Camera-Based Barbell Velocity

§1 Purpose and scope

Every velocity-based training product claims accuracy. None publish a characterisation. This is one, for a measurement that reads bar speed from an ordinary phone camera.

A dropped barbell accelerates at g — the net acceleration the Earth imparts to objects, standard value 9.80665 m/s² — which makes free fall the only test in this domain whose correct answer is not itself a measurement. It exercises the entire chain at once — the plate-diameter calibration that turns pixels into inches, the frame timing that turns them into seconds, and the tracker between them. A reading of 9.81 m/s² certifies all three together; any other reading condemns them together.

Scope. One measurement system, one operator, one bar and plate, one venue, one camera, one format. Repeatability and bias only. Clause 8 states what this does not cover, and that list is longer than this one.

§2 Method

Five drops of a barbell loaded with an Olympic bumper plate onto a mat, filmed at 4K 120 fps in portrait from a tripod square to the drop, camera position unchanged throughout. The plate face is what the tracker follows, and its 450 mm diameter is what converts pixels to inches. Each of the five recordings was then analysed three times, each pass beginning with an independent placement of the calibration circle.

Focus and exposure were left on automatic. Locking both would have been the better protocol — autofocus can alter apparent scale mid-clip — and this records what was done rather than what should have been.

That structure is the point. The circle is placed by a human and is the system's only ruler, so the operator sits inside the measurement rather than beside it. Analysing each clip once would confound the operator's contribution with the drop's; three independent placements separate them.

Pass 3 was taken after the placement interface gained full-screen and 12× zoom, to test whether operator contribution was limited by what the operator could see.

§3 Acceptance criteria

Declared before any data was collected, and derived from the decisions the number drives rather than from what looked achievable.

§4 Results

Fifteen measurements. Every one returned a value; none was rejected.

The measurement is a least-squares fit of the plate's vertical position against time to y = at² + bt + c, over the frames between release and first impact. Acceleration is |2a|; b is the speed at the first fitted frame and c its position. R² is the fit of that parabola to those positions, and it was 1.000 in all fifteen — reported chiefly to show that it carries no information about accuracy, see §7.

The three pass means span 0.11% of reading. Three independent re-placements of the ruler, on five separate drops, reproduce the same answer to a tenth of a percent. Whatever is displacing this measurement from 9.81 is not random and is not the operator.

Variance decomposition

§5 Isolating the operator's contribution

Placement is the largest single component in Table 3, and it is the one a human controls, so it was the first candidate for the bias. Between passes 2 and 3 the placement screen gained full-screen presentation and 12× zoom. In a 260-point preview a 4K plate spans roughly three image pixels per screen point, so a pixel of calibration was not reachable because it was not visible.

Across all five recordings the paired change is −0.03% (t = −0.06, 4 df). That is the result this clause exists for: placement is eliminated as the source of the bias. A cause removed by measurement is worth more than a cause suspected, and it redirects the remaining search onto the fixed camera geometry in §6.

§6 Causes eliminated

Each of the following was excluded by derivation or by measurement, not by assumption.

The three that remain open share a property that fits the data: they are fixed in the camera's geometry, and the tripod did not move for the duration of the study. A constant setup error produces exactly what Table 2 shows — a bias stable to a tenth of a percent across every recording and every pass.

§7 Two instrument findings

R² is not a quality gate

R² read 1.000 on all fifteen measurements, including the one that was 4.8% wrong. The quadratic term dominates the variance of a falling body, so a fit can be arbitrarily misplaced and still describe a parabola. Every fit-quality figure in this class of product has the same defect. The instrument now reports the worst frame's distance from the curve in millimetres instead — a physical quantity that does not flatter itself.

Reliability, before and after

Before the corrections in Appendix B, four drops yielded one usable measurement. After, fifteen of fifteen returned a value. A measurement that works half the time cannot support an accuracy claim however close its successes land, and that failure was invisible because failed measurements were being hidden rather than displayed.

What two of those failures looked like

Both of the cards below are pre-correction. They are not a before and an after — they are two different defects, each caught and fixed before the measurements in Table 2 were taken. Neither behaviour survives in the shipped tool.

§8 Limitations

What follows is not covered by any figure above. It is scoped to what a known mechanism could plausibly do — an untested case with no mechanism behind it is not a limitation, it is a sentence.

• Linearity is partly covered, and the uncovered part is one-sided. Each drop begins at rest and sweeps continuously to 4.4 m/s, so the 0.1–1.0 m/s lifting band is inside every measurement rather than extrapolated to — about 20% of the frames in each fit are taken there. And the mechanisms that degrade a camera measurement all worsen with speed: the bar travels 37 mm between frames at the bottom of a drop against 4 mm during a squat, so the drop is roughly nine times the tracking difficulty of the lift it stands in for. For any error that grows with speed, the figures above are an upper bound. No mechanism is known that would run the other way — that would take an error growing as the plate slows — so nothing identified is left unbounded.
• Reproducibility across operators is not assessed. One operator, so the between-operator term is structurally absent. Its likely size is constrained from two directions: the within-operator figure is 0.82%, and the placement step now carries full screen, 12× zoom and axis ticks specifically to narrow the room two people have to differ.
• Stability over time is not assessed. All fifteen measurements come from one session.
• The plate was not independently verified. Its diameter is carried as the specification with tolerance — 450 ± 2 mm, Type B — rather than as a measured value, because nothing on hand could measure a chamfered rubber rim to better than the tolerance itself. At 0.26% against 0.82% from placement, the term does not repay a better instrument.
• One configuration throughout — one bar and plate, one camera, one venue, one format, one camera position. Generalisation to other plates, phones and rooms is untested.
• The offset is characterised but not attributed. It reproduces to ±0.46% and §6 narrows it to the camera's fixed geometry or the frame time base. It is not corrected in the product, and that is a decision rather than an omission: the question was whether the measurement is accurate within a delta that matters for training, and at 3.00% — cancelling entirely for velocity loss — it is.

§9 Conclusion

The question was simple: can an ordinary phone measure bar speed accurately enough for training? It can, and now there is a number attached rather than an assurance.

Fifteen measurements against a constant that does not care what anyone hoped for. 1.25% combined standard uncertainty — against a 2% criterion fixed in writing before a single drop was filmed — and 3.00% from standard gravity with no correction applied at all. For velocity loss, which is what most velocity-based training decisions actually rest on, that offset divides out exactly and the repeatability figure stands unmodified.

What makes those numbers worth anything is what it took to get them. Nine defects in Appendix B were found by running this study and fixed before the data in Table 2 existed — a bounce fitted as part of a fall, a card whose own arithmetic did not close, a calibration drawn at half the available resolution. R² read 1.000 through most of them. None of it was visible until the measurement was pointed at an answer that was already known.

What the offset means for a velocity reading. Absolute velocity carries the same repeatability and an offset of between 1.51% and 3.00%. Acceleration is a distance over a time squared, so the two possible causes do not carry through equally: a spatial error passes into velocity in full, while a timing error passes as the square root of the acceleration ratio — √0.9700 = 0.9849, or 1.51%. Separating them needs a measurement this study did not take.

No further characterisation is planned, and that is a decision. The measurement clears the bar it was given, by a margin, for the job it does today. Appendix C lists what would be worth running next and what would have to be true for it to be worth running — every entry there is gated on a feature that does not exist yet, not on a doubt about this one.

§10 Appendix A — instrument outputs

All fifteen, unselected, in acquisition order. Clip C is visibly the poorest in every pass: a single-frame tracker excursion mid-plot, the worst residuals of the set at 13–18 mm, and the lowest reading of the set each time. It was not excluded from any figure in this report.

§11 Appendix B — defects this study found and fixed

Every one of these was found by running this study, and every one was fixed before the measurements in Table 2 were taken. None of them reached a user.

They are published because the list is the point. A feature that measures something can be shipped on a plausible-looking screenshot, or it can be taken apart until the numbers survive a reference that does not care what anybody hoped for. Every one of these was only visible because the measurement was pointed at an answer known in advance — R² read 1.000 through most of them.

The study was run to establish confidence in the measurement. It also repaired the product, which was not its purpose and is the more useful outcome. The right column says which: four of the nine reached the shipping bar-path feature that measures your set, three were confined to the drop-test tool itself, and two fixed a detector that was later removed on other grounds.

The four that landed are the load-bearing ones. Frame rate is the whole temporal resolution of a rep — at 30 fps a turnaround is located to ±17 ms and a five-frame peak spans 167 ms of the lift; at 120 fps that becomes ±4 ms and 42 ms. And the tracker is not a detail of the drop test, it is what follows the plate through every rep you film.

And what it changed that was never a defect

These were working as designed. The study is what showed the design was the limit — which is a different and more useful thing for a study to do than find bugs.

§12 Appendix C — studies not yet run

Each of these answers a question the current feature does not ask. They are listed so the scope of this report is unambiguous, and so that if any of them becomes relevant it is already designed.

Until then the honest position is the one the app already takes: rather than publishing an accuracy figure and asking anyone to believe it, it ships the check itself. Any user can drop a barbell and measure their own setup, in the tool this study was written from.