A Rigid-Body Pendulum Model for Plyometric Push-Up Biomechanics: Analytical Derivation and Numerical Quantification of Flight Time, Arc Displacement, Maximum Height, and Mechanical Power Output

Aim: Conventional free-fall kinematic models applied to plyometric push-up assessment treat the upper body as a vertically translating point mass, a simplification that ignores the curvilinear, arc-like trajectory imposed by the ankle pivot and systematically biases flight-time and height estimates. This study developed and analytically validated a novel rigid-body pendulum model to quantify plyometric push-up performance, deriving closed-form expressions for flight time, arc displacement, maximum height, and mean mechanical power at both the hand and whole-body center-of-mass reference levels. Methods: A planar rigid pendulum pivoting about the ankle axis was formulated using two independent derivation pathways, static moment equilibrium and a gravitational-torque center-of-mass coordinate approach, yielding the effective pendulum length L=(MW/M)×LOS. All performance indices were derived analytically from conservation of mechanical energy. Numerical simulations were conducted in R across seven pendulum arm lengths (LOW=0.50–2.00 m) and 500 uniformly spaced initial hand velocities per length, using adaptive Gauss-Kronrod quadrature with relative tolerance 10-10 and independent ODE cross-validation (maximum inter-method discrepancy <2.5×10-7 s). Free-fall and pendulum model predictions were compared parametrically across the full physiologically admissible parameter space. Results: Both derivation pathways operationalize identical static rotational equilibrium conditions and yield the effective pendulum length (below); the geometric deviation between dOG and L remains below 4% for θ₀ ≤ 16°. Flight time equivalence between hands and center of mass (tH=tG) was formally established. The free-fall model systematically overestimated flight time by up to 18.82% (Δt=0.096 s at LOW= 0.50 m, VH,0=2.50 m/s) and maximum height by up to 28.43% (Δh=0.087 m at LOW= 0.50 m, t=0.50 s), with both errors increasing nonlinearly with initial velocity and flight time. Overestimation in height was proportionally greater at shorter pendulum arm lengths, reaching 18.18% at t=0.30 s for LOW=0.50 m versus 10.91% for LOW=1.00 m under identical conditions. Conclusion: The pendulum model provides a physically consistent, analytically tractable, and computationally validated framework for plyometric push-up performance assessment. It resolves the structural overestimation errors of the free-fall simplification, requires only four anthropometric measurements obtainable in field conditions, and supplies geometry-adjusted performance indices that improve measurement accuracy, particularly for athletes with shorter effective arm lengths or high take-off velocities.