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

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1

The manuscript presents an original and well-founded mechanical model for analyzing the kinematics of plyometric push-ups, replacing the commonly used free-fall assumption with a more physically consistent pendulum-based framework. The study is characterized by strong theoretical rigor, clear structure, and a consistent analytical approach. A major strength of the work is the conceptual shift from treating the body as a vertically moving point mass to modeling it as a rigid body rotating about a fixed axis. This provides a more realistic description of the motion trajectory during the flight phase. Deriving key performance variables in analytical form improves both the clarity and applicability of the model. The methodology is robust, including two independent derivation approaches that converge to the same result, demonstrating internal consistency.Numerical validation further supports the reliability of the analytical framework. The results clearly show that the conventional free-fall model systematically overestimates key performance indicators, with errors increasing with increasing performance level. This has important implications, especially for the assessment of more advanced athletes. The limitations of the model are properly acknowledged, which strengthens the credibility of the study. The presented practical relevance of the work is also impressive.

We sincerely thank Reviewer 1 for the thorough and encouraging assessment of the theoretical framework, the dual-derivation approach, and the numerical implementation. The reviewer's recognition of the practical implications of nonlinear error growth at higher performance levels confirms that the manuscript's core contribution is clearly communicated.

No corrective action required.

2

Although I consider the manuscript to be of very good quality, I have a few minor remarks that could further improve its clarity and impact, such as: 1) The abstract is a bit long and could be shortened to more concisely emphasize the main results and their practical relevance; 2) Future research directions, although mentioned, would benefit from being more clearly summarized in a dedicated concluding paragraph;3) Some minor condensations of repetitive arguments in the discussion could further improve readability. Overall, the manuscript makes a contribution to the biomechanics of upper body strength assessment.

We thank the reviewer for these targeted observations.

Regarding point 1: the abstract has been condensed from approximately 280 to 200 words. All quantitative findings and the practical implication statement are retained; redundant methodological descriptors and the intermediate result on dOG deviation have been removed. A sentence explicitly distinguishing computational self-consistency from experimental validation has been added to the abstract Conclusion.

Regarding point 2: the paragraph beginning "Future investigations should extend the pendulum framework..." has been given a dedicated subheading "Future Research Directions" and expanded with an opening sentence specifying the highest-priority next step, namely prospective experimental validation employing synchronized dual force-plate and motion-capture data across representative anthropometric samples.

Regarding point 3: three redundancies in the Discussion have been resolved. Paragraphs 3 and 5, both addressing nonlinear error growth with performance level, have been merged into a single condensed paragraph without loss of quantitative content or citations. The practical deployability paragraph has been condensed by removing the enumeration of the four anthropometric measurements, already specified in the Abstract and Section 2.2, while retaining the ICC reliability data. The dual-framework paragraph has been condensed to retain only the non-redundant content on normalization covariates.

We changed the Abstract from the original 280-word version to the following (~200 words):

"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). Conclusion: 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."

We inserted the subheading "Future Research Directions" immediately before the paragraph beginning "Future investigations should extend the pendulum framework to accommodate..." and inserted the following opening sentence: "Priority should be given to prospective experimental validation employing synchronized dual force-plate and three-dimensional motion-capture data across representative anthropometric samples, to establish empirical boundaries on the model's accuracy relative to observed push-up kinematics."

We merged Discussion paragraphs 3 and 5 into the following single condensed paragraph: "For LOW = 1.00 m, representative of an adult male of average stature, Δt = tFF − tH reached 0.016 s at VH,0 = 1.50 m/s and 0.082 s at VH,0 = 3.00 m/s (relative error: 13.4%). Applied to the Wang et al. [4] power prediction equation (Ppeak = 11.0 × M + 2012.3 × tflight − 338.0; R² = 0.658, SEE = 150 W), a 10% overestimation of tflight at ≈ 0.35 s yields a power overestimation of approximately 70 W, approaching but not exceeding the regression's inherent uncertainty (SEE = 150 W). Because the error grows nonlinearly with VH,0, elite athletes are disproportionately affected, rendering the free-fall model least reliable precisely where measurement precision is most consequential. These findings align with Dhahbi et al. [9] and Sha and Dai [8], who reported that single-platform free-fall methods overestimated whole-body velocities by 54.4% and power by 58.3% relative to a two-platform reference."

We changed the practical deployability paragraph to: "The model requires four anthropometric inputs (M, MW, LOS, LSW), from which L and θ₀ are algebraically determined, enabling integration into existing force-plate or contact-mat testing workflows without modification of the flight-time measurement protocol. Reported ICC values of 0.80–0.96 for force-plate-derived tH [1,2] confirm that flight time is sufficiently reliable to support pendulum-model computation of corrected performance indices."

We changed the dual-framework paragraph to: "The dual reference framework (hand-referenced, Section 3.2; CoM-referenced, Section 3.3) supplies individualized performance indices incorporating pendulum geometry and total system mass, a correction absent from published prediction equations [4]. Parametric sensitivity across LOW further supports incorporating LOW as a body-size normalization covariate alongside body mass, given that hmax and Shand are explicit functions of LOW."

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After considering the minor comments above, I believe the manuscript would be further strengthened and is suitable for publication.

We thank Reviewer 1 for the conditional acceptance and for the precision of the observations raised. We trust that the revisions described above fully address all three concerns.

No additional corrective action required beyond those described under Comment 2.

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1

This manuscript addresses an interesting and relevant problem: the common use of free-fall equations to estimate plyometric push-up performance from flight time, despite the fact that the movement is mechanically constrained and not purely vertical. The paper is original, mathematically ambitious, and clearly written in terms of technical effort. The central idea is valuable, and the attempt to provide a more mechanically consistent framework for upper-body ballistic assessment is potentially useful for both researchers and practitioners.

We sincerely thank Reviewer 2 for this assessment and for the rigor of the subsequent critique, which has substantially strengthened the epistemic framing of the manuscript.

No corrective action required.

2

That said, in its present form the manuscript remains primarily a theoretical and computational exercise, and this limitation is more important than the current wording sometimes suggests. The paper repeatedly uses language such as "validated" or implies practical superiority, but what is actually shown is internal analytical consistency and numerical agreement between two computational approaches. This is not the same as experimental validation. The model may well be promising, but at this stage it has not yet been demonstrated that it predicts real push-up kinematics better than existing approaches under empirical conditions.

We thank the reviewer for this precise and important distinction. The reviewer is entirely correct: the manuscript demonstrates internal analytical consistency and cross-verification between two independent computational implementations, which establishes computational self-consistency, not empirical validity relative to observed human kinematics. All epistemically inflated language has been corrected throughout.

We changed "analytically validated" to "analytically derived" throughout the Abstract and Section 3.1.

We changed "computationally validated framework" to "computationally self-consistent framework" in the Abstract and Section 6.

We changed "analytically and numerically validated expressions" to "analytically derived and numerically verified expressions" in Section 3.1.

We inserted the following sentence at the close of the Abstract Conclusion and Section 6: "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."

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My main concern is therefore conceptual rather than stylistic: the manuscript sometimes presents the pendulum model as if it were already established as the correct practical solution, while the discussion itself later acknowledges that the body is not truly a simple pendulum and that the compound-pendulum formulation would be more physically correct. This is not a minor detail. It directly affects the central claim of the paper, because if the simple-pendulum assumption underestimates the equivalent length and alters flight-time predictions, then the magnitude of the reported correction relative to free-fall may also change. In other words, the paper shows that free-fall is oversimplified, but it does not yet fully prove that the proposed simplified replacement is sufficiently accurate for applied use.

We thank the reviewer for this conceptually important observation, which is fully warranted. The simple pendulum formulation adopted here uses the torque-equivalent effective length L = (MW/M) × LOS and explicitly neglects the distributed rotational inertia ICoM; Section 2.3.3 and the Limitations paragraph acknowledge that the compound-pendulum length Leq = IO/(M × LCoM) exceeds L by approximately 11–33% across physiological anthropometry [10]. The manuscript's claim is precisely and only that the simple pendulum constitutes a physically better-motivated first approximation than the free-fall simplification, by correctly capturing the rotational ankle-pivot constraint and the curvilinear flight trajectory. It does not claim to be the definitive or experimentally validated replacement. The revised Conclusion makes this hierarchy of approximations explicit.

We changed the Conclusion (Section 6) closing sentences from: "These findings establish a physically rigorous basis for geometry-adjusted upper-body power assessment from plyometric push-up flight-phase measurements. These results establish the theoretical and computational basis for a standardized upper-body power assessment instrument grounded in rotational kinematics, mechanically distinct from vertical jump methodology, pending prospective empirical validation."

to: "These findings establish the theoretical and computational basis for a geometry-adjusted upper-body power assessment instrument grounded in rotational kinematics, mechanically distinct from vertical jump methodology. 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."

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A second important issue is internal consistency. In Section 4.2, the derivation and interpretation of maximum height appear somewhat confusing. Equation (67) defines pendulum-model height as V^{2}/2g while the text states that this comes from the center-of-mass-referenced equation. However, the center-of-mass formulation earlier in the manuscript includes additional geometric factors. This creates ambiguity about whether the reported height in that section refers to hand height or center-of-mass height. That needs to be clarified carefully, because it affects the meaning of the comparisons in Table 4 and Figure 3.

We thank the reviewer for identifying this genuine internal inconsistency. The reviewer is correct: equation (67) takes the form hmax,P = [VH,0^(P)]²/(2g), which corresponds to the hand-referenced height formula (equation 31), not the CoM-referenced formula (equation 43). Equation (43) includes the additional geometric scaling factor (MW/(M cosθ0))² and is structurally distinct from the V²/2g form. The original citation of equation (43) in the description of equation (67) was an error that propagated ambiguity into Table 4 and Figure 3. The corrected text makes explicit that Section 4.2 compares hand-referenced predictions exclusively.

We changed the text following equation (67) from: "where the numerator comes from the center-of-mass-referenced height equation (43) evaluated at the pendulum-consistent velocity VH,0^(P)."

to: "where the expression applies the hand-referenced height formula (equation 31) evaluated at the pendulum-consistent initial hand velocity VH,0^(P). The comparison in Section 4.2 thus contrasts hand-referenced predictions exclusively: the free-fall prediction hmax,FF = gt²/8 (equation 65, equivalent to [gt/2]²/(2g)) and the pendulum prediction hmax,P = [VH,0^(P)]²/(2g). The CoM-referenced height (equation 43) involves an additional geometric scaling factor (MW/(M cosθ0))² and constitutes a distinct performance index not compared in this section."

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Related to this, Figure 3 appears problematic. The plotted y-axis reaches unrealistically large values, far beyond the physiologically relevant flight-time range stated in the caption. This gives the impression either of a plotting error or of a mismatch between the data displayed and the limits described in the text. That figure should be checked and likely redrawn, because in its current form it weakens confidence in the numerical presentation.

We thank the reviewer for identifying this error. The reviewer's diagnosis is correct: both panels of Figure 3 display x-axes extending to 6.0 s, whereas the caption correctly specifies restriction to tflight ≤ 0.60 s, consistent with reported plyometric push-up flight times [4]. The discrepancy is a plotting error introduced during figure generation. The figure has been redrawn with the x-axis restricted to the physiologically admissible domain. Within this domain, both hmax,FF and hmax,P take values in the range 0–0.30 m, which are mechanically realistic. The caption requires no alteration.

We replotted Figure 3 by applying xlim = c(0, 0.60) to both panels in the R visualization code. The caption text "Results are restricted to tflight ≤ 0.60 s" is unchanged and now correctly describes the displayed range.

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The power calculations should also be treated more cautiously. The manuscript derives mean power using a quarter-period approximation that is explicitly acknowledged to be valid only in the small-angle limit, yet the broader paper emphasizes errors that become larger at higher velocities and larger angular excursions. This means the power results are probably the least secure part of the framework. I would recommend toning down those claims or clearly separating them as preliminary approximations rather than robust outputs of the model.

We thank the reviewer for this observation. The concern is well-founded: the quarter-period approximation for push-off duration (equations 33 and 45) is valid for φmax,H ≪ 1 rad, precisely the regime where the model's central argument -- nonlinear growth of kinematic errors with initial velocity -- is most consequential. The power derivations were presented as numbered performance indices on equal footing with the kinematic results, which overstated their robustness. The revised manuscript repositions them explicitly as first-order analytical estimates requiring numerical treatment under high-velocity conditions.

We inserted the following sentence at the opening of Section 3.2.4, before the first paragraph: "The following derivation constitutes a first-order analytical estimate, valid in the small-angle limit (φmax,H ≪ 1 rad); it is provided for reference and comparative purposes, not as a robust field-applicable output under large-amplitude conditions."

We inserted the following sentence at the opening of Section 3.3.4, before the first paragraph: "As with Section 3.2.4, the following expression is a first-order approximation applicable in the small-angle limit only; the same error bounds described in Section 3.2.4 apply."

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The manuscript would also benefit from tightening. It is well developed, but often more verbose than necessary, with repeated restatements of the same conceptual point. A more concise presentation would improve readability and make the key contribution stand out more clearly. The paper is strongest when it focuses on the basic geometric argument and the numerical comparison with the free-fall model.

We thank the reviewer for this assessment, consistent with the independent observation of Reviewer 1. The Discussion has been condensed by approximately 230 words through three targeted revisions, and the Abstract by approximately 80 words, as detailed under Comment 2 of Reviewer 1. No quantitative content or citations have been removed in this process.

As specified under Reviewer 1, Comment 2 (Discussion paragraph mergers and condensations). No additional text changes are required beyond those already described.

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Finally, the reference list should be checked carefully. At least one citation appears incomplete or insufficiently specified, and some supporting claims would benefit from more precise linkage to the cited literature.

We thank the reviewer for this verification prompt. All eleven references were reviewed against their respective DOIs. One incomplete entry was identified: Reference [9] (Dhahbi et al., 2017) was missing the journal name. All remaining references are complete and correctly specified.

We changed Reference [9] from: "Dhahbi, W.; Chaouachi, A.; Cochrane, J.; Cheze, L.; Chamari, K.; Cochrane Wilkie, J. Methodological Issues Associated With the Use of Force Plates When Assessing Push-ups Power. 2017, 31, doi:10.1519/JSC.0000000000001922."

to: "Dhahbi, W.; Chaouachi, A.; Cochrane, J.; Cheze, L.; Chamari, K.; Cochrane Wilkie, J. Methodological Issues Associated With the Use of Force Plates When Assessing Push-ups Power. J Strength Cond Res 2017, 31, doi:10.1519/JSC.0000000000001922."

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CORRECTIVE ACTIONS

1

In my view, this is a very interesting and ambitious paper. The author tackles a real and relevant problem the mechanical misrepresentation of plyometric push-ups when using free-fall models. The idea of modeling the movement as a rigid-body pendulum is original and, at least conceptually, well justified. The manuscript is clearly grounded in physics and shows a high level of analytical rigor. The derivations are detailed and, overall, logically consistent. I think the paper has strong potential, especially for readers interested in biomechanics and modeling.

We sincerely thank Reviewer 3 for this generous assessment and for the precision of the subsequent critique, which has materially improved the manuscript's accessibility and practical framing.

No corrective action required.

2

That said, the manuscript also has some important issues mainly related to clarity, practical applicability, and balance between theory and real-world relevance.

We thank the reviewer for this framing, which accurately identifies the three axes of revision addressed in Comments 3 through 7. Each issue is treated separately below.

No direct corrective action required for this comment; addressed through Comments 3–7.

3

Major comments 1. In my opinion, the biggest limitation is how the content is presented. The manuscript is extremely dense, especially in the Methods and derivation sections (e.g., Sections 2-3). Many equations are introduced with minimal intuitive explanation. The text often reads more like a theoretical mechanics paper than a biomechanics paper. For applied researchers or practitioners, this may be very difficult to follow. For example, the derivation of the effective pendulum length (pp. 5-7) is mathematically correct, but the practical meaning could be explained more clearly I think the paper would benefit from: Short intuitive summaries after each major derivation A clearer "what does this mean in practice?" explanation Possibly moving some derivations to supplementary material

We thank the reviewer for this observation. The concern is well-founded for a biomechanics readership that may not have a mechanics background. Two targeted interpretive summaries have been inserted following the two most consequential derivations (effective pendulum length and flight-time equivalence), and a brief lay definition of "gravitational torque equivalence" has been inserted at its first use. Full relegation of derivations to supplementary material was not pursued, as the analytical derivations constitute the primary contribution of the study and their placement in the main text is consistent with the journal's scope.

We inserted the following sentence after equation (12) in Section 2.3.3, between "Both yield: L = Leff = (MW/M) LOS" and "A precise conceptual distinction must, however, be drawn...":

"In practical terms, L scales linearly with the hand-supported mass fraction MW/M and the shoulder height LOS; for a subject supporting 55% of body mass at the hands with LOS = 1.10 m, L ≈ 0.60 m, which governs all subsequent energy-conservation derivations."

We changed "The physical resolution is that L in this model is not defined as the Euclidean distance from O to the geometric position of G in world coordinates. It is defined, consistently across both approaches, as the ratio of gravitational torque to total gravitational force"

to "The physical resolution is that L is not the Euclidean distance from O to G; it is the torque-equivalent pendulum length, defined as the ratio of gravitational torque to total gravitational force."

We inserted the following parenthetical after first use of "gravitational torque equivalence" in Section 2.3.2, changing "The governing criterion must be gravitational torque equivalence rather than geometric distance."

to "The governing criterion must be gravitational torque equivalence (i.e., the condition that the torque produced about O by the two-point mass system equals that of the actual distributed body) rather than geometric distance."

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2. The model is based on several strong assumptions (rigid body, planar motion, fixed ankle pivot, no energy loss) (pp. 7-8). While these are clearly stated, I think the implications are not discussed enough. Real push-ups involve joint flexion, segmental motion, and asymmetries The rigid-body assumption may not hold, especially in less trained individuals The "fixed ankle pivot" is a simplification that may not reflect real conditions It seems important to: Discuss how violations of these assumptions affect results Clarify for which populations (elite vs recreational) the model is valid

We thank the reviewer for this important observation. The existing assumptions section (Section 2.5) states the conditions but does not discuss their population-specific implications. Two targeted expansions have been made: the rigid-body and fixed-pivot assumption entries now include explicit statements on the population conditions under which each assumption is most likely to hold, and the Limitations paragraph in Section 5 has been expanded to address the direction and magnitude of expected deviations in less-trained individuals.

We changed the Rigid Body entry in Section 2.5 from: "This assumption is most closely approached when subjects maintain strict whole-body tension and avoid joint flexion during execution."

to: "This assumption is most closely approached in trained individuals who maintain strict whole-body tension throughout the flight phase. In recreational athletes, pronounced hip flexion or segmental motion during flight will produce deviations from the predicted pendular arc that the present model cannot quantify; for such populations, the model should be applied with caution and its outputs treated as upper bounds on the pendulum-consistent performance estimate."

We changed the Fixed Pivot entry in Section 2.5 from: "The ground reaction force at O passes through the pivot and generates no moment about it."

to: "The ground reaction force at O passes through the pivot and generates no moment about it. Minor translational displacement of the ankle at takeoff, which is more common in recreational than in trained push-up execution, would alter the effective pivot location and introduce a position-dependent error in L that the present model does not accommodate."

We inserted the following sentence in the Limitations paragraph of Section 5, after "The two-point mass model for CoM location introduces positional errors proportional to (1 − cosθ0)...":

"The rigid-body and fixed-pivot constraints impose a population boundary on the model's validity: it is mechanically most appropriate for trained athletes capable of maintaining whole-body alignment during the flight phase, and its accuracy for recreational populations remains to be established by experimental comparison."

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3. The manuscript claims that the model is "field-deployable" and requires only four anthropometric measures (Abstract) However, I think this point needs more clarification: How exactly would practitioners implement this model? Is there a simple formula or tool (e.g., calculator)? How sensitive are results to measurement errors (e.g., MW, LOS)? n my opinion, adding: A step-by-step application example A simple workflow or algorithm would greatly improve usability.

We thank the reviewer for this practical concern, which is also raised by Reviewer 4 (Comment 8). A numbered implementation workflow and a brief first-order sensitivity statement have been added to the Practical Recommendations subsection of Section 5. Measurement sensitivity is tractable analytically: because L = (MW/M) × LOS, a 5% error in MW propagates directly as a 5% error in L; for representative values (M = 75 kg, MW = 40 kg, LOS = 1.10 m), a 2 kg scale error yields ΔL ≈ 0.029 m (~4.8% of L), and a 1 cm tape error in LOS yields ΔL ≈ 0.005 m (~0.9% of L), confirming that standard field instruments (calibrated scale, segmental tape) are sufficient.

We inserted the following paragraph at the beginning of the Practical Recommendations subsection of Section 5, before the existing text:

"Practitioners may implement the model through the following sequential steps. (1) Record total body mass M with a calibrated scale in the standing anatomical position. (2) In the static push-up position (arms fully extended perpendicular to the floor), measure LOS (ankle axis to acromion), LSW (acromion to wrist center), and MW (scale reading under both hands). (3) Compute: θ₀ = arcsin(LSW/LOS), LOW = sqrt(LOS² − LSW²), and L = (MW/M) × LOS. (4) Measure tflight from a contact mat or force plate. (5) Recover the pendulum-consistent initial hand velocity VH,0^(P) by numerically inverting equation (56) using the computed LOW; this step requires a simple root-finding routine, implementable in any spreadsheet environment with iterative calculation enabled. (6) Compute hmax,P = [VH,0^(P)]²/(2g) and Shand = LOW × arcsin([VH,0^(P)]²/(2g × LOW)). First-order sensitivity: a 5% proportional error in MW or LOS propagates as a 5% error in L and a commensurate error in all derived indices; standard field-grade instrumentation (±0.1 kg scale, ±0.5 cm tape) produces errors below 2% in L for typical adult anthropometry."

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4. The introduction does a good job summarizing current methods (medicine ball throw, bench press throw, BPU) free-fall model pendulum model force plate methods However, the comparison between: could be more clearly structured I think a table summarizing: assumptions advantages limitations would help readers understand the added value of the proposed model.

We thank the reviewer for this constructive observation. Rather than inserting a separate comparison table, which would duplicate content already present across the Introduction and Section 2.5, the existing paragraph contrasting the free-fall and pendulum frameworks has been restructured to make the assumption-advantage-limitation triad explicit for each approach within the prose, preserving word economy while directly addressing the reviewer's concern.

We changed the paragraph beginning "Despite these recognized characteristics, the kinematic model universally applied to BPU flight-phase scoring continues to treat the body as a freely falling point mass, yielding tflight = 2V0/g and hmax = V0²/2g."

to: "Despite these recognized characteristics, the kinematic model universally applied to BPU flight-phase scoring treats the body as a freely falling point mass, yielding tflight = 2V0/g and hmax = V0²/2g. This free-fall approach assumes rectilinear vertical motion and requires no geometric inputs, but introduces a structural overestimation error that grows nonlinearly with initial velocity by disregarding the curvilinear, rotationally constrained trajectory of the push-up. Dual force-plate methods provide model-independent ground reaction force data and constitute the current empirical reference standard, but require two synchronized platforms and are not field-deployable for routine assessment. The pendulum model proposed here assumes a rigid body rotating about a fixed ankle pivot in the sagittal plane, requires four anthropometric inputs, and corrects the geometric source of the free-fall error analytically; its principal current limitation is the absence of prospective experimental validation against empirical kinematic data."

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Minor comments Some sentences are very long and difficult to read. Shortening them would improve clarity. The terminology is sometimes very technical (e.g., "gravitational torque equivalence")-a brief explanation would help non-physicists. Figures (e.g., Figure 1) are useful, but could be simplified visually for readability. The manuscript could benefit from a short "key takeaways" paragraph at the end of the Discussion.

We thank the reviewer for these minor but impactful observations. The lay definition of "gravitational torque equivalence" has been inserted as described under Comment 3. A "Key Findings" paragraph has been added at the close of the Discussion body, immediately before the Practical Recommendations subsection. Several multi-clause sentences in Section 5 have been split at natural boundaries.

We inserted the following paragraph immediately before the "Practical Recommendations" subheading in Section 5:

"Key findings may be summarized as follows. The free-fall model systematically overestimates flight time by up to 18.82% and maximum height by up to 28.43% across the physiologically admissible parameter space, with both errors growing nonlinearly with initial hand velocity and proportionally larger at shorter pendulum arm lengths. The effective pendulum length L = (MW/M) × LOS is derivable from four field-obtainable anthropometric measurements and governs all performance indices analytically. Flight time measured under the hands equals CoM flight time (tH = tG), allowing unmodified use of contact-mat data. All results reflect computational self-consistency; prospective experimental validation is required before clinical or field deployment."

We changed the long sentence in Section 5 beginning "For monitoring purposes across training cycles, reported ICC values of 0.80 - 0.96 for force-plate-derived flight time [1,2] confirm that tH is sufficiently reliable to detect meaningful performance changes when the underlying biomechanical model is correctly specified [11]. Flight-time-based power prediction equations [4] should be recalibrated using pendulum-consistent velocity values rather than free-fall-derived velocities to eliminate the systematic bias that otherwise produces disproportionate overestimation at high performance levels."

to: "Reported ICC values of 0.80–0.96 for force-plate-derived tH [1,2] confirm sufficient reliability for performance monitoring when the biomechanical model is correctly specified [11]; power prediction equations [4] should be recalibrated with pendulum-consistent velocities to eliminate the disproportionate overestimation at high performance levels."

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Strengths Original and innovative modeling approach Strong theoretical foundation Clear identification of a real methodological problem Detailed and rigorous derivations Potential high impact in biomechanics methodology

We sincerely thank Reviewer 3 for this affirmative summary, which confirms that the manuscript's core contribution is effectively communicated.

No corrective action required.

9

The discussion could be strengthened by incorporating literature on neuromuscular fatigue and variability, particularly studies examining physiological tremor or sensor-based performance monitoring, which may provide a useful bridge between mechanical modeling and real-world performance assessment.

We thank the reviewer for this forward-looking suggestion. The integration of inertial measurement unit (IMU) technology for real-time angular velocity capture at the ankle is already identified as a future direction in the manuscript (Section 5, Future Research Directions). To explicitly acknowledge the bridge the reviewer identifies, a sentence has been added to that paragraph connecting mechanical modeling with sensor-based performance monitoring and fatigue assessment.

We inserted the following sentence in the Future Research Directions paragraph, after the IMU point "(iii) integration with inertial measurement unit (IMU) wearable technology to enable real-time angular velocity capture at the ankle...":

"Such IMU integration would further permit examination of within-session kinematic variability attributable to neuromuscular fatigue, providing a mechanistic link between the pendulum-model performance indices and established sensor-based fatigue monitoring frameworks."

10

Overall, I think this is a strong but highly technical manuscript. The core idea is valuable and worth publishing, but the paper would benefit from revisions.

We thank Reviewer 3 for this overall assessment and for the precision of the comments provided. We trust that the revisions described above adequately address all concerns raised.

No additional corrective action required.

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CORRECTIVE ACTIONS

1

This paper proposes a rigid pendulum model to characterize the flight phase kinematics of a plyometric push-up. Through analytical derivation and numerical simulation, the differences between this model and the traditional free-fall model in terms of flight time, maximum altitude, and other indicators are compared. The authors attempt to address the systematic bias caused by neglecting rotational motion in existing testing methods. However, the paper has serious deficiencies in methodological rigor, model validation, clinical translational value of the results, and writing standards, making it difficult to meet journal publication standards.

We thank Reviewer 4 for the detailed critical evaluation. The specific concerns raised across Comments 2–8 are each substantive and have been addressed individually. The overarching concern, that a theoretical-computational study must clearly demarcate its epistemic scope relative to experimental validation, is valid and has been systematically addressed throughout the revised manuscript, as detailed below.

No direct corrective action required for this comment; addressed through Comments 2–8.

2

1. The study simplifies the human body as a rigid pendulum rotating around the ankle joint, assuming the entire body remains perfectly rigid during the flight phase. However, relative motion between the shoulder, hip, and trunk is inevitable during the plyometric push-up, an assumption that severely contradicts real biomechanical characteristics. The authors did not provide any experimental data (such as motion capture or electromyography) to verify the validity of the rigid body assumption, raising questions about the model's ecological validity.

We thank the reviewer for this fundamental observation. The reviewer is correct that inter-segmental motion at the shoulder, hip, and trunk occurs during plyometric push-up execution and constitutes a departure from the rigid-body idealization. This limitation is acknowledged in Section 2.5 (Rigid Body assumption) and the Limitations paragraph of Section 5. The revised manuscript strengthens both entries to explicitly state that the rigid-body constraint applies exclusively from hand take-off to hand landing, that the approximation is most defensible in trained individuals, and that its applicability to recreational populations remains to be established experimentally. The absence of motion-capture or EMG data reflects the study's defined scope as a theoretical-computational investigation; the Limitations and Future Research Directions sections now explicitly identify dual force-plate and three-dimensional motion-capture validation as the required next step.

As specified under Reviewer 3, Comment 4 (Section 2.5 expansions and Limitations paragraph insertion). No additional changes required beyond those already described.

3

2. The study validated the model solely through numerical integral consistency (maximum difference between two numerical methods) without comparing it with actual human experimental data. No subject data (comparison of measured flight times with model predictions) was provided, making it impossible to determine the model's accuracy in predicting real-world motion.

We thank the reviewer for this precise characterization. The reviewer is entirely correct, and this distinction is now explicit throughout the revised manuscript. As addressed under Reviewer 2, Comment 2, all instances of "validated" have been replaced with epistemically accurate terms ("derived," "verified," "self-consistent"). The manuscript's numerical cross-validation (maximum discrepancy < 2.5 × 10⁻⁷ s between Gauss-Kronrod quadrature and ODE-based methods) establishes computational self-consistency between two independent implementations of the same mathematical model, not empirical agreement with human kinematic data. This distinction is now stated explicitly in the Abstract Conclusion, Section 4.1.2, Section 5 Limitations, and Section 6. Prospective experimental validation employing synchronized dual force-plate and three-dimensional motion-capture data is identified as the primary future research priority.

As specified under Reviewer 2, Comment 2, and Reviewer 1, Comment 2 (Future Research Directions paragraph). No additional changes required beyond those already described.

4

3. The discussion section acknowledges that simple pendulum dynamics neglect the rotational inertia contribution I_CoM and points out that the equivalent pendulum length should be 11- 33% larger than the current L. However, this correction was not incorporated into the model derivation and numerical simulation, leaving the direction and magnitude of the systematic deviations of all performance indicators (power, height, arc length) unknown, which seriously weakens the theoretical rigor of the model.

We thank the reviewer for this specific and important critique. The reviewer is correct that the simple pendulum formulation, which uses IO = M × L², underestimates the dynamically correct equivalent length Leq = IO/(M × LCoM), where IO = MCoM + M × LCoM². A uniform-rod approximation of the body yields Leq/L ≈ 1.11–1.33 [10]. The directional consequence is determinate: because Leq > L, the compound pendulum oscillates more slowly than the simple pendulum for any given amplitude; therefore, for a fixed VH,0, the compound pendulum predicts a longer flight time than the simple pendulum model, which partially offsets (but does not cancel) the overestimation attributed to the free-fall model. The net bias of the simple pendulum relative to the compound pendulum is in the direction of underestimating flight time and maximum height, opposite to the direction of the free-fall error. Quantification of the net residual bias requires a compound-pendulum simulation incorporating segmental inertia parameters [10], which is identified as a priority for future work. This directional analysis has been added to the Limitations paragraph.

We changed the Limitations paragraph sentence beginning "More consequentially, the use of simple pendulum dynamics, where IO = M·L², neglects the rotational inertia contribution ICoM of the distributed body mass about its own center of mass. The physically correct equivalent pendulum length, Leq = IO / (M·LCoM), exceeds L by ICoM / (M·LCoM); for a uniform-rod approximation of the body this implies Leq /L ≈ 1.11–1.33 across physiological anthropometry [10]. This systematic underestimation of Leq by the simple pendulum formulation results in underestimation of the predicted flight time relative to the physically correct compound pendulum, partially offsetting but not canceling the overestimation attributed to the free-fall model."

to: "More consequentially, the simple pendulum formulation (IO = M × L²) underestimates the compound-pendulum equivalent length Leq = IO/(M × LCoM), where IO = ICoM + M × LCoM². For a uniform-rod body approximation, Leq/L ≈ 1.11–1.33 [10]. Because Leq > L, the compound pendulum oscillates more slowly than the simple pendulum; for a given VH,0, this produces a longer flight time and greater maximum height than the simple pendulum predicts, partially offsetting but not canceling the free-fall overestimation. The net residual bias relative to empirical data therefore depends on the balance between the free-fall overestimation (quantified here) and the compound-pendulum correction (not yet incorporated), and its magnitude requires simulation with measured segmental inertia parameters [10] as a priority for future work."

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4. The parameter range lacks a physiological basis, and the power output formula contains conceptual errors.

We thank the reviewer for this comment, which comprises two distinct sub-issues addressed separately below.

Regarding the parameter range: the physiological basis for the LOW range (0.50–2.00 m) and the velocity ceiling (VH,0 ≤ 3.0 m/s) is already stated in Section 4.1.2 by citation of Wang et al. [4] and Sha and Dai [8]. A clarifying sentence has been added to make the anthropometric rationale for the LOW range explicit.

Regarding the power formula: the reviewer does not specify which formula or error is intended. The most plausible concern is the quarter-period approximation used as the denominator in equations (33) and (45), which is valid only in the small-angle limit yet applied in a context where large angular excursions are relevant. This concern was identified independently by Reviewer 2 (Comment 6) and has been addressed by inserting explicit first-order approximation framing statements at the opening of Sections 3.2.4 and 3.3.4, repositioning the power expressions as preliminary estimates rather than robust performance indices.

We inserted the following sentence in Section 4.1.2 after "Seven pendulum arm lengths were analyzed, spanning the anthropometrically relevant range: LOW ∈ {0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00} m":

"This range spans the physiologically plausible interval for adult push-up geometry: the lower bound (0.50 m) corresponds to individuals with short effective arm reach or high LSW/LOS ratios, and the upper bound (2.00 m) approximates the maximum anatomically feasible horizontal hand-to-ankle distance in a full push-up position for tall adults. The velocity grid upper bound of VH,0 ≤ 3.0 m/s is consistent with reported take-off velocities in trained populations [8]."

Power formula changes: as specified under Reviewer 2, Comment 6 (framing statements inserted at the opening of Sections 3.2.4 and 3.3.4). No additional changes required.

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5. The citations were insufficient and inappropriate; 11 references were simply not enough to support the research.

We thank the reviewer for this observation. The reference count reflects the targeted scope of the study: a theoretical-computational derivation whose empirical anchors are a small number of directly relevant experimental papers, each precisely linked to a specific claim. The revisions undertaken in response to the full reviewer panel introduce two categories of new citation opportunities: (i) the expanded Limitations and Future Research Directions paragraphs create citation points for compound-pendulum biomechanics [10] and IMU-based validation methodology; and (ii) the restructured Introduction paragraph contrasting the free-fall, pendulum, and dual force-plate frameworks creates additional anchor positions for empirical references on force-plate methods and upper-body ballistic assessment. All existing references have been verified against their respective DOIs and one incomplete entry (Reference [9]) has been corrected, as detailed under Reviewer 2, Comment 8.

As specified under Reviewer 2, Comment 8 (Reference [9] correction). The restructured Introduction paragraph introduced under Reviewer 3, Comment 6, creates additional citation positions without requiring new table infrastructure.

7

6. The results show over-inference, directly mapping flight time error to power error. This calculation is based on the regression equation of Wang et al., but the equation itself is established under the assumption of free fall, and using it to evaluate model differences carries the risk of circular reasoning.

We thank the reviewer for identifying this methodological circularity, which is correct. The Wang et al. [4] regression equation (Ppeak = 11.0 × M + 2012.3 × tflight − 338.0) was derived from force-plate data under a free-fall kinematic framework; applying it to quantify the power consequences of a free-fall flight-time error implicitly uses the free-fall model both as the source of the error and as the evaluation instrument, introducing circular inference. The Discussion has been revised to explicitly acknowledge this limitation, and the power overestimation figure (~70 W) is now presented as an illustrative order-of-magnitude estimate rather than a validated correction.

We changed the sentence in the merged Discussion paragraph (introduced under Reviewer 1, Comment 2) reading "Applied to the Wang et al. [4] power prediction equation (Ppeak = 11.0 × M + 2012.3 × tflight − 338.0; R² = 0.658, SEE = 150 W), a 10% overestimation of tflight at ≈ 0.35 s yields a power overestimation of approximately 70 W, approaching but not exceeding the regression's inherent uncertainty (SEE = 150 W)."

to: "As an illustrative order-of-magnitude estimate (noting that this calculation is inherently circular, since the Wang et al. [4] regression was itself derived under free-fall assumptions), a 10% overestimation of tflight at ≈ 0.35 s applied to their equation (Ppeak = 11.0 × M + 2012.3 × tflight − 338.0; R² = 0.658, SEE = 150 W) yields a power overestimation of approximately 70 W. This figure approaches but does not exceed the regression's inherent uncertainty (SEE = 150 W), and should be interpreted as a directional indicator only; precise quantification requires a regression recalibrated on pendulum-consistent velocity data."

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7. The computational details of the numerical simulation are disconnected from clinical operability. The descriptions of R code implementation, integral tolerance, root-finding algorithm, etc., are too technical, but no simplified formulas or calculation tables are provided for practitioners, which limits the value of the model for widespread application.

We thank the reviewer for this observation, which parallels the concern of Reviewer 3 (Comment 5). The numerical implementation detail in Section 4.1.2 serves the methodological transparency and reproducibility requirements of a computational study and has therefore been retained. A six-step practitioner implementation workflow with explicit algebraic formulas and a first-order sensitivity statement has been added to the Practical Recommendations subsection of Section 5, as specified under Reviewer 3, Comment 5. This workflow enables field deployment without direct engagement with the R code or quadrature algorithms; the root-finding step (step 5) is implementable in standard spreadsheet software using iterative calculation.

As specified under Reviewer 3, Comment 5 (Practical Recommendations workflow paragraph). No additional changes required beyond those already described.

9

In summary, this study has fundamental shortcomings in terms of model hypothesis verification, rotational inertia processing, experimental verification, and clinical translation. If the authors intend to continue this research, they should first complete a human motion capture experiment to verify the applicability of the rigid pendulum hypothesis, incorporate rotational inertia corrections for distributed mass, and provide a comparative analysis with actual measurement data.

We thank Reviewer 4 for this summary, which identifies the four axes of the critique with clarity. We respectfully note that the reviewer's concerns, while legitimate, apply categorically to all first-generation theoretical models in biomechanics; the present study is explicitly positioned as a theoretical-computational foundation, not as a completed validation study. The three corrective steps recommended by the reviewer (motion-capture validation, compound-pendulum correction, and comparative analysis against measured data) are now explicitly incorporated as the highest-priority future research directions in the revised manuscript, with the compound-pendulum correction additionally analyzed for directional effect in the Limitations paragraph. The revised framing throughout the manuscript consistently presents the model as a physically rigorous candidate framework pending prospective empirical validation, which accurately reflects both its contributions and its current limitations.

As specified collectively under Comments 2, 3, and 4 of this reviewer. No additional changes required beyond those already described.

N°

REMARK

RESPONSE

CORRECTIVE ACTIONS

1

Thank you and congratulations

We sincerely thank Reviewer #3 for the thorough evaluation and positive assessment of the manuscript.

None required

N°

REMARK

RESPONSE

CORRECTIVE ACTIONS

1

Thank the authors for their efforts in improving the quality of their papers. The quality of the article has already improved a bit with the revisions.

We thank Reviewer #4 for the continued constructive engagement across review rounds. The remaining technical and stylistic concerns have been fully addressed in the responses below.

None required

2

The discussion has mentioned that relative movement of the trunk and joints may occur, but the specific magnitude of this deviation’s impact on flight time or altitude prediction errors has not been assessed. It is recommended to supplement this with estimates for one or two typical scenarios (such as the effect of 5° shoulder flexion), or to explicitly define this as a direction for subsequent experimental verification.

We thank the reviewer for this targeted recommendation. In response, we have augmented the rigid-body Limitations paragraph in §5 with a first-order analytical estimate for a representative scenario. For a 5° shoulder-flexion perturbation during flight, the CoM deviates from the nominal pendular arc by approximately L·cos(θ₀)·δ ≈ 0.60 × cos(12°) × 0.087 ≈ 0.05 m for representative anthropometry (L = 0.60 m, θ₀ = 12°), a displacement of comparable order to the height discrepancy between models at short flight times (Δh = 0.020 m; t_flight = 0.30 s, L_OW = 0.50 m; Table 4). Precise quantification of inter-segmental deviation effects on t_H and h_max requires prospective motion-capture measurements and is registered as a priority for future experimental validation.

In §5, Limitations, rigid-body paragraph:

We changed “In recreational athletes, pronounced hip flexion or segmental motion during flight will produce deviations from the predicted pendular arc that the present model cannot quantify; for such populations, the model should be applied with caution and its outputs treated as upper bounds on the pendulum-consistent performance estimate.”

To “In recreational athletes, pronounced hip flexion or segmental motion during flight produces deviations from the predicted pendular arc that the present model cannot quantify. As a first-order illustration, a 5° shoulder-flexion perturbation during flight displaces the CoM from the nominal pendular arc by approximately L·cos(θ₀)·δ ≈ 0.05 m for representative anthropometry (L = 0.60 m, θ₀ = 12°), a displacement of comparable order to the height discrepancy between models at short flight times (Δh = 0.020 m; t_flight = 0.30 s, L_OW = 0.50 m; Table 4). Precise quantification of inter-segmental deviation effects on t_H and h_max requires prospective motion-capture measurements and is registered as a priority for future work; for non-rigid subjects, model outputs should be interpreted as upper bounds on the pendulum-consistent performance estimate.”

3

Lines 39-47: “Upper-body muscular power is a fundamental determinant of athletic performance…”, to provide more effective evidence, the authors may consider referring to the following updated relevant studies: Effects of Loading Positions on Lower Limb Biomechanics During Lunge Squat in Men with Different Training Experience (https://paahjournal.com/articles/10.5334/paah.489); New insights optimize landing strategies to reduce lower limb injury risk (https://doi.org/10.34133/cbsystems.0126).

We thank the reviewer for drawing attention to these recent publications. Both studies address lower-limb biomechanics, specifically lunge-squat loading mechanics and landing-strategy optimization for lower-limb injury prevention. Lines 39–47 contextualize upper-body muscular power as a determinant of athletic performance across disciplines including combat sports, swimming, and throwing events. Incorporating lower-limb references at this specific location would misrepresent the evidentiary basis for the stated claim and introduce a topical inconsistency with the paragraph’s scope. The existing citation cluster [1–4] collectively substantiates the role of upper-body power across the cited athletic disciplines and constitutes an appropriate reference base for this passage. The existing reference structure at lines 39–47 is therefore retained without modification.

None required

4

The systemic bias caused by the simplification of rotational inertia has not been adequately corrected: The authors point out in the discussion that the simple pendulum model neglects the rotational inertia of the body about its center of mass and provides a range for the uniform rod approximation. However, all performance indicators in the main text are still not based on Leq calculations. It is recommended that the conclusion explicitly state that, for higher accuracy requirements, rotational inertia correction should be introduced.

We thank the reviewer for this precise observation. The Discussion (§5, Limitations) already acknowledges that L_eq/L ≈ 1.11–1.33 under uniform-rod approximation and that the compound-pendulum correction partially offsets, without canceling, the free-fall overestimation. Restructuring all performance-index derivations around L_eq would constitute a substantive re-derivation inconsistent with the minor-revision scope confirmed in Comment 6. In direct response to the reviewer’s specific recommendation, an explicit statement identifying rotational inertia correction as a necessary extension for higher-accuracy applications has been added to the Conclusion (§6).

In §6, Conclusion:

We inserted “For applications requiring higher kinematic accuracy, the simple-pendulum formulation (I_O = M·L²) should be extended to incorporate the compound-pendulum equivalent length L_eq = I_O/(M·L_CoM), which exceeds L by a factor of 1.11–1.33 under uniform-rod body approximation; this correction constitutes a priority for future model development.”

Between “...mechanically distinct from vertical jump methodology.” and “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.”

5

Some language expressions could be further refined; for example, phrases like “mathematically obligatory” are somewhat absolute and could be changed to “mathematically consistent.” Additionally, several instances of formula citations within parentheses lack spaces between them and the main text; consistent proofreading is recommended.

We thank the reviewer for these precise stylistic observations. The phrase “mathematically obligatory” has been replaced with “mathematically consistent” in §2.3.3 as recommended. A manuscript-wide proofreading pass has been conducted to enforce uniform spacing between all in-text equation references and surrounding prose.

In §2.3.3, we changed “Both derivation pathways operationalize the same static rotational equilibrium condition about O, so their convergence is mathematically obligatory rather than independently confirmatory.”

To “Both derivation pathways operationalize the same static rotational equilibrium condition about O, so their convergence is mathematically consistent rather than independently confirmatory.”

We applied a global spacing correction: all instances where equation references in parentheses appear without a preceding space (e.g., “L(equation 6)”) have been corrected to include a single space (e.g., “L (equation 6)”) throughout all sections.

6

Accept after minor revision.

We sincerely thank Reviewer #4 for the recommendation and for the rigorous expert scrutiny applied to this manuscript across both review rounds.

None required