Matteo Campagna
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The archer's paradox: nearly forty years of measuring the arrow as it flexes

That the arrow flexes as it leaves the bow is now a measured fact rather than an intuition. This page follows the studies that measured it from 1987 to 2025, from an accelerometer on the riser to high-speed footage of sixteen world-class archers, from mathematical models to shooting machines, along with the questions about spine that were left open along the way.

Studies in the registry
29
On archers
1
Years
1987-2025

The question

At full draw a recurve arrow does not point at the centre of the target: resting against the side of the riser, for a right-handed archer it lines up slightly to the right, and it still gets there. The usual explanation is that the shaft flexes and bends around the riser rather than leaving straight. In the registry this is a corroborated claim since 2012, but for a quarter of a century it stayed at reported, with one archer measured at a time.

This page follows the studies in chronological order. The physics for anyone tuning an arrow is in the articles on the archer's paradox and spine and on bow tuning; here the subject is which instruments measured the flex and how the question moved, from the arrow that bends to the arrow that gets chosen.

1987-1990: the same sequence seen from the bow and from the arrow

The first measurement in the registry looks at the bow. In 1987 an Italian group fitted an accelerometer to the riser of a national-team archer and read a clear sequence from the phases of the signal: for the first 22 milliseconds the flexing shaft pushes on the button, between 22 and 26 ms the button pushes it back, and between 26 and 30 ms the arrow has already left the riser while still on the string.

Three years later the same sequence was seen from the arrow's side. Footage of a Polish national-team archer, at up to 2,500 frames per second, showed the shaft flexing as the string drove it, and a shooting machine produced the same shape with no hand on the string. The machine also showed the opposite case: a shaft that was too weak struck the bow with its fletching. The model that went with the footage led to a nomogram that took in the equipment of eighteen of the twenty-two best Polish archers, and the registry's claim on the rule linking spine, draw weight and length starts there.

1997-2001: the models put numbers on it

In 1997 a model that treats the shaft as a vibrating beam reproduced the shapes filmed in 1990 but questioned the original explanation: even with a matched shaft, the nock did not stay still sideways for long before it left the string. A year later a fuller model showed that contact with the button breaks off and resumes, and that the nock crosses back over the plane of the bow at about the moment the string releases it.

In 2001 an independent model gave the condition in figures: the arrow clears the riser if it completes a full vibration cycle while still on the string. In the worked example the stiffer shaft stayed in contact for 16.2 ms against a period of 15.6 and cleared, while the weaker one stayed 15.5 ms against 18.9 and touched the riser. The same calculation opened two questions. The first is that the static test and the dynamic analysis choose different shafts for the same bow, because mass plays no part in the static test: this is the claim on static and dynamic spine, which still rests on models alone. The second concerns buckling: the arrow loses stability at a dimensionless force of 14.79, while the peak thrust of an average target bow comes to 12.3, so compression increases the flex without crossing the threshold.

2011-2013: sixteen archers and the vibration modes

The status rose to corroborated in 2012, with sixteen world-class recurve archers filmed at 4,000 frames per second at the London 2011 test event. At the release the string pushes the nock an average of 13.1 mm towards the body, from 9 to 20 mm across archers; the shaft is straight again at 8.6 ms and leaves the string at 14.6, while contact with the button ends within 8 ms. In passing, every archer's spine was the one the manufacturer's chart gives for their draw weight and length.

The same group built a recurve model calibrated on the bench. With the sight centred on a target at 20 m, the arrow at full draw lay about 220 mm to the right, against the 226 predicted. In the model the next spine size is worth about 11 N of draw weight; taking half a gram off the fletching shifts the draw weight that tunes the arrow by 3 N, taking 1.3 g off the point by half a newton. On tail weight this remains a calculation, with no arrows shot to check it. Also in 2012, laser vibrometry on free shafts placed the second mode at about three times the frequency of the first, 241 Hz against 80.2, and the footage of the sixteen archers often showed a small second-mode component, as an S shape; how much it matters for the shot remains open in the claim on vibration modes.

Before and after 2012 the same author marked out the edges of the phenomenon. In 2011 a compound shot with a mechanical release gave a few millimetres of flex before release, against the 10-15 expected on a recurve; in 2013 the same model applied to the vertical plane found flex about ten times smaller than in the lateral plane. The paradox plays out in the plane of the riser, and the largest flex is that of a recurve shot with the fingers.

2014-2022: flex without an archer

In 2014 a bow fixed on a machine flexed four types of arrow by 4.63 to 5.53 mm: the flex is not a flaw of the release. The same year a Japanese laboratory launched the same arrows four ways: 1.22 mm of flex with a mechanical release, 6.86 with two national-level archers, 14.8 with a mechanical finger rig. In the archers' shots air resistance was the most consistent: the standard deviation of the drag coefficient was 0.073, against 0.267 with a compressed-air launcher that does not make the arrow flex.

In 2015 a robot with mechanical fingers reproduced the flex of an international-level archer, 6.94 mm against 6.86, and showed that opening the fingers faster does not increase the flex in proportion. In 2022 automatic image analysis of 35 arrows shot by a robot put a number on the flex that carries on in flight, at around 75 Hz.

From the arrow that bends to the arrow that gets chosen

Meanwhile the machines were moving the question towards shaft selection. Two studies on ProTour shafts measure that stiffer shafts vibrate at a higher frequency, 74, 77 and 80 Hz in one and 78.3, 80.2 and 83.3 in the other, but in 2019 a rig that shakes the shafts could not tell carbon spine 500 from 600. On straightness, a compound test in 2017 gave groups from 27 to 48 mm between straight shafts and shafts bent on purpose, whereas in 2025 fifty-nine X10 shafts of one spine, shot by machine at 70 m, showed no effect, because they were straight to 0.007 mm on average against a tolerance of 0.025. There, mass, mean spine and the variation of stiffness around the shaft were what counted, and the arrows, not matched by weight, stayed within a vertical spread of 8.8 cm against the 12.2 of the ten ring: this is the claim on matching by weight and spine. A 114 kg machine had already shown in 2020, in a test of six arrows, that defective arrows land outside the group, and on the compound three different spines group in much the same way. Finally, in 2025, the 2012 model, rewritten, reproduced the commercial charts but found them up to four sizes too stiff at the extremes, with no test shots.

What remains open

The flex is well measured in top archers, and it appears when a machine does the shooting too; in the 2012 footage the tail clears the button with room to spare. What remains open is the weight of the second flex, the S shape, and whether the thrust of the string ever takes the arrow to buckling, which so far has been calculated or tested on the bench with kyudo shafts. The figures, too, apply to a finger release. Almost every test of shaft selection uses machines or calculations: nobody has yet compared the groups of archers shooting matched and unmatched shafts, or checked with real shots that static and dynamic spine rank the same shafts differently.

How we came to know it

  1. 1987OriginLeonardi et al. 1987 Reduction of Lateral Bow Displacement Using Different Torque Flight Compensators and Stabilizers in ArcheryAn accelerometer on the riser of an Italian national-team archer: for the first 22 ms the flexing shaft pushes on the button, and between 26 and 30 ms it has already left the riser while still on the string. The sequence read from the bow's side.
  2. 1990ExtensionPekalski 1990 Experimental and theoretical research in archeryFootage at 1,000 to 2,500 frames per second of a Polish national-team archer and of a machine: the shaft flexes the same way with the archer and with the machine, and a model reproduces its shape. A shaft that is too weak touches the bow with its fletching.
  3. 1997ContradictionKooi et al. 1997 On the mechanics of the arrow: Archer's ParadoxA model that treats the shaft as a vibrating beam reproduces the shapes filmed in 1990 but questions the explanation given then: even with a matched shaft, the nock does not stay still sideways for long before it leaves the string.
  4. 1997ExtensionKobayashi et al. 1997 Dynamics of Japanese Bows and Arrows : Analysis of "Tsunomi" with Link Model and ExperimentThe Japanese bow: three grip-torque profiles with the same integral over the 26 ms of the launch send the arrow in different directions, while its speed changes by less than 1%. Contact between arrow and bow depends on how the grip is twisted.
  5. 1998ExtensionKooi 1998 Bow-arrow interaction in archeryA fuller model: contact with the button breaks off and resumes, and the nock crosses back over the plane of the bow at about the moment the string releases it. Allowing for compression of the shaft predicts more flex.
  6. 2001ExtensionZanevskyy 2001 Lateral deflection of archery arrowsThe condition in figures: the arrow clears the riser if it completes a full vibration cycle while on the string. In the same calculation the static test and the dynamic analysis point to different shafts for the same bow.
  7. 2009ExtensionZaniewski 2009 Modeling of the Archery Bow and Arrow VibrationsThe author of the 2001 model, under the Polish spelling of his name, moves it to the vertical plane: while on the string the shaft completes about one two-node bending cycle, 18.4 ms calculated against the 18-19 of a manufacturer's demonstration film, the only experimental comparison. The amplitude of around 5% of shaft length is taken from that film.
  8. 2011ExtensionPark 2011 The Behaviour of An Arrow Shot from a Compound Archery BowThe compound with a mechanical release: peak flex before release is a few millimetres, against the 10-15 expected on a recurve, and flash photographs show a second harmonic that can give the shaft an S shape.
  9. 2012ReplicationPark et al. 2012 High-speed video analysis of arrow behaviour during the power stroke of a recurve archery bowThe measurement across many archers: sixteen world-class recurve archers filmed at 4,000 frames per second. The string pushes the nock 13.1 mm on average, the button releases the shaft within 8 ms and the string at 14.6 ms.
  10. 2012ExtensionPark 2012 Arrow behaviour in the lateral plane during and immediately following the power stroke of a recurve archery bowA recurve model calibrated on the bench: the arrow does not leave in the direction it points, and shafts of the same nominal spine but of different types need different draw weights to leave without rotating.
  11. 2012MethodRieckmann et al. 2012 Modelling the three-dimensional vibration of composite archery arrows under free–free boundary conditionsLaser vibrometry on free shafts: where the two still points of the first bending mode lie, and the second mode at about three times the frequency of the first. Models and measurements agree, with no bow and no archer.
  12. 2013ExtensionPark 2013 Arrow behaviour in the vertical plane during and immediately following the power stroke of a recurve bowThe same model in the vertical plane: flex is about ten times smaller than in the lateral plane. The flex that matters for the paradox is horizontal.
  13. 2014ReplicationKim et al. 2014 Measurement of Archer's Paradox Size using Multiple FramesA bow fixed on a machine, four arrow types: with no fingers and no archer the shaft still flexes sideways, by 4.63 to 5.53 mm. The flex does not come from the release.
  14. 2014ExtensionAndo et al. 2014 J0520102 Aerodynamic properties of an archery arrow : Influence of flexural oscillationThe same arrows launched four ways: 1.22 mm of flex with a mechanical release, 6.86 with two archers, 14.8 with mechanical fingers. Air resistance is most consistent in the archers' shots.
  15. 2014ExtensionJeong et al. 2014 A Study of Arrow Performance using Artificial Neural NetworkSeventy-nine arrows from one production run, shot by machine at 25 m: straightness and measured spine are enough for a neural network to predict a dispersion index for each arrow, though the index rewards a regular triangle of three impacts more than a small one. The move from physics to shaft selection.
  16. 2015MethodWakatabe et al. 2015 2A2-T08 Development of Archery Robot : Development of finger mechanism for controllable the bending vibrationA robot with mechanical fingers reproduces the flex of an international-level archer: 6.94 mm against 6.86. Opening the fingers faster does not increase the flex in proportion.
  17. 2015ExtensionMeyer 2015 Applications of Physics to ArcheryOn a compound with a spine 500 arrow the peak axial load is estimated at about 60 N, against a critical load of about 100: the power stroke compresses the shaft without taking it to lateral buckling.
  18. 2017ExtensionPark et al. 2017 Effect of arrow shaft straightness on arrow groupingA compound on a machine at 25 m, with straight shafts and shafts bent on purpose: in aluminium the group grows from 27 to 48 mm between one and six thousandths of an inch of deviation.
  19. 2019ReviewPark 2019 The impact of the atmosphere on target archeryA review sets out the shape of the phenomenon: from a recurve, large lateral flex and small vertical flex. Among compounds, design differences rule out a general prediction.
  20. 2019ContradictionFish et al. 2019 Dynamic Characterization of Arrows through Stochastic PerturbationShafts shaken on the bench rather than loaded: in carbon, spine 500 and 600 cannot be told apart, 34.5 against 35.3 Hz. A course project on an uncalibrated rig.
  21. 2020MethodOhara et al. 2020 Development of an Archery Robot for the Selection of ArrowsA 114 kg machine shoots six arrows at 70 m: the two defective ones land outside the group, each always on the same side. The machine separates the arrow from the archer.
  22. 2022ExtensionMurakami et al. 2022 Automatic image analysis of an archery arrow in free flight using high-speed video camerasAutomatic image analysis of 35 X10 arrows shot by a robot: the flex carries on in flight, at around 75 Hz.
  23. 2024ExtensionSuetsugu et al. 2024 Buckling Fracture Behavior of Duralumin Arrows used for Japanese ArcheryDuralumin kyudo shafts tested on the bench buckle at about 50 and 65 N, through instability rather than material overload, in the case where the arrow is squeezed between string and bow.
  24. 2025ContradictionKuch et al. 2025 Improving the arrow selection process in archery: the stiffness variation pattern mattersFifty-nine X10 shafts of one spine, measured one by one and shot by machine at 70 m: mass, mean spine and the variation of stiffness around the shaft matter, while straightness, already well inside tolerance, enters no model.
  25. 2025ExtensionWilliams 2025 Modelling Arrow Dynamics for Optimised Recurve Archery Arrow SelectionThe 2012 model rewritten to build selection charts: overall it reproduces the manufacturers' charts, but at the extremes finds them up to four sizes too stiff. Not tested with shots.

Studies in this line

Related claims

Claims in the registry with at least one of these studies among their evidence.

85 claims · 27% with evidence on archers

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Cite
APA

Campagna, M. (2026). The archer's paradox in research: from 1987 to today. In Research. matteocampagna.com. Retrieved 16 October 2026, from https://matteocampagna.com/en/research/archers-paradox-arrow-dynamics

BibTeX
@misc{campagna2026_enresearcharchersparadoxarrowdynamics,
  author = {Campagna, Matteo},
  title = {The archer's paradox in research: from 1987 to today},
  year = {2026},
  howpublished = {Research, matteocampagna.com},
  url = {https://matteocampagna.com/en/research/archers-paradox-arrow-dynamics},
  note = {accessed 2026-10-16}
}