Beyond Imbalance: An Elasticity Framework for the Distance-averaged Force-Velocity Relationship in Vertical Jump

This study aimed to (1) establish the distance-averaged F-V relationship framework and (2) develop elasticity metrics that quantify how F-V relationship variables govern jump height and inform training prescription. Theoretical derivation and experimental validation across 108 F-V relationship models derived from 1578 jumps (countermovement jump and squat jump at three knee angles; 20 well-trained subjects) yielded a standard error of 2.1% and a nearly perfect correlation (r = 0.96, p < 0.001) between measured and predicted jump height. Four elasticity metrics were formulated: force elasticity (F_{e}), the elasticity of jump height to maximal force (F_{0}); velocity elasticity (v_{e}), the elasticity of jump height to maximal velocity (v_{0}); the force-velocity elasticity norm {(\mathrm{F}-\mathrm{V}}_{\mathrm{EN}}=\sqrt{F_{e}^{2}+v_{e}^{2}}), reflecting the overall sensitivity of jump height to changes in F-V relationship variables; and the force-velocity elasticity ratio {(\mathrm{F}-\mathrm{V}}_{\mathrm{ER}}=F_{e}{\div v}_{e}), indicating which variable dominates the jump height response. Simulations and experiments revealed that F_{e} bore an inverse relationship to F_{0}, and v_{e} was inversely related to v_{0}, reflecting diminishing marginal returns. At a fixed jump height, simulations showed {\mathrm{F}-\mathrm{V}}_{\mathrm{EN}} and {\mathrm{F}-\mathrm{V}}_{\mathrm{ER}} displayed a U-shaped relationship; a balanced profile ({\mathrm{F}-\mathrm{V}}_{\mathrm{ER}}=1) did not always correspond to the lowest {\mathrm{F}-\mathrm{V}}_{\mathrm{EN}}. The distance-averaged F-V elasticity framework offers a physically grounded and quantitative tool for linking F-V relationship variables directly to jump performance, providing a basis for informing individualized training decisions.