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, ignoring the curvilinear trajectory imposed by the ankle pivot and systematically biasing flight-time and height estimates. Methods: A planar rigid-body pendulum pivoting about the ankle axis was formulated via two independent derivation pathways (static moment equilibrium and a gravitational-torque coordinate approach), yielding effective pendulum length L = (MW/M) × LOS. Closed-form expressions for flight time, arc displacement, maximum height, and mean mechanical power were derived analytically from energy conservation and compared against free-fall predictions across seven pendulum arm lengths (LOW = 0.50–2.00 m) and 500 initial hand velocities per length, using adaptive Gauss–Kronrod quadrature (relative tolerance 10−10) with ODE cross-validation (maximum discrepancy < 2.5 × 10−7 s). Results: Flight time equivalence (tH = tG) was formally established. The free-fall model overestimated flight time by up to 18.82% (Δt = 0.096 s; LOW = 0.50 m, VH,0 = 2.50 m/s) and maximum height by up to 28.43% (Δh = 0.087 m; LOW = 0.50 m, tflight = 0.50 s), with both errors growing nonlinearly with initial velocity. Overestimation in height was proportionally larger at shorter pendulum arm lengths (18.18% at tflight = 0.30 s for LOW = 0.50 m vs. 10.91% for LOW = 1.00 m). Conclusions: The pendulum model provides a physically consistent, analytically tractable framework for geometry-adjusted upper-body power assessment from four field-obtainable anthropometric inputs. These results reflect computational self-consistency; prospective experimental validation against force-plate kinematics is required before applied deployment. Prospective empirical validation against dual force-plate and motion-capture reference data is required to establish the model’s accuracy boundaries under real push-up kinematics.