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 decisions. 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 systematic error of - 2.1% and a strong linear association (r = 0.96, 95% CI [0.93; 0.97]) between measured and predicted jump height. Four elasticity metrics were formulated: force elasticity ([Formula: see text]), the elasticity of jump height to maximal force ([Formula: see text]); velocity elasticity ([Formula: see text]), the elasticity of jump height to maximal velocity ([Formula: see text]); the force-velocity elasticity norm [Formula: see text], reflecting the overall sensitivity of jump height to changes in F-V relationship variables; and the force-velocity elasticity ratio [Formula: see text]), indicating which variable dominates the jump height response. Simulations and experiments revealed that [Formula: see text] bore an inverse relationship to [Formula: see text], and [Formula: see text] was inversely related to [Formula: see text], reflecting diminishing marginal returns. At a fixed jump height, simulations showed [Formula: see text] and [Formula: see text] displayed a U-shaped relationship; a balanced profile ([Formula: see text] = 1) did not always correspond to the lowest [Formula: see text]. 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.