multiple sprint protocol (Rabita
et al., 2015) as the reference method. The input variables
of the present model are basic anthropometric (body
mass and stature) and spatiotemporal data: split times (as
used here) or running velocity measurements (as could
be obtained from radar guns, e.g., di Prampero et al.,
2005; Morin et al., 2006, or laser beams, e.g., Bezodis
et al., 2012) during one single sprint. Because the aim of
the present study was to validate the whole approach and
the proposed equations, using split times from several
sprints allowed us to compare exactly the same exercises
between the two methods. Using radar data would have
forced us to compare mechanical variables obtained
from one single sprint using the proposed method to data
obtained from five sprints with the reference method,
which could have added a bias that would have only been
associated to the validation protocol itself. However, the
proposed method was also tested in the present protocol
from data measured using a radar (Stalker ATS System,
Radar Sales, 46.875 Hz) during the best sprints of the 30
and 40-m trials. Results were very similar to those
obtained from split times, with slightly higher bias
values (absolute bias from 3% to 7%) due to the above-
discussed point, which supports the validity of the
proposed method from running velocity measurements.
As partly noticed by Furusawa et al. (1927), the present
macroscopic biomechanical model shows that, whatever
the kind of locomotion, when displacement velocity
changes with time follow an exponential function
(as described by eqn. [1] and previously reported during
maximal sprint acceleration from recreational to
highly trained sprinters; di Prampero et al., 2005; Morin
& Seve, 2011; Morin et al., 2012), the relationship
between horizontal GRF component and velocity is
quasi-linear.
Reliability
The reliability of the proposed method was tested here in
a second protocol as the first mentioned protocol already
required seven maximal sprints to compare the proposed
to the reference method. For all the mechanical vari-
ables, low CV and SEM values (<5%) associated to
change in the mean close to 0 showed the high Test-retest
reliability of the proposed method. The present level of
reliability is in accordance with reliability previously
reported during isoinertial all out tests (e.g., Hopkins
et al., 2001; Samozino et al., 2008). This high reliability
led to low SWC values for both intra- and inter-
individual comparisons for each variable and strongly
that this simple method is of great interest for sport
practitioners and clinicians to detect training or rehabili-
tation effects, or for scientists when using progressive
magnitude statistics.
Practical applications
In light of the above-discussed points, the present study
proposed an accurate, reliable, valid, and simple method
to determine F–v, P–v, and RF–v relationships during
overground sprint running from variables easily obtained
in field conditions and with a precision similar to that
obtained with specific laboratory devices (force plates;
Rabita et al., 2015). This new field method is based on
the previously validated simple method focusing on the
same kind of mechanical properties during jumping
movements and based on a similar biomechanical mac-
roscopic approach (Samozino et al., 2008, 2014). The
present proposed method only requires individual basic
anthropometric data (body mass and stature) and ~5 split
times or instantaneous velocity measurements obtained
during one single sprint acceleration until maximal
velocity. Note that split times or instantaneous velocity
have become more accessible with new technologies
such as position trackers, GPS, or accelerometer-based
systems. Anthropometric and split times or instanta-
neous velocity data can then be used as inputs in the
basic data processing to determine vHmax and τusing
least-square regression method from eqns. [1] or [2] and
then to compute different mechanical variables (details
in the Methods section “Proposed computation method:
material and specific data processing” and “Common
data analysis for both methods”). This will contribute
to generalize such evaluations for both scientific and
training purposes as it has the potential to be easily
reproduced by coaches, sport practitioners, or physio-
therapists in their daily practice. Such a testing session
could take only 15–20 min after a regular warm-up with
two or three sprints by athlete, only the best one being
analyzed using the method proposed here.
The variables obtained from F–v, P–v, and RF–v rela-
tionships give key information about force, velocity, and
power output capabilities, and about the effectiveness of
force application, which are of great interest to optimize
sprint running acceleration performance by comparing
P–F–v qualities of different athletes, orienting and indi-
vidualizing training loads exercises, and monitoring
training or rehabilitation in sports using sprint accelera-
tions (e.g., track-and-field events, team sports). Indeed,
as previously mentioned, sprint performance has been
shown to be related to these mechanical variables,
notably Pmax,v0, and DRF (Morin et al., 2011a, 2012;
Rabita et al., 2015). Furthermore, these mechanical vari-
ables seem to be sensitive to training modalities (notably
through in-season variations, unpublished personal
data). For instance, using weighted sled towing improves
horizontal force production and force application effec-
tiveness in sprint running (Cronin et al., 2008;
Kawamori et al., 2014). Note that methods using force
plate system with several sprints or instrumented sprint
treadmill would give the same kind of information, but
are currently impossible to set in training practice for
Simple method to compute sprint mechanics
9
most sport practitioners. That said, they are still very
interesting for studying inter-step variability or for intra-
step analyses, contact and aerial times, step length/
frequency and force impulse, and rate of development
during sprint running, or for other analyses requiring
additional and synchronized laboratory measurements.
Perspectives
This study proposed an accurate, reliable, and valid
simple method to evaluate mechanical properties of
overground sprint running propulsion and validated it
through a very good agreement with gold standard force
plate measurements. The proposed method is based on a
macroscopic biomechanical model using an inverse
dynamic approach applied to the runner’s CM. This
method is convenient for field use by sport practitioners
and clinicians as it requires only anthropometric (body
mass and stature) and spatiotemporal (split times or
instantaneous velocity) variables easy to obtain out of a
laboratory during sprint running acceleration. This
method could be further used to increase the understand-
ing in the mechanical determinants of sprint acceleration
performance in many sports, to study the adaptations of
the mechanical properties of lower limb neuromuscular
system in a variety of sprint running propulsion, and to
optimize sprint performance by individualizing and ori-
enting training or rehabilitation programs.
Key words: Acceleration, maximal performance, evalu-
ation, sprint mechanics, lower limb explosive capabilities,
ratio of force application.
Acknowledgements
We are grateful to Dr Pascal Edouard and Philippe Gimenez (Uni-
versity of Saint-Etienne, France), Pedro Jiménez-Reyes (UCAM,
Murcia, Spain), Matt Brughelli (AUT, Auckland, New Zealand),
Antoine Couturier, Gaël Guilhem, Caroline Giroux, Stevy Farcy,
and Virha Despotova (INSEP, France) for their valuable participa-
tion to this project and their enthusiastic and friendly collabora-
tion. We would like to thank Guy Ontanon, Dimitri Demonière,
Michel Gilot, Pierre Carraz, and the athletes of both the French
Institute of Sport (INSEP) and the Athletic Sport Aixois who
voluntarily gave their best performance for this protocol.
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