A calibration-free method for calculating vertical jump mechanics from video data using the constant of gravity

1. Introduction

An abundant body of biomechanical literature documents the diverse methodologies used to evaluate vertical jump performance. Recent systematic reviews and meta-analyses—most notably the comprehensive review by Eythorsdottir et al. (2024), appropriately titled "The Battle of the Equations," alongside critical reviews by Xu et al. (2023), Kozinc and Pleša (2022), and Moir (2008)—have rigorously mapped the mathematical landscape comparing force platforms, linear position transducers, contact mats, and optical systems, documenting substantial discrepancies between methods that carry direct implications for field-based testing. These evaluations demonstrate that despite the maturity of vertical jump testing, existing field-based and laboratory measurement modalities remain constrained by distinct structural or practical trade-offs:

Contact-to-Contact Method: Contact mats and flight-time algorithms bypass spatial calibration by estimating jump height solely from total flight time ($t_{flight}$). Specifically, these models estimate center-of-mass displacement from the instant of takeoff to touchdown—formally classified in biomechanical taxonomy alongside initial-standing-to-apex displacement (JH-1) as takeoff-distance or flight-distance (JH-2) (Xu et al., 2023):

$$h=\frac{g\cdot t_{flight}^2}{8}$$

Originating from classical formulations (Asmussen & Bonde-Petersen, 1974; Bosco et al., 1983), this JH-2 model assumes identical center-of-mass (COM) height at takeoff ($z_{takeoff}$) and touchdown ($z_{landing}$). In practice, systematic reviews show that the standard flight-time method systematically overestimates impulse-momentum jump height by 2% to 6% (0.6 cm to 4.1 cm) in unloaded bodyweight jumps (Eythorsdottir et al., 2024; Xu et al., 2023). Across literature, ascent duration is universally treated as a derived assumption ($t_{ascent}=\frac{t_{flight}}{2}$) rather than a directly observed timestamp (Eythorsdottir et al., 2024). Athletes routinely alter landing posture via ankle dorsiflexion, knee flexion, or lower arm positioning—with arm lowering alone contributing a $-0.017\text{ m}$ COM displacement—artificially extending measured flight time and creating a gameable "cheating" effect (Yamashita et al., 2020). Simulation models confirm that a modest 10°–20° change in ankle joint position introduces an approximately 6.5% overestimation in jump height, scaling up to 59.6% error in extreme postural shifts (Gonçalves et al., 2024).

Video Analysis & Motion Capture: Conventional two-dimensional video analysis and motion capture kinematics rely on physical scaling objects (e.g., 1-meter calibration grids) positioned in the movement plane. Perspective artifacts, lens distortion away from the optical center, and camera distance changes introduce spatial scaling errors (Weberling et al., 2021).

Force Platforms: The laboratory gold standard calculates jump height by double integrating vertical ground reaction force data to determine center-of-mass displacement. While highly accurate, this method requires exact static body weight determination before the jump. Over the duration of the movement, any minor weighing errors or sensor drifts are subject to mathematical bias accumulation (integration drift), with even small errors in the weighing phase propagating into substantial jump height discrepancies (Moir, 2008; Street et al., 2001). Because of this sensitivity to bias accumulation, the contact-to-contact (flight time) method is frequently used alongside it as a robust cross-validation metric in force plate systems, as time-in-air avoids integration drift entirely (Eythorsdottir et al., 2024; Linthorne, 2001).

To resolve spatial scaling errors, landing posture confounds, and integration bias without requiring force platform hardware, this paper presents a mathematical framework that replaces total flight-time tracking with direct, ascent-only temporal event logging.

---

2. Key Event Logging

Rather than tracking pixel coordinates across physical space, the proposed method logs three discrete chronological timestamps along the vertical trajectory of the athlete's center of mass ($z(t)$):

1. Amortization Turnaround ($t_0$): The lowest boundary of the countermovement squat, where vertical velocity $v(t_0)=0$.
2. Takeoff ($t_1$): The precise frame of ground separation.
3. Apex ($t_2$): The peak vertical position of the jump trajectory, where vertical velocity $v(t_2)=0$.

2.1 Dual Event Detection Modalities

Because this model requires identifying temporal frame indexes rather than measuring physical distances, event markers can be extracted through two independent modalities:

• Manual Human Annotation: Because the methodology requires identifying only three discrete event frames, human annotators can easily extract $t_0$, $t_1$, and $t_2$ manually using standard video playback software. An annotator records the specific frame numbers corresponding to the lowest depth, foot takeoff, and peak jump apex, converting frames to time via the camera's frame rate ($\mathrm{\Delta }t=\frac{1}{\text{fps}}$).
• Computational Extraction: Markerless pose estimation or feature-tracking algorithms track sacral keypoints or segment estimates (Aderinola et al., 2023). The system identifies $t_0$ and $t_2$ by locating velocity zero-crossing points ($\frac{dz}{dt}=0$), while $t_1$ is identified via foot-ground separation thresholds or acceleration spikes.

By omitting the landing phase entirely, both human annotators and automated algorithms bypass the most visually ambiguous and variable phase of the jump. Recent validation of markerless monocular systems demonstrates that while pose estimation accurately tracks velocity and jump height, contact-time and landing event detection remain the pipeline's primary weak link (Aderinola et al., 2023). By relying exclusively on takeoff and apex as the flight-phase boundaries, this framework fundamentally bypasses the landing detection problem. These timestamps define two clean time intervals:

• Concentric Push Duration ($t_{con}$): $t_{con}=t_1-t_0$
• Ascent Duration ($t_{ascent}$): $t_{ascent}=t_2-t_1$

---

3. Mathematical Derivation

3.1 Resolving Jump Height ($h$)

During the unconstrained flight phase, the athlete's center of mass is subject only to gravitational acceleration ($-g$). Because vertical velocity at the apex is exactly zero ($v(t_2)=0$), the true JH-2 jump height ($h$) can be calculated exclusively from the ascent duration ($t_{ascent}=t_2-t_1$) using the kinematic equation for spatial displacement:

$$h=\frac{1}{2}g\cdot t_{ascent}^2$$

Unlike the traditional contact-to-contact method ($h=\frac{g\cdot t_{flight}^2}{8}$), this formulation derives height strictly from the upward trajectory, eliminating any assumptions or mathematical confounds associated with landing posture.

3.2 Resolving Takeoff Velocity ($v_t$)

With ascent time known, vertical velocity at the moment of takeoff ($v_t$) resolves directly from the constant acceleration of gravity. Integrating vertical acceleration from takeoff ($t_1$) to apex ($t_2$) yields:

$$v(t_2)=v_t-g\cdot (t_2-t_1)$$

Because $v(t_2)=0$ at peak apex and $t_2-t_1=t_{ascent}$, vertical takeoff velocity resolves to:

$$v_t=g\cdot t_{ascent}$$

This calculation relies strictly on duration ($t_{ascent}$) and gravitational acceleration ($g$), eliminating spatial displacement vectors, camera distance dependencies, and landing posture errors entirely. (Note: This identity is internally consistent by construction with spatial energy conservation, $v_t=\sqrt{2gh}$).

3.3 Exact Time-Averaged Concentric Force ($\overline{F}$)

The Impulse-Momentum Theorem states that net vertical impulse applied to mass ($m$) equals its change in linear momentum (Linthorne, 2001):

$${\int }_{t_0}^{t_1}(F(t)-m\cdot g)dt=m\cdot v_t-m\cdot v(t_0)$$

Because $v(t_0)=0$ at the amortization point:

$${\int }_{t_0}^{t_1}F(t)dt-m\cdot g\cdot t_{con}=m\cdot v_t$$

By definition, time-averaged force ($\overline{F}$) over the concentric duration ($t_{con}$) is:

$$\overline{F}=\frac{1}{t_{con}}{\int }_{t_0}^{t_1}F(t)dt$$

Substituting $\overline{F}\cdot t_{con}$ into the impulse equation:

$$\overline{F}\cdot t_{con}-m\cdot g\cdot t_{con}=m\cdot v_t$$

$$\overline{F}\cdot t_{con}=m\cdot v_t+m\cdot g\cdot t_{con}$$

Dividing by $t_{con}$ and substituting $v_t=g\cdot t_{ascent}$:

$$\overline{F}=\frac{m\cdot g\cdot t_{ascent}}{t_{con}}+m\cdot g$$

Factoring out body weight ($m\cdot g$):

$$\overline{F}=m\cdot g\cdot (\frac{t_{ascent}}{t_{con}}+1)$$

This expression provides an exact analytical identity for time-averaged force. Because it relies purely on the impulse-momentum relationship, it requires absolutely no assumptions about force-curve linearity or trajectory shape during the concentric phase.

3.4 Estimated Average Concentric Power ($\overline{P}$)

While time-averaged force ($\overline{F}$) is mathematically exact, it is tempting to assume that average mechanical power ($\overline{P}$) can be derived with equal precision. However, doing so introduces a fundamental assumption: it requires concentric acceleration to remain perfectly constant throughout the push phase, producing a linear velocity profile where mean velocity (${\overline{v}}_{con}$) simplifies to:

$${\overline{v}}_{con}=\frac{v(t_0)+v_t}{2}=\frac{g\cdot t_{ascent}}{2}$$

Multiplying time-averaged force by this linear mean velocity yields an estimate for average concentric power:

$$\overline{P}\approx \overline{F}\cdot {\overline{v}}_{con}=[m\cdot g\cdot (\frac{t_{ascent}}{t_{con}}+1)]\cdot \left(\frac{g\cdot t_{ascent}}{2}\right)$$

$$\overline{P}\approx \frac{1}{2}m\cdot g^2\cdot t_{ascent}\cdot (\frac{t_{ascent}}{t_{con}}+1)$$

In real human movement, force-time curves are highly non-linear, meaning instantaneous force and velocity vary continuously and asynchronously throughout the concentric drive (Cormie et al., 2009; Aragón-Vargas & González-Lutz, 2023). Because the average of a product does not equal the product of the averages ($\overline{F\cdot v}\ne \overline{F}\cdot \overline{v}$), this power calculation is by no means an exact physical identity. Nevertheless, it provides a valuable, computationally trivial raw estimation for field profiling and comparative analysis when force plate data is unavailable.

---

4. Sensitivity to Temporal Discretization Error

To evaluate how video frame rate affects kinetic accuracy, we establish the maximum discretization error limits across all three event markers ($t_0,t_1,t_2$). For time-averaged force, taking the partial derivatives with respect to timing errors ($\mathrm{\Delta }t$) yields:

$$\frac{\partial \overline{F}}{\partial t_{ascent}}=\frac{m\cdot g}{t_{con}}$$

$$\frac{\partial \overline{F}}{\partial t_{con}}=-m\cdot g\cdot \frac{t_{ascent}}{t_{con}^2}$$

Because $t_{ascent}=t_2-t_1$ and $t_{con}=t_1-t_0$, the total force calculation is governed by three distinct event frame selection uncertainties ($\pm 0.5$ frames per boundary). Writing the relative error explicitly:

$$\frac{\mathrm{\Delta }\overline{F}}{\overline{F}}=\frac{\mathrm{\Delta }{t}_{ascent}}{t_{ascent}+t_{con}}-\frac{t_{ascent}\cdot \mathrm{\Delta }{t}_{con}}{t_{con}(t_{ascent}+t_{con})}$$

Substituting $\mathrm{\Delta }{t}_{ascent}={\delta }_2-{\delta }_1$ and $\mathrm{\Delta }{t}_{con}={\delta }_1-{\delta }_0$, where ${\delta }_i$ denotes the selection error at each boundary, yields a distinct sensitivity coefficient per marker:

1. Amortization turnaround ($t_0$): Enters through $t_{con}$ in the denominator, with coefficient $\frac{t_{ascent}}{t_{con}(t_{ascent}+t_{con})}$.
2. Takeoff instant ($t_1$): Acts as a correlated boundary appearing in both terms, and therefore carries the largest coefficient, equal to the sum of the other two.
3. Jump apex ($t_2$): Enters through $t_{ascent}$ alone, with coefficient $\frac{1}{t_{ascent}+t_{con}}$.

Critically, these coefficients are unequal, so combined error cannot be obtained by simply counting contributing frame boundaries. Because the three selection errors arise from independent frame-snapping decisions, we combine them in quadrature rather than by linear summation, treating each boundary as contributing a half-frame half-width ($\pm 0.5$ frames):

$$\left|\frac{\mathrm{\Delta }\overline{F}}{\overline{F}}\right|=\frac{\mathrm{\Delta }{t}_{frame}}{2}\sqrt{c_0^2+c_1^2+c_2^2}$$

This root-sum-square treatment reports the expected combined uncertainty rather than the absolute bound obtained when all three boundaries simultaneously reach maximum error with worst-aligned signs. The linear worst-case bound is roughly $1.6$ times larger, but requires a sign coincidence with low probability of occurrence in practice.

Quadrature combination assumes the three selection errors are mutually independent. This holds well for automated pipelines, where each boundary is resolved by a separate detection criterion. It is a weaker assumption for manual annotation, where a single annotator may exhibit systematic bias across boundaries: an observer who consistently identifies events late introduces correlated errors that partially cancel in $t_{ascent}=t_2-t_1$ while propagating differently through $t_{con}$. Under full positive correlation the quadrature estimate understates true uncertainty, and the linear sum given above should be read as the governing bound. Reported values should therefore be interpreted as expected uncertainty for computational extraction, and as a lower bound for human annotation.

Furthermore, absolute jump height error ($\mathrm{\Delta }h\approx g\cdot t_{ascent}\cdot \mathrm{\Delta }{t}_{ascent}$) scales linearly with true jump height. Higher jumps require longer ascent times, meaning the same absolute temporal error yields a slightly larger absolute centimeter error, while simultaneously reducing the relative percentage error.

For manual human annotation, jump height depends only on the $t_1$ and $t_2$ boundaries, giving a combined ascent uncertainty of $\sqrt{2}\times 0.5$ frame intervals, while force additionally inherits the $t_0$ boundary through $t_{con}$.

4.1 Interpreting Relative Force Error

Relative error figures for $\overline{F}$ require one important qualification. Because $\overline{F}=mg(\frac{t_{ascent}}{t_{con}}+1)$, the additive body weight term is exactly known and carries no timing uncertainty whatsoever. For the reference jump, this constant accounts for $54.8\mathrm{\%}$ of the reported force magnitude, meaning only the remaining $45.2\mathrm{\%}$ is subject to discretization error at all.

Consequently, relative error expressed against total force is diluted by an error-free constant. Practitioners comparing the net concentric force ($\overline{F}-mg$), which isolates the propulsive contribution above body weight, should expect relative uncertainties larger by a factor of $2.21$: approximately $\pm 15.1\mathrm{\%}$ at 30 fps and $\pm 3.8\mathrm{\%}$ at 120 fps for discrete annotation. We report total force throughout for consistency with force platform conventions, but note that net force, impulse ($J=mg\cdot t_{ascent}$), and jump height each carry distinct error characteristics despite deriving from the same three timestamps. Net vertical impulse is the most favourable of these, inheriting only the $t_1$ and $t_2$ boundaries and scaling linearly with $t_{ascent}$, and is therefore the most robust single kinetic output of the framework.

4.2 Computational Enhancement: Sub-Frame Parabolic Curve Fitting

While human annotators cannot reliably estimate sub-frame events, computational pipelines can easily bypass the discrete apex boundary error ($t_2$) entirely. Because airborne movement is governed strictly by uniform gravity, vertical center-of-mass position across the flight phase follows a deterministic second-order polynomial:

$$z(t)=A{t}^2+Bt+C$$

Applying a least-squares quadratic regression across uncalibrated vertical keypoint position coordinates ($z_i$) recorded during the flight phase allows the system to calculate the continuous parabolic vertex ($t_2$) at the exact analytical zero-derivative point ($\frac{dz}{dt}=0$):

$$t_2=-\frac{B}{2A}$$

This sub-frame interpolation converts the apex timestamp ($t_2$) into a continuous floating-point calculation ($\mathrm{\Delta }{t}_{apex}\approx 0$ frames). Removing $t_2$ reduces the combined ascent uncertainty by a factor of $\sqrt{2}$, since jump height then depends on a single discrete boundary. Its effect on force is smaller still, because the eliminated boundary carries the smallest of the three sensitivity coefficients and enters in quadrature: for the reference jump, combined relative force error falls to approximately $0.89$ times the discrete value.

The table below reports combined (root-sum-square) error margins, comparing normal discrete selection (human annotation) against parabolic curve fitting (computational analysis) standardized on a reference 30 cm vertical jump ($t_{ascent}=247.3\text{ ms}$, $t_{con}=0.300\text{ s}$).

Table 1. Discretization Error Matrix, Root-Sum-Square Combination (Reference Jump Height: 30 cm)

Under this combination model, parabolic curve fitting at 60 fps yields spatial accuracy slightly better than a human annotator clicking discrete frames at 120 fps, recovering roughly one doubling of capture rate for jump height at no hardware cost. The gain is more modest for kinetics: because the concentric boundaries ($t_0,t_1$) remain discrete regardless of apex treatment, and because $t_1$ dominates the quadrature sum, force accuracy continues to scale primarily with capture rate. For computational pipelines combining 120 fps video with parabolic curve fitting, total system jump height error is suppressed to approximately 1 cm ($\pm 3.4\mathrm{\%}$), with force derivation error at $\pm 1.5\mathrm{\%}$.

---

5. Practical Applications from Performance to Clinical Rehabilitation

The foundational output of this calibration-free framework is a highly accurate, tamper-resistant measurement of vertical jump height. By establishing this baseline strictly through temporal events, the methodology provides sports scientists and clinicians with a robust toolset that accommodates both high-performance and rehabilitation environments:

• Tamper-Proof Jump Height and Clinical Robustness: In talent identification and scouting environments, athletes frequently attempt to "game" standard flight-time jump mats by altering landing mechanics or pulling their legs up (Yamashita et al., 2020). By ignoring the descent and landing phases entirely, this framework provides a tamper-resistant mechanical profile of raw explosive capacity. Furthermore, standard flight-time calculations suffer from disproportionately high percentage errors during low-amplitude jumps, as absolute landing posture errors make up a larger fraction of total flight time (Gonçalves et al., 2024). Restricting measurement strictly to the ascent phase maintains high relative accuracy even for small jumps, making the methodology highly viable for clinical rehabilitation setups where patients frequently exhibit restricted explosive capacity due to injury.
• Comprehensive Kinetic Profiling: Beyond simple displacement, when a subject's body mass ($m$) is known and inputted into the equations, the temporal framework instantly unlocks comprehensive mechanical outputs. While average power ($\overline{P}$) must be treated as a practical approximation due to non-linear force-velocity interactions, time-averaged concentric force ($\overline{F}$) provides a mathematically exact ground reaction force metric. This allows practitioners to evaluate true kinetic output alongside scalable power estimates without investing in laboratory-grade force plates.
• The Mass-Independent Dimensionless Index: Finally, even if a subject's mass is unknown or fluctuates over time, the mathematical isolation of the temporal ratio $\frac{t_{ascent}}{t_{con}}$ serves as a universal, dimensionless index of movement strategy (Gathercole et al., 2015). Because the mass variable cancels out of this ratio entirely, it instantly distinguishes "force-reliant" jumpers (who utilize a deep, prolonged countermovement, increasing $t_{con}$) from "velocity-reliant" jumpers (who utilize a stiff, rapid amortization, minimizing $t_{con}$). This permits seamless inter-subject comparisons and intra-subject longitudinal tracking, allowing coaches and clinicians to objectively evaluate neuromuscular efficiency without ever needing a scale.

---

6. Conclusion and Future Work

By shifting the computational paradigm from spatial pixel tracking to chronological event logging, this paper demonstrates that mechanical force outputs and precise jump height can be derived without spatial calibration grids, optical depth scaling, or force plates. The base methodology requires identifying only three specific discrete temporal events ($t_0,t_1,t_2$), making it highly accessible for manual frame annotation. Furthermore, for automated computer vision pipelines, integrating parabolic curve fitting across the flight phase resolves the apex timestamp ($t_2$) with sub-frame precision, mathematically excising the apex boundary error and establishing an uncalibrated, camera-agnostic framework for jump mechanics.

6.1 Future Work

Future iterations of this framework will explore expanding the temporal event logging model beyond the current three-marker system ($t_0,t_1,t_2$). Specifically, upcoming work will investigate:

1. The Eccentric Initiation Marker ($t_{ecc}$): Incorporating the onset of the unweighting phase (where velocity first drops below zero) will allow for mathematical derivation of braking phase duration, active stiffness metrics, and modified Reactive Strength Index ($\text{RSI-mod}=\frac{h}{t_{push}}$).
2. Touchdown and Landing Amortization ($t_{touchdown},t_{land\mathrm{\_}min}$): While the present study purposefully excludes the descent phase to eliminate flight-time overestimation, capturing the touchdown instant ($t_3$) and subsequent landing turnaround ($t_4$) independently would enable uncalibrated assessment of landing attenuation strategies, eccentric landing forces, and bilateral stiffness without corrupting ascent mechanics.

Integrating these additional event boundaries will yield a complete end-to-end kinetic and kinematic profile of the entire jump cycle—from initial movement initiation through shock absorption—while preserving the core calibration-free, ascent-isolated advantages demonstrated in this paper.

---

Declarations

Acknowledgements

This work was not preregistered, as it is a purely mathematical and theoretical derivation involving no data collection.

Author Contributions

R.V. is the sole author of this manuscript and was responsible for conceptualization, mathematical derivation, error analysis, and writing of the original draft and subsequent revisions. The author read and approved the final manuscript.

Funding

This work received no external funding. The author is a cofounder of Factorial Biomechanics, which supported the work internally.

Conflicts of Interest

The author is cofounder and Chief Technology Officer of Factorial Biomechanics, a company developing markerless video-based biomechanics software. The method described here is relevant to that commercial activity. No aspect of the derivation or error analysis depends on proprietary data or software, and all results presented are analytically reproducible from the equations given.

Ethics Approval and Informed Consent

Not applicable. This manuscript is a theoretical and methodological derivation. No human or animal participants were recruited, no experimental data were collected, and no identifiable personal data were processed. Ethics approval and informed consent were therefore not required.

Data and Supplementary Materials Accessibility

No empirical datasets were generated or analyzed. All numerical values reported in Table 1 are derived analytically from the equations presented in Sections 3 and 4, using the stated reference jump parameters ($h=0.30\text{ m}$, $t_{con}=0.300\text{ s}$, $g=9.81{\text{ m/s}}^2$), and are fully reproducible from those expressions alone.

---

References