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Force-Velocity Relationship and Stretch-Shortening Cycle Function in Sprint and Endurance Athletes

Harrison, Andrew J.; Keane, Sean P.; Coglan, John

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Journal of Strength and Conditioning Research: August 2004 - Volume 18 - Issue 3 - p 473-479
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Many factors influence performance in athletic activities and determine whether an athlete will excel at power/sprint or endurance events. Of these factors, 2 in particular deserve special consideration, namely, the force-velocity relationship of muscle contraction and the utilization of the stretch-shortening cycle (SSC) phenomenon.

The force-velocity (F-V) and power-velocity (P-V) relationships of muscle contraction are crucial in describing the muscle's functional capacity. In an isolated muscle contraction, the F-V relationship can be described by the Hill equation (11) as

(P + a)(V + b) = constant,

where P = force of contraction, V = velocity of shortening, and a and b are constants.

Constant a describes force and is proportional to the (physiological) cross-sectional area (CSA) of the muscle. Constant b relates to velocity, and it should be proportional to the length of the muscle (11). The nature of the F-V relationship is determined at least in part by the size of the muscle, and evaluation of the effectiveness of muscle contraction requires appropriate correction for the effects of muscle size. Full correction of the F-V relationship is complex because it requires simultaneous correction of force by CSA and velocity by muscle length. Direct in vivo measurement of muscle force is difficult and isokinetic measures of joint torques are often used instead, assuming the local moment arm variations to be negligible (2). Because power is the product of force and velocity (or torque × angular velocity), it is logical that muscle power should be proportional to the volume of the muscle. Barrett and Harrison (6) have shown that the P-V relationship lends itself to a single and effective correction by thigh volume. While several studies have examined the F-V (or the joint torque-angular velocity) relationship and its implications for performance in power- or endurancerelated activities (14, 27–29), many of these studies have obtained contradictory findings due to methodological problems such as lack of correction for muscle size or gravity effect torque.

The importance of the SSC to sprinting and jumping performance is well established (21, 23); however, its benefits are not restricted solely to sprinting and jumping. Cavagna et al. (8) estimated that SymbolO2 during running might be 30–40% higher without contributions from the SSC. The SSC is typically characterized by an eccentric muscular contraction (or stretch) followed immediately by a concentric muscular contraction. Utilizing a stretch immediately before a concentric contraction has been shown to augment the concentric phase, resulting in increased force production and power output (8, 18, 30). SSC effectiveness is influenced by the rate and magnitude of the stretch, the level of activation resulting stiffness of the muscle tendon unit prior to the concentric phase, the change in muscle length during the stretch, and the time lag between the completion of the stretch and the initiation of the concentric contraction (1).


Performance of the SSC is often measured by prestretch augmentation, i.e., comparing the increase in performance in a countermovement jump (CMJ) or drop-rebound jump (DJ) with a squat jump (SJ) (4, 18, 21). Komi and Gollhofer (19) argue that, while it can be demonstrated that the CMJ produces higher jumping than the SJ, “for many neurophysiological and mechanical aspects” the CMJ is not a suitable model of an efficient SSC and more natural movements such as hopping and running represent better models of the SSC. Quantification of SSC effectiveness can, however, be difficult in these more natural movements due to problems of controlling the eccentric loading of the muscle and isolating muscle actions. An effective solution to this lack of control is the use of sledge jumping (12, 23).

The control of muscle tendon unit stiffness also plays an important role in utilizing the full benefits of a SSC movement. Similarly, it is recognized that performance in running is strongly influenced by the control of muscle stiffness. It has been demonstrated that running in humans may be likened to a simple spring-mass system consisting of a single linear leg spring and a point mass equivalent to body mass (9, 25). The stiffness of this spring-mass system is thought to have 2 distinct components, namely vertical stiffness (Kvert) and overall legspring stiffness (Kleg). Vertical stiffness provides the mechanism by which the downward velocity of the body is reversed during limb contact and can be calculated from the ratio of the peak vertical force (Fypeak) to the vertical displacement of the center of mass (Δy) during contact. McMahon and Cheng (25) and Farley and Gonzalez (9) have shown that both Fypeak and Δy occur simultaneously during running and hopping. Although Kvert does not correspond to any physical spring in the model, it is important in determining how long the spring-mass system remains in contact with the ground. Furthermore, in activities where the leg spring acts vertically, such as hopping or running on the spot, Kvert = Kleg.

Experimental evidence on how the leg-spring model behaves with changes in running velocity or stride rate seems to be equivocal. Farley and Gonzales (9) used a spring-mass model proposed by McMahon and Cheng (25) and reported that kleg in running is independent of running velocity but does increase with stride or hopping frequency or with hopping height. Ferris and Farley (10) suggested that a stiffer leg spring allows humans to run with a higher stride frequency at the same forward speed. More recently, Arampatzis et al. (3) reported that the knee joint is the main determinant of Kleg and up to a velocity of 6.5 m·s-1, knee spring stiffness increases with running speed. Arampatzis et al. (3) reasoned that the discrepancy of their findings compared with other studies was due to differences in calculation techniques. Arampatzis et al. (3) used direct measurements of changes in the leg-spring length during running and argued that the theoretically calculated length changes used in the Mc-Mahon and Cheng (25) model overestimated the actual length change and consequently underestimated legspring stiffness during running.

Experimental research to date clearly indicates the importance of the F-V relationship, SSC, and control of muscle stiffness to sprinting and running in general. It is also clear that research in these areas has been subject to inappropriate corrections for variations in muscle size and lack of control in measurements of SSC function and muscle stiffness. This study had 2 main aims. The first aim was to examine the knee joint extension torque-velocity and P-V relationships in sprint and endurance athletes and to ascertain whether any qualitative differences exist in the function of the knee extensors when the performance outputs are normalized for variations in muscle size. The second aim was to examine the stretch-shortening cycle function and control of leg-spring stiffness of sprint and endurance athletes while performing controlled squat, countermovement, and drop-rebound jumps on an inclined sledge apparatus.

Materials and Methods

Experimental Approach to the Problem

This study involved 2 experiments carried out concurrently on endurance and sprint-trained athletes. The first experiment determined whether significant differences existed between sprint and endurance athletes in the knee extension torque-velocity and P-V relationships. Isokinetic knee extension torque and power were obtained at 10 angular velocities. The experimental design involved a mixed model with 2 independent groups (sprint and endurance) and 10 repeated measures (knee extension velocities). The second experiment determined whether differences existed between the sprint and endurance groups in SSC function and vertical leg stiffness. SSC function was examined by comparing relative jumping performance in squat, countermovement, and drop jumps performed on a sledge apparatus. Vertical leg stiffness was assessed by examining the ratio of maximum force/maximum crouch displacement in both countermovement and drop jumps. Again, a mixed model design was used with 2 independent groups and 3 different jump types.


Fourteen men participated in this study. All subjects were competitive athletes and were divided into 2 groups. Group 1 consisted of 7 sprinters and Group 2 consisted of 7 endurance athletes. The sprinters had 100-m personal best times in the range 10.45–11.20 seconds, and the endurance athletes were of competitive national league standard and competed in different events ranging from 1500 to 10,000 m. All tests took place during the period of January-March, which was an indoor competition period for sprinters and cross-country competitive period for endurance athletes. Subjects participated in their normal training regimes and were not subject to any special behavioral or dietary controls. Ethical approval for this study was obtained from the University research ethics committee and written informed consent was obtained from all subjects prior to their participation. Table 1 provides details of age, height, and mass of the subjects by group.

Table 1
Table 1:
Physical characteristics of the subjects.


All subjects completed 3 performance tests and the results of these were used to verify the subject's sprint or endurance capacity. The performance tests were a 30-m sprint test from a standing start, a 30-m sprint test from a running start (i.e., flying 30 m), and 20-m progressive shuttle run test (24).

For both the 30-m sprint and flying 30 m, performance times were recorded using a Brower photo electronic timing device. Each subject completed 3 trials, with the fastest time being recorded for further analysis. Subjects were given 5 minutes rest between each trial. The 20-m progressive shuttle run was conducted using a prerecorded cassette tape (NCF, Leeds, UK) and the performance score was the number of shuttles the subject completed.

Thigh volume was measured on the subject's dominant leg (i.e., preferred kicking leg). Anthropometric measurements comprising a series of circumference and length measurements were made using a flexible steel tape. The circumference sites were at the gluteal furrow, one third of the subischial level, and the maximum circumference around the knee joint space. The distances between the adjacent circumference sites were also measured. Skinfold thickness was measured at the anterior and posterior thigh in the midline at the one-third subischial level with a Harpenden fat calliper. The circumference measurements were corrected for skinfold thickness using the method of Jones and Pearson (15). The lean-thigh volume was calculated as the sum of 2 adjacent truncated cones using the method of Katch and Weltman (16). Thigh CSA was calculated from the lean circumference measurements at the one-third subischial level.

The volumes of the truncated cones were calculated using the following equation:

V = [h(C12 + C22 × (C1 + C2))]/12π,

where V is the volume of the truncated cone, C1 and C2 are the circumferences at the top and bottom of the cone, and h is the distance between C1 and C2.

In vivo measurement of the torque-velocity and P-V relationships using isokinetic dynamometry is a well-established and accepted technique (2, 27–29). Isokinetic concentric knee extension (Con KE) torque on the subject's dominant leg was determined on a Con-Trex isokinetic dynamometer (CVH AG, Duübendorf, Switzerland). Before each test session, the dynamometer was calibrated using the manufacturer's instructions. All subjects performed a warm up, which consisted of a 5-minute cycle on a Monarch 814E cycle ergometer (Varberg, Sweden). All subjects also completed a short habituation and practice session on the isokinetic dynamometer. During the tests, the subjects were stabilized at the thigh, pelvis, and trunk with Velcro straps. The axis of rotation of the dynamometer lever arm was aligned with the posterior aspect of the lateral femoral epicondyle. The distal shin pad of the dynamometer was placed 3 cm proximal to the medial malleolus. Subjects were instructed to place their arms across their chest during the testing procedure. Gravity effect torque was recorded on each subject and this was used to correct torque measurements during tests.

Maximum Con KE torque was measured at 10 different velocities (ranging from 30°·s-1 to 300°·s-1, in 30°·s-1 increments). A submaximal extension-flexion movement immediately preceded each maximal effort trial. This helped ensure the muscle contracted maximally throughout the measured concentric knee extension exercise. The sequence of the velocities was varied between subjects to negate possible effects of fatigue on the results. Three to 5 minutes rest was given between each effort. Each subject was given the same level of encouragement during trials. Subjects performed 5 trials at each velocity and Con KE torque was recorded continuously throughout the full range of motion. A visual inspection of the angle-time and torque-time graphs of each trial was made to ensure that peak torque occurred at constant angular velocity. Trials were rejected if the angle-time graph was nonlinear at the instant of peak torque.

To examine SSC function and obtain indices of muscle stiffness, each subject performed 5 countermovement jumps, 5 squat jumps, and 5 drop jumps on a sledge apparatus as described by Horita et al. (12). The apparatus consisted of 3 main components: a sledge frame, a sliding chair, and a force platform (see Figure 1). The sledge frame was constructed from box steel with sledge rails inclined at 30°. The chair was mounted on the rails on low-friction steel rollers. The force platform (AMTI OR6–5) was mounted at right angles to the sledge apparatus and sampled at 500 Hz. The subjects were secured to the chair with a harness and Velcro straps at the waist and shoulders to prevent any upper-body movement during the jumps. The subjects performed all jumps with maximum effort and with their arms held across their chests. The CMJs were performed with a maximum knee flexion angle of 90°, the DJs from a drop height of 0.30 m, and the SJs from a static crouch with knee flexion at 90°. A series of pilot tests on the experimental set-up obtained high agreement between measured impact velocity and predicted velocity based on the height of drop. An intraclass correlation coefficient (absolute agreement) of R = 0.996 was obtained for 4 repeated drops at a variety of heights ranging from 0.3 to 0.9 m; the single measure intraclass correlation coefficient was R = 0.982. Reflective markers were attached to the sledge and rails and sagittal plane. Super-VHS (50 Hz) video recordings were obtained of the motion of the sledge during the jumps. Ground reaction force measurements were obtained for each jump and the force platform was synchronized with the video using a Peak event and video control unit (Peak Performance Technologies, Englewood, Colorado). The flight time of the jumps was obtained by inspection of the vertical ground reaction force (Fy) data. The video records were digitized using Peak Motus (Peak Performance Technologies) and the displacement of the sledge was calculated from the video records using a general cross validatory quintic spline algorithm (31).

Figure 1.
Figure 1.:
Photograph of the set-up of the sledge apparatus and force platform.

Statistical Analyses

The ability of subjects to utilize SSC function was estimated by obtaining a SSC performance index for CMJ and DJ. This was obtained by dividing the subject's respective CMJ or DJ flight time by their average SJ flight time. Estimates of muscle stiffness were obtained from the force and video data for the CMJ and DJ. The stiffness model consisted of a mass and a single linear mass less leg spring (9, 17, 25). The stiffness of the leg spring was defined as the ratio of the force in the spring to the displacement of the spring at the instant of maximal compression (i.e., bottom of the crouch). The leg-spring muscle stiffness index (SI) was obtained using the following equation:

where Fcrouch = maximum force at the bottom of the crouch and Dsledge = maximum displacement of sledge at the bottom of the crouch.

Peak torque values were corrected for thigh CSA by dividing them by the area of the thigh at the one-third subischial level. The corrected torque and velocity values for each subject were fitted to the Hill equation by substituting known values of torque and velocity into a system of 3 simultaneous equations. This was done using the midrange velocities of 120, 150, and 180°·s-1. The values calculated for the constants a and b were then substituted back into the equation for the remaining velocities, yielding estimates of torque. The resulting torque predictions were compared with the actual measures obtained on the dynamometer to check the validity of the derived constants. Concentric knee extension power was calculated as the area beneath the torque-velocity curve at each velocity. Power values were then corrected by dividing by lean-thigh volume (6).

Student t-tests were performed on the mean Hill equation constants of the 2 groups to determine significant differences in the F-V relationship between sprint and endurance groups. Differences in corrected torque and power values, flight time, and stiffness indices between sprint and endurance groups were evaluated using a 2-way analysis of variance (ANOVA) with repeated measures. For torque and power, the general linear model (GLM) had 1 within-subjects factor, namely, velocity (with 10 levels) and 1 between-subjects factor, namely, group (with 2 levels: sprint and endurance). For SSC function and muscle stiffness, the GLM ANOVA had 1 between-subjects factor, namely, group, with 2 levels (sprint and endurance) and 2 within-subjects factors, namely, jump type, with 3 levels (SJ, CMJ, and DJ), and trial, with 5 levels. The dependent variables were flight time of the jump and stiffness index.


Table 2 presents the results of the sprint and endurance performance tests. The results show very clear differences between the sprint and endurance groups and justify the classification of these groups.

Table 2
Table 2:
Mean scores (±SD) in performance tests for sprint and endurance groups.

Table 3 shows the results of the anthropometric measurements of the subject's thigh volume, CSA, and length. There were significant differences between the sprint and endurance groups in thigh volume (p = 0.042) and thigh CSA (p = 0.009) but not for thigh length.

Table 3
Table 3:
Mean estimates of thigh volume, cross-sectional area (CSA), and length.

Figure 2 shows the average corrected Con KE torque velocity curves for sprint and endurance groups.

Figure 2.
Figure 2.:
Comparison of the mean concentric knee extension torque-angular velocity curves (corrected for muscle cross-sectional area) in sprint and endurance groups.

The graphs show that sprint athletes generated higher Con KE torque at all velocities, with an average difference in CSA-corrected torque across all velocities of 0.15 ± 0.05 N·m·cm-2. The GLM ANOVA found a significant between-subjects (i.e., group) main effect on corrected Con KE torque (p = 0.011) and a significant within-subjects interaction effect (i.e., group × velocity) on Con KE torque (p = 0.035). The Hill equation constants a and b were derived from the corrected torque values for each subject, and these mean Hill equation constants are shown in Table 4. A Student t-test on constants a and b revealed a significant difference between groups for the constant a but not b. This indicated that, across all velocities, sprinters generated higher torque.

Table 4
Table 4:
Mean Hill equation constants for sprint and endurance groups cross-sectional area (CSA) on corrected torque data.

Figure 3 shows the mean P-V graphs for both groups. Due to the velocity limitation of the dynamometer, the peak power values were not achieved. This is a common limitation of isokinetic dynamometers. Despite this, the graphs indicate that the sprint group achieved higher volume-corrected power output at all velocities.

Figure 3.
Figure 3.:
Mean power-angular velocity curves (corrected for muscle volume) in sprint and endurance groups.

The results of the GLM ANOVA indicated a significant between-subjects (i.e., group) main effect for volumecorrected power (p = 0.006) and a significant within-subjects interaction effect for group × velocity (p = 0.013). This result shows that the corrected P-V relationships were significantly different between the 2 groups of athletes.

Stretch-shortening Cycle Function and Muscle Stiffness Indices

Figure 4 shows the mean flight times in SJ, CMJ, and DJ in sprint and endurance groups. These data show that sprinters jumped higher than endurance athletes in all types of jumps. The results of the GLM ANOVA indicated significant between-subjects main effects for group and for jump type. The group × jump-type interaction effect was not significant, which suggests a similar pattern of differences in SJ, CMJ, and DJ between the groups and, therefore, similar ability to utilize the SSC.

Figure 4.
Figure 4.:
Mean jump performance scores (flight times) in sprint and endurance groups for countermovement jump (CMJ), drop-rebound jump (DJ), and squat jump (SJ). Data show significant differences in flight time between jump types and also between groups. The interaction effect jump × group was not significant.

Figure 5 shows the mean scores for SSC performance indices in CMJ and DJ in sprint and endurance athletes. These data show that there was no significant difference in SSC function between the sprinters and endurance athletes on either CMJ or DJ.

Figure 5.
Figure 5.:
Stretch-shortening cycle performance indices based on countermovement jump/squat jump (CMJ/SJ) and drop-rebound jump/squat jump (DJ/SJ) ratios for sprint and endurance groups.

Figure 6 shows the estimated leg-spring stiffness indices in CMJ and DJ in sprint and endurance groups. The results show that the sprint group leg-spring stiffness was significantly higher than the endurance group using both CMJ and SJ stiffness indices. The results of the GLM ANOVA indicated significant main effects for group and jump type (p < 0.001).

Figure 6.
Figure 6.:
Leg stiffness indices for sprint and endurance groups based on countermovement jump (CMJ) and drop-rebound jump (DJ) performance. Data show significant differences in vertical stiffness between jump types and also between groups. The interaction effect jump × group was not significant.


The isokinetic dynamometer tests provide Con KE torque-velocity and P-V curves that conform to the typical shape of the Hill-type F-V curve. The data show that the Con KE torque-velocity and P-V relationships were significantly different between sprint and endurance groups. The differences remained significant even when corrections were made for thigh CSA on torque and thigh volume on power. Table 2 indicates a significant difference in thigh volume between the sprint and endurance groups; therefore, the corrected P-V curves provide the most effective indication of a functional difference between the groups that is independent of muscle size.

The analysis of the results of the Hill equation constants indicated significant differences between the sprint and endurance groups in constant a but not for constant b. This is consistent with the results of the GLM ANOVA on torque. However, care is advised in interpreting these data because the Hill equation was derived for an isolated muscle, whereas the data in this study refers to an in vivo muscle group. Furthermore, the Hill equation describes muscle force with respect to velocity, not joint torque, and therefore, despite the fact that such techniques have been employed in other studies, the direct evidence for deriving Hill equation constants from isokinetic joint torque measurements is somewhat questionable. Despite this conceptual flaw, the torque velocity data does conform to the Hill-type F-V curve and the t-test results on derived Hill equation constants are consistent with the ANOVA results. This suggests that the Hill equation does have application in vivo.

The isokinetic test data suggest that differences in muscle function exist between the groups, with sprint athletes able to generate higher volume-corrected power output over the range of 30–300°·s-1. It is likely that the functional differences observed between the groups were related to qualitative factors such as fiber type and muscle-contraction velocity. The ability of the sprint athletes to generate higher power output per unit volume could be due to a higher percentage of FT (Type II) muscle fibers in this group. Barany (5) and Buchtal and Schmalbruch (7) demonstrated that FT fibers had a faster contraction time due to greater myosin ATPase activity, enabling them to generate greater power. This could also explain why the differences in corrected power output between the 2 groups became more marked as angular velocity increased (see Figure 3).

The tests of SSC performance indicated that the sprint group performed significantly better than the endurance group on all types of jumps (see Figure 4). Analysis of the SSC performances indices, however, appear to show that both groups utilized the SSC equally to augment performances in both CMJ and DJ (see Figure 5). This result appears to contradict the findings of Kubo et al. (21), who found that the CMJ/SJ ratio of long-distance runners was significantly lower than untrained control subjects. Caution is advised in interpreting these data for a number of reasons. First, as Komi and Gollhofer (19) pointed out, the CMJ is not an ideal model for a SSC; therefore, the results of Kubo et al. (20) are not obtained from an ideal model. Second, the SSC indices obtained in the present study using sledge jumps provided better isolation of muscle action and control of impact velocity in the DJ. Third, it is important to remember that the SSC performance indices merely describe the relationship between CMJ or DJ performance to SJ. An artefact of this calculation procedure is that relative reductions in SJ performance can misleadingly elevate the SSC performance indices. It is more important to note that, in both CMJ and DJ activities, which involve SSCs, the sprint group significantly outperformed the endurance group. While the sprint group demonstrated significantly better overall performance than the endurance group on SSC jumps, the similar SSC indices show that the ability to augment performance using a prestretch is retained in endurance athletes. These data suggest that prestretch augmentation may still be an important factor in submaximal speed running. This is consistent with the findings of Izquierdo et al. (13), who indicated that strength training could improve maximal endurance capacity in middle-aged and older men.

The results of the test of leg-spring stiffness showed that sprint athletes performed countermovement jumps and rebound jumps using a significantly stiffer leg spring. The Kvert values obtained in this study were much lower that the stiffness values obtained by Farley and Gonzalez (9), who reported Kvert values between 15.1 and 52.4 kN·m-1 depending on stride rate, and Arampatzis et al. (3), who reported even higher Kvert values. The differences in Kvert can be explained by the different test protocols used in this study. The measures of Kvert using the inclined sledge CMJ and DJ obtained much lower peak ground reaction forces and higher displacement values compared with the peak ground reaction forces and vertical displacement in running. Stiffness values obtained using the sledge jump protocols cannot be directly compared with measures obtained during running because of the differences in loading and displacement between the 2 protocols. In contrast with running, however, the sledge jump protocols provide greater control of impact velocities and isolate the leg action from interferences such as upper body movement and therefore provide a well-controlled and valid comparison of the leg-spring stiffness of sprint and endurance athletes. The data from this study show clearly that, in activities where the impact loading is controlled, sprint athletes utilize a stiffer leg spring than endurance athletes. Arampatzis et al. (3) have shown that running velocity influences leg-spring stiffness, but it is not clear if endurance and sprint athletes running at the same speed would use the same or different leg-spring stiffness. Paavolainen et al. (26) have shown that low-caliber and high-caliber endurance athletes differed significantly in their neuromuscular characteristics while running at a controlled speed and suggested that the higher caliber athletes utilized a stiffer leg spring. Because the results of this study show that sprinters generally use a stiffer leg spring when impact velocity is controlled, this suggests that sprinters would use a stiffer leg spring at any given speed, although further research would be required to confirm the validity of this interpretation.

Practical Applications

It is important from a practical point of view that distinctions of muscle performance in athletic activities should take appropriate account for individual variations in muscle size. The corrected torque and P-V data provide clear indications of the functional differences between sprint and endurance athletes with correction for size variations and show that sprint athletes generated higher power, especially at higher speeds. This emphasizes the need for sprint and jump training to be carried out at higher contraction speeds. The results suggest that, while sprinters perform better than endurance athletes in jumping activities, the SSC indices of both groups were similar. This suggests that the ability to utilize the SSC is important in endurance athletes as well as sprinters and therefore endurance runners may benefit from the inclusion of SSC-related activities in their training. The results also suggest that muscle-stiffness control is perhaps the main factor that distinguished the sprint and endurance runners. Leg-stiffness indices tend to increase as the range of the movement during the eccentric phase decreases, and this suggests that training for high legstiffness responses should emphasize short and quick movements during the eccentric phase of a SSC movement. It is not clear from the present study whether the ability to produce high leg stiffness is a factor that limits running speed or efficiency, but this may be a useful avenue for further research.


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    biomechanics; isokinetics; running; jumping; leg stiffness; Hill equation

    Copyright © 2004 by the National Strength & Conditioning Association.