Journal of Strength and Conditioning Research, 2007, 21(3), 967-972'K 2007 National Strength & Conditioning Association Technical Note

A DETERMINISTIC MODEL OF THE VERTICAL JUMP:IMPLICATIONS FOR TRAINING

DANIEL J. HAM,' WADE L. KNEZ/^ AND WARREN B . YOUNG*

'University ofBallarat, School of Human Movement and Sports Science, ML Helen, Victoria, Australia; -Instituteof Sport and Exercise Science, James Cook University, Cairns, Queensland, Australia.

ABSTRACT. Ham, D.J., W.L. Knez. and W.B. Young. A deter-ministic model ofthe vertical jump: implications for training, J.Strength Cond. Res. 21(3).-967-972. 2007.—Increasing verticaljump height is a critical component for performance enhance-ment in many sports. It takes on a numher of different formsand conditions, including double and single legged jumps andstationary and run-up jumps. In an attempt to understand thefactors that influence vertical jump performance, an extensiveanalysis was undertaken using the deterministic model. Onceidentified, practical training strategies enabling improvement inthese factors were elucidated. Our analysis showed that a suc-cessful vertical jump performance was the result of a complexinterplay of run-up speed, reactive strength, concentric actionpower ofthe take-off leg(s), hip flexors, shoulders, body position,body mass, and take-ofF time. Of special interest, our analysisshowed that the concentric action power ofthe legs was the crit-ical factor affecting stationary double leg vertical jumps, where-as reactive strength was the critical component for a single legjump from a run-up.

KEY WORDS, muscular power, performance optimization, train-ing methodology

INTRODUCTION

ertical jumping (VJ) is an important feature ofmany sports, and increasing the height towhich an athlete can reach may be advanta-geous to sport performance (10). Much researchhas been conducted on this activity, which has

provided insights into possible strategies for training (6,23, 32). However, previous studies have been limited tostanding double legged take-offs, where contact times are>500 ms, whereas some sports involve run-ups and singleleg take-ofTs with an associated take-off time that can be<200 ms. Subsequently, it is necessary to determine whatfactors influence performance in these specific jumps, sothat further insights into appropriate training practicesmay be elucidated. Therefore, the purpose of this articleis to develop a detailed deterministic model to indicatethe factors that govern VJ performance and discuss prac-tical training strategies and technique adaptations tbatmay be adopted to improve jumping performance. Such amethodical approach is yet to be completed in detail andis intended to assist the strength and conditioning pro-fessional by providing a map of factors that contribute tooptimal performance of a skill, allowing them to focustraining methods on identified weaknesses. It should benoted that specific training exercises and coaching meth-ods are beyond the scope of this paper, as this is seen asthe responsibility of coaches and strength and condition-ing professionals.

Deterministic Model

In this study, a form of qualitative analysis, not dissimi-lar to the mechanical method outlined by Hay and Reid

(10), has been used to develop a deterministic model forthe VJ. This method involves systematically examiningthe performance of a skill to determine what contributoryfactors may be improved to enhance overall performance(12). This is achieved by identifying the bottom line fac-tors, which are the determinants ofthe skill and the per-formance in tbeir most purest form. For example, jumpheight is determined by take-ofF height, flight height ofthe center of gravity, and reach height. Reach height isin turn determined by body position and limb lengths,which are not determined by any other factors and aretherefore termed bottom line factors. This type of analysisaims to supplement whatever experience the sport sci-entist or coach might have and to channel or direct theanalysis in a logical, systematic fashion (12), eliminatingthe risk of overlooking influential factors. In this study,bottom line factors from a detailed deterministic modelwere analyzed from a biomechanical and strength andconditioning perspective and will be used as a precursorfor practical training strategies and technique adapta-tions for the VJ.

Vertical Jump. The VJ is frequently used in a varietyof sports and has a number of components that requirespecific abilities that are not necessarily interrelated. TheVJ can take the broad form of a single leg jump for reachheight (basketball lay-up), a single leg jump for clearanceheight (high jump), a double leg jump from a stationaryposition (volleyball block), and a double leg jump with arun-up (volleyball spike). In this analysis, a deterministicmodel was developed to cover all of tbese types of jumpsand to explore specific practical training applications andtechnique adaptations. Inter-jump-type differences arealso discussed in detail, and further recommendations forthese techniques are given.

Generic Jump Deterministic Model. The model depict-ed in Figure 1 represents a generic deterministic modeladapted from the model developed by Hay and Reid (10).Some factors are not relevant for every jump type. In-deed, neither run-up speed nor hip flexor power is rele-vant for a standing countermovement jump, but becausethey are important to a single leg run-up jump, they havebeen included in the model.

Research DesignIn this study, the pathway from VJ height is followedalong the model branch to the extremities, or bottom linefactors, which have been highlighted in Figure 1. Thesebottom line factors represent the determinants of VJ per-formance in their simplest form—contributing factorsthat are not contributed to by any other factors. Further-more, these bottom line factors have been divided into 2categories, with the exception of limb lengths, which havebeen left out because of their unchangeable nature. Thefirst category is specifically relevant to strength and con-

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968 HAM, KNEZ. AND YOUNG

FlcuRK 1. Deterministic model of performance for a verticaljump (bottom line factors are in bold).

ditioning, and the second represents important areas fortechnique adaptation and body size considerations.

CATEGORY 1—STRENGTH AND CONDITIONING

Run-up SpeedFor sports such as high jump, it is believed that a fastapproach run improves the height of the jump (6); how-ever, it has been shown that athletes have an individualoptimum run-up speed (31). Indeed, the faster the ap-proach velocity, the more the jumper is dependent on thestrength ofthe take-off leg (28). If an approach run is veryfast, and the take-off leg is not strong enough, these jointswill be forced to flex excessively, or "buckle," during thefirst half of the take-off pbase, and this will weaken thesubsequent extension ofthe leg (6). Tbe ability to resistthis buckling of the leg during the eccentric phase of thetake-off is considered vital for successful jumping (30),and jumpers must therefore be able to tolerate highstretch loads to achieve optimum jump heights (31).Moreover, the jumpers who used the fastest run-upsachieved the greatest vertical velocities at take-off (6),which suggests that better performing jumpers werethose with greatest dynamic strength. Therefore, greaterdynamic strength will allow an athlete to use a fasterrun-up speed and subsequently achieve a greater take-offvelocity and jump higher.

Training ApplicationsOptima! technique will involve using the fastest approachrun possible without reaching the threshold for buckling.If optimum run-up speed is slow, but close to bucklingthreshold, appropriate strength training should be car-ried out to enable tbe athlete to resist excessive fiexionduring the first half of the take-off phase. Because buck-ling occurs in the first phase of the foot plant, and themuscle contraction in this phase is eccentric, appropriatetraining should focus on the legs ability to resist largeamounts of force during eccentric contraction.

Reactive Power of the LegReactive strength is the ability to quickly switch from theeccentric to concentric phase in a stretch shortening cycle(SSC) (31) and can add to power production during theconcentric action ofthe takeoff leg(s) that will ultimatelyincrease jump height. It is thought to be a relatively in-dependent strength quality that is of importance in jumpsinvolving short contact times and large take-off forces(27). Indeed, even long SSCs 00.250 seconds) and short

SSCs (<0.250 seconds) have been found to be indepen-dent skills, and therefore, it is possible for an athlete toexcel in one and not the other (30). Reactive strength cor-relates best with short SSCs, such as run-up jumps,where contact time is low and force is high (32). Thestretch component ofthe SSC refers to the eccentric mus-cle action, whereas the shortening component refers tothe concentric muscle action; elastic energy is stored inthe tendomuscular system during the eccentric phase andreleased during the concentric phase (25).

Studies examining different characteristics ofthe SSChave shown increases in force and power production bydecreasing the duration ofthe amortization phase (elec-tromechanical delay between eccentric and concentric ac-tions), a direct result of an increase in reactive strength,and increasing the speed of eccentric action (25). Recentstudies have shown that the series elastic elements/mus-cle-tendon complex in the lower leg (below the knee) wereable to store and release elastic energy even during a ver-tical squat jump that consisted of solely concentric musclecontraction (3, 19). Furthermore, increasing tensionplaced on these elements/structures through greater mus-cle contraction increased the amount of elastic energythat could be stored and released. Considering that muchhigher intrinsic muscle forces are able to be developedduring fast eccentric contractions (15), it is not surprisingthat reactive strength is an important factor in jumpsthat involve high eccentric loads. Indeed, it has beenfound that fast eccentric contractions and small amorti-zation phases, associated with a short SSC. increases theamount of stored energy that can be released during thesubsequent concentric phase of jumps. This is especiallytrue in fast twitch fibers (2).

Reactive strength is a very specific skill that is char-acterized by the high forces and low contact times indic-ative of jumps involving a run-up. It is linked with theeffective functioning of a short SSC (<0.250 seconds) andcan be improved by increasing the amount of energy thatis stored in the tendomuscular system during the eccen-tric phase and ensuring maximum release of this energyby decreasing the length of the amortization phase.

Training Applications

To increase reactive strength and optimize the use ofstored energy for power production, plyometric exercises,such as depth jumps with a focus on maximum height andminimum contact time, should be used (25), because theyfocus heavily on the rapid change from eccentric to con-centric contractions. Depth jumps have been shown to bemore effective in increasing power output, force produc-tion, and jumping performance than other VJ tests suchas countermovement jumps (25). Quality of training is im-portant for the development of reactive strength, becausean increase or decrease from an individual's optimaldepth jump height has been found to decrease perfor-mance and therefore improperly overloading the function-ing ofthe SSC (25). Furthermore, specificity is a crucialfactor, because research has shown that eccentricstrength can be improved by 3 times as much througheccentric training than concentric training (13). Further-more, eccentric training protects muscle from injury dur-ing subsequent eccentric contractions (5, 17, 22, 26).Therefore, preceding plyometric training with specific ec-centric training may decrease the risk of injury and de-crease the time needed for recovery because of decreasedmuscle damage.

VERTICAL JUMP AND TRArNiNC, IMPLICATFONS 969

Concentric Action Power of the Legs

Increasing the ability to produce power (P = F X d/t) dur-ing the concentric action of the take-off phase of a VJtakes on 2 main pathways: (a) strengthening specificmuscle groups and (b) alteration of specific neuromuscu-lar parameters (23). A recent simulation study found thatstrengthening of the knee fiexor muscle group and in-creasing the neuromuscular property f̂ .,, had the largesteffect on VJ performance compared with increasing plan-tar flexors or hip extensors and increasing maximum ve-locity (V,,̂ ,J or activation capacity, respectively (23). In-terestingly, strengthening the plantar flexors, hip exten-sors, and knee fiexors and increasing F^,^^, V,,,̂ ,̂ and ac-tivation capacity simultaneously corresponded to agreater increase in jump height than the sum of the in-dividual muscles or parameters (23). This study was,however, based solely on the squat jump, and therefore,its application to jumps requiring a run-up is limited. In-deed, this is especially pertinent when considering that apoor coiTelation exists between ^.^^yweight and both adouble-leg countermovement jump with no run-up and asingle-leg jump from a run-up (32). Conversely, Jaric etal. (14) found a significant correlation between both rateof force development (RFD), which is similar to V̂ ,̂ ^ andF,„^ of different muscle groups, and counter-movementjump. However, no significant correlations were found be-tween RFD and F,,,,,.,, reiterating the independence ofstrength qualities. Despite these inconsistencies, the con-tribution of f̂ mx to jumps involving a SSC is not to beunderestimated, because an 80-ms isometric contractionseparates tbe stretch and shortening phases (9). In sup-port of this, a number of training studies have shown thatthe gains in VJ performance are similar for heavy resis-tance strength and high-velocity SSC training methods(32). Although it is likely that increases in F,,,^^ of between20 and SO'/e after strength training programs lasting be-tween 2 and 6 months are possible (8, 13, 18, 24), anyweight gains caused by increased muscle cross-sectionalarea may counter the effects of increased F,̂ .,̂ on VJ per-formance. Although an increase in weight is a negativeside affect of hypertrophy training, an increase in musclesize should not always be avoided. Increases in the cross-sectional area of vastus lateralis are paralleled by in-creases in muscle fibre pennation angle, whicb places themuscle fiber in a mechanically superior position and in-creases the muscles capacity to produce force (1).

Although it is clearly advantageous to increase theconcentric action power of the legs, it is important to notethat, for the benefits of increased strength and power inindividual muscles to be fully realized in VJ performance,reoptimization of the muscle activation patterns is re-quired {4, 23).

Training Applications

Temporarily discounting the effect of body weight on VJheight, it would seem that increasing the muscle cross-sectional area of the ankle plantar fiexors, knee exten-sors, and hip extensors to increase F,,,̂ , and, in some cas-es, increase tbe pennation angle for mechanical superi-ority should be a primary focus of training for squat jumplike movements. Although the role of F,,,̂ ^ in jumps in-volving a SSC is not as significant as it is in a squat jump,it is likely that this strength quality is of at least somerelevance because of the presence of isometric contrac-tions in the SSC. To achieve optimum improvement inconcentric action power of the legs, all relevant muscle

groups and strength qualities must be improved in simi-lar proportions. The most important training applicationwhen attempting to increase concentric action power ofthe legs and subsequent improvement in VJ height is thatmuscle activation patterns must be reoptimized for trans-fer of increased strength qualities to increased VJ perfor-mance. Reoptimization can be achieved through incorpo-ration of jumping movements throughout the trainingprogram so that the athlete can learn how to use thestrengthened muscles.

Hip Flexor PowerThe effect of hip flexor power on VJ performance is rele-vant for sports involving a single leg jump, sucb as higbjump; however, because this type of jump is prevalent inmany sports, it is an important factor to consider. In astudy on single and double leg jumps, all subjects wereahle to jump 75% of tbeir standing double leg VJ heightwith right and left legs separately witb tbe use of bothan arm and free leg swing (33). Altbougb others have re-corded single leg jumps that were only 58.5% of doubleleg scores, the use of arms and the free leg was restricted(29). The large discrepancy between these values may beexplained by the benefits derived from the swing of thefree leg. Although the effect of the free leg was quite ap-parent during a stationary single leg jump, its infiuenceon VJ height during the much more sport-related run-upjump is not as clear. In a study investigating the relativecontribution of tbe free limbs to wbole body vertical mo-mentum, the free limbs were judged to contribute 7.1%;however, it was suggested that the arms provided a great-er contribution to this value than the lead leg (21).

Training ApplicationsAlthough it is likely that hip-flexors of the free leg con-tribute to performance in a single leg run-up jump, thecontribution is likely to be fairly small In sports such ashigh jump, where every edge for performance is critical,it is recommended that athletes spend some time devel-oping the hip fiexors; bowever, there are other trainablefactors that will elicit far greater improvements in per-formance. For this reason, development of hip fiexor pow-er should not be a primary focus of training for increasedVJ performance. In contrast, the free leg's contribution totake-off height through elevation should not be underes-timated.

Shoulder PowerSeveral studies have undergone investigation into the in-fiuence of arm swing during different types of VJ. Armswing bas been found to increase jump height by 10-27%,depending on the type of jump and type of athletes se-lected in the protocol (7, 11, 16, 20, 27). Tbere have beena number of reasons given for the apparent increase inVJ height associated with an arm swing. The fact thatarm swing increases vertical ground reaction impulse(VGRI) is widely accepted; however, theories on bow tbisis accomplished vary. Kinematic changes occurring withan arm swing are also thought to contribute to increasedVJ performance. Unlike VGRI, the contribution of kine-matic changes to increased VJ height is both supported(11) and refuted (16) throughout the literature.

Research has shown that arm swing increases the ten-sion placed on each segment of tbe jump, allowing forimproved functioning of the SSC, particularly in the legextensors, and a greater total VGRI (11, 16). Although itis clear that arm swing increases VGRI, observations of

970 HAM, KNEZ, AND YOUNG

jumpers have shown that, in most cases, the arms returnto zero vertical velocity relative to the rest of the hody astbey approach the fully raised position before take-off,meaning the net effect on VGRI should be about zero (11).Despite the use of an arm swing contributing very littleto VGRI, it can place the muscles of the take-off leg(s)under faster eccentric conditions and slower concentricconditions depending on the period of the take-off phase(6), which is advantageous when considering the f'orce-velocity relationship of muscle contraction (11). However,if the athlete's take-off leg is already near the thresholdfor buckling, stronger arm actions could take the athletebeyond the optimum and result in a suboptimal jump per-formance (6). The upward/backward swing of the armspartially offsets the downward acceleration of the non-arm body mass, causing a decrease in unweighting duringthe countermovement (11). When the muscles crossingthe hip and knee are in the most advantageous positionto exert vertical ground reaction force (VGRF), the up-ward acceleration of the arms creates a downward forceat the shoulders on the rest of the body, slowing the con-traction of the large quadriceps and gluteal muscles tovelocities at which they can exert more force (11). By thetime the arms decelerate near the end of their swing,wbich makes the quadriceps and gluteals contract at amore rapid speed and consequently diminishing forcegeneration capability, tbe knees and hips are almost fullyextended, and the muscles around them are not in a po-sition to generate much positive VGRF (11). Arm swingmay also contrihute to an increased VJ height through anincrease in pre-take-off total body center of mass (TBCM)displacement. Interplay between peak pre-take-off TBCMrise and increased VGRI has been suggested because ofincreases in VJ height after an arm swing that exceedincreases in VGRI alone (11).

Training ApplicationsUltimately, arm swing during a countermovement jumpincreases the TBCM rise; however, the interplay betweenkinematic changes and tbe increase in VGRI has yet tobe quantified. Shoulder power training should be includ-ed in an exercise program with a focus of increasing VJheight. However, in sports with a single-leg jump from arun-up, such as high jump, where the leg extensors arealready placed under considerable eccentric load, tbe ex-tent to which shoulder power can increase jump height isultimately reliant on the eccentric strength of the ath-lete's legs and should therefore be reflected in training.

CATEGORY 2—SKILLS COACH

Body Position—Take-off and Reach HeightThe capacity to jump and reach for height is an importantattribute in many sports (32) and can be improvedthrough a number of methods. Alteration of body positionduring both the pre-take-off, whicb has been discussedpreviously, and post-take-off phases can significantly im-prove VJ performance. During the post-take-off pbase ofa VJ-and-reach test an athlete will thrust the nontoucb-ing arm downward while reaching up with the other arm(11). Lowering 1 arm and fully extending the legs willlower the position of TBCM within the body and henceallow the raised arm to be further from the TBCM andhigher from the ground. This redistribution of body massduring the flight phase of the VJ can only be enabledthrough body position alterations during the pre-take-off'phase. Raising of the free limhs (the arms for a double

leg jump and the arms and non-take-off leg for a singleleg jump) causes a positive vertical shift in the TBCMwithin the body and is an imperative prerequisite forpost-take-off internal TBCM lowering.

Training ApplicationsAlthough the positive effects of increasing pre-take-ofFTBCM have yet to be quantified, it is clearly beneficial toVJ height (11). Raising the limbs during the pre-take-offphase and lowering of all but 1 arm (where possible) dur-ing the post-take-off phase also increases VJ reachheight. Although tbese benefits may be small, trainingthis technique is likely to be far less time consuming thanmaking physiological adaptations and should be includedin training where applicable. Obviously the extent towhich body position can be used to increase the TBCMdepends on the requirements of the jump witbin thesporting context. Indeed, a block in volleyball will not al-low for these adjustments.

Lowering Center of Mass Before Take-offSome investigators have proposed that maintaining a lowcenter of mass at the end of the approach run also im-proves the beigbt of a VJ (6). The effect of lowering thecenter of mass hefore take-off was found by Dapena et al.(4) to be very similar in effect to increasing run-up speed.If the approach is very low and the take-off leg is notstrong enough, the joints of the take-off leg will be forcedto fiex excessively during the first half of tbe take-offphase, and this will weaken the subsequent extension ofthe leg (6). Therefore, the extent to whicb an athlete isable to lower his/her center of gravity at the end of tbeapproacb run without forcing the take-off leg to buckle isdependent on the strengtb capabilities of the take-off leg.These findings refer specifically to single leg jumps froma run-up, and it is unclear whether tbey will be practi-cally applicable for double leg jumps.

Training ApplicationsBecause run-up speed and lowering of the center of massbefore take-off both rely on strengtb capabilities of thetake-off leg, it would seem logical that there be an opti-mum combination of run-up speed and center of massheight for individual athletes. If the optimum combina-tion of run-up speed and center of mass height is slowand high, strength qualities, particularly eccentric, mustbe worked on, because it means the take-off leg is weak.

Body MassAlthough hypertrophy of the leg muscles has been shownto improve VJ performance, it is important to note thatany gain in mass without an associated increase instrength could negatively influence performance. Jump-ing for maximum height is a powerful action, dominatedby fast twitch fiber contraction, therefore it is importantto avoid excessive hypertrophy of slow twitch fibers, es-pecially in jumps involving low contact times. The capac-ity to produce force during a maximal voluntary contrac-tion is determined by the cross-sectional area and acti-vation ability of the relevant muscles (29); it is thereforeimportant to accompany gains in muscle size with main-tenance or increase in activation percentage of the asso-ciated muscles.

Training ApplicationsNon-force-producing tissue is useless for the VJ andshould be minimized. Diet should be monitored along

VERTICAL JUMP AND TRAINING IMPLICATIONS 971

witb training to ensure a low body fat percentage. Train-ing programs should aim to maximize the activation per-centage of all relevant muscles, especially after hypertro-phy training. Furthermore, hypertrophy of irrelevantmuscle groups and slow twitch fibers should be avoided.

Take-off Time

Although, in theory, a greater take-off time would allowfor a greater impulse (force x time = impulse) in relationto reactive strengtb, a short contact time is imperativefor storing and reusing elastic energy during the SSC injumps with a run-up, particularly if the run-up is fast.Noteworthy, contact time increases from single leg jumpswith a run-up to double leg countermovement and squatjumps, tbe relative importance of the reuse of elastic en-ergy decreases (3, 19), and tbe importance of the musclesability to produce force during the concentric musclephase increases.

Training Applications

Take-off times during plyometric exercises are imperativeto specific training of different strength qualities andmuscle functions. As previously stated, long and shortSSCs are independent muscle functions, and training for1 type will not necessarily improve performance in theother. Subsequently, take-off times used in trainingshould mimic tbose in competition.

PRACTICAL APPLICATIONS

Training Applications for Double Leg JumpsFor double leg VJs from a stationary position, the relativeinvolvement of the SSC is small, and training shouldtherefore focus on increasing the concentric action powerof the ankle plantar fiexors, knee extensors, and hip ex-tensors simultaneously, with knee extensors being themajor contributor. This should be achieved throughweight training that increases maximum force throughincreased cross-sectional area.

Training Applications for Single Leg Jumps

The ability to jump with a single leg take-off from a run-up can be improved by increasing the threshold for buck-ling that effectively changes hody position and alters theheight of tbe center of mass. The limiting physiologic fac-tor is the ability of the take-off leg to resist and store largeamounts of force during the eccentric phase and the ef-fective reuse of this stored energy in the subsequent con-centric phase. This can be achieved through specific ec-centric training coupled with plyometric exercises, suchas depth jumps that optimize the use of stored energythrough a focus on the rapid change form eccentric to con-centric contraction conditions. Also, preceding plyometrictraining with specific eccentric training may decrease therisk of injury and increase recovery time. Further im-provements can be achieved by raising and subsequentlowering of the arms and free leg during the pre- andpost-take-off phases, respectively. Tbis increases VJheight by increasing the pre-take-off TBCM rise and in-creasing reach height.

The VJ is a physiologically and biomechanically com-plex movement. It relies on the interplay of many physi-cal properties that combine and alter the magnitude oftheir contribution to affect different jump types. Trainingaimed at increasing VJ height needs to be bighly specificto the type of jump required in athletic competition.Through proper strength and power training and tech-

nique adaptations, VJ performance can be improved sub-stantially.

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Address correspondence to Dr. Daniel J. Ham, d.ham®ballarat.edu.au.