{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [
    "Beijing Polytechnic College"
  ],
  "article_type": "New results",
  "arxiv": null,
  "author_total": 6,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Z. Li"
    },
    {
      "affs": [
        0
      ],
      "name": "M. Li"
    },
    {
      "name": "X. Zhang"
    },
    {
      "name": "Z. Chen"
    },
    {
      "name": "L. Yang"
    },
    {
      "affs": [
        0
      ],
      "name": "Q. Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f7k6bx2ppd5vq.bib",
  "cited_by": 0,
  "doi": "10.64898/2026.09.01.748530",
  "fulltext_url": "https://www.biorxiv.org/content/10.64898/2026.09.01.748530v1.full",
  "id": "f7k6bx2ppd5vq",
  "issns": [],
  "landing_url": "https://www.biorxiv.org/content/10.64898/2026.09.01.748530v1",
  "language": null,
  "licence": "CC-BY-NC",
  "licence_label": "CC BY-NC",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "biorxiv",
      "biophysics"
    ],
    [
      "openalex",
      "2732|Sports Performance and Training"
    ]
  ],
  "openalex": "W7207757721",
  "pdf_url": "https://www.biorxiv.org/content/10.64898/2026.09.01.748530v1.full.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f7k6bx2ppd5vq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:37Z",
      "native_id": "10.64898/2026.09.01.748530",
      "source": "biorxiv",
      "url": "https://www.biorxiv.org/content/10.64898/2026.09.01.748530v1"
    }
  ],
  "subjects": [
    {
      "field": "biochemistry",
      "origin": "biorxiv:biophysics",
      "subfield": "biophysics"
    }
  ],
  "title": "Beyond Imbalance: An Elasticity Framework for the Distance-averaged Force-Velocity Relationship in Vertical Jump",
  "title_text": "Beyond Imbalance: An Elasticity Framework for the Distance-averaged Force-Velocity Relationship in Vertical Jump",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:37Z",
  "url": "https://research.draconitic.com/paper/f7k6bx2ppd5vq",
  "venue": "bioRxiv",
  "versions": []
}