As it leaves the bow, the arrow flexes and snakes around the riser instead of leaving straight

The summary today

The shaft does flex as it leaves the bow, and that is no longer an inference from the laboratory. High-speed films of 16 world-level archers show the string driving the nock 13 mm towards the body in just over five milliseconds, after which the shaft straightens again before it passes the riser. The pressure button, the lateral support the arrow rests against at full draw, lets go of it within 8 ms, while the string releases it only at 14.6 ms: by the time the tail reaches the riser the arrow is no longer resting on anything and passes without touching it. Films of a Polish national-team archer, twenty years earlier, give the same picture, and an accelerometer mounted on the bow of an Italian national-team archer describes the same sequence from the other end, that is from the force the shaft unloads into the bow. No study in the bibliography points the other way, and on the bench the arrow flexes even when a machine does the shooting, so the flexing is not a fault of the release. Two independent models also give the condition for the shaft to pass without touching: it has to complete a full cycle of vibration while still on the string, and that depends on spine. Two limits remain: the shaft snaking around the riser without touching it is predicted by the models rather than filmed, and the figures hold for the finger release, since with a compound bow and a mechanical release the flexing drops to a few millimetres.

Also phrased as: "archer's paradox"; "the arrow bends as it leaves the bow"; "why the arrow does not hit the riser"; "arrow flex at release"; "how much the arrow flexes on release"

Evidence

• 1987
  For Equipment tests Published later

  One Italian national-team archer, three stabiliser set-ups of 15 shots each, the bow's lateral acceleration measured with an accelerometer on the riser and sampled at 2,500 Hz. From the phases of the signal the authors read that over the first 22 ms the flexing shaft pushes the button to the end of its travel, between 22 and 26 ms the button pushes back, and between 26 and 30 ms the arrow has already left the riser while still on the string. It is the same sequence, read from the bow's side.

  Reduction of Lateral Bow Displacement Using Different Torque Flight Compensators and Stabilizers in Archery · L. M. Leonardi, C. Gallozzi, A. Pace, A. Dal Monte

• 1990
  For Archers Published later

  High-speed films (1,000 to 2,500 frames per second) of a Polish national-team archer, plus a shooting machine: the shaft flexes sideways while the string drives it, in the same way with the archer and with the machine, and the mathematical model reproduces its shape. On the machine the case where it does not come off is also visible, since a shaft that is too weak ends up striking the bow with its fletching after separation. Olympic and world medallists were also filmed, but their data are not reported.

  Experimental and theoretical research in archery · Roman Pekalski

• 1997
  Context Model Published later

  A mathematical bow and arrow model applied to ancient bows: the arrow leaves the string when the accelerating force falls to zero, which happens after the centre of the string has passed the braced position, because the string stretches. It fixes the instant of separation that all the other works use as a reference, without measuring the flexing.

  An approach to the study of ancient archery using mathematical modelling · B. W. Kooi, C. A. Bergman

• 1997
  For Model Published later

  A model of the arrow as a vibrating beam, pushed by the string and held by the button, with the bow and arrow parameters of a Polish national-team archer (a 66 inch, 34 lb Hoyt Pro Medalist and an Easton 1714X7). It reproduces the shapes filmed in 1990, and at 10 and 12 ms better than the model that came with them: the shaft bends three times before the nock passes the button, the second time the opposite way as the string reaches brace height, at around 15 ms, and the third like the first, which helps the vanes clear the bow. In the model the arrow leaves the string just as the nock crosses back over the plane of the bow; a weaker shaft strikes the button. The comparison with the films is by eye only.

  On the mechanics of the arrow: Archer's Paradox · B.W. Kooi, J.A. Sparenberg

• 1998
  For Model Published later

  A fuller model, checked against the 1990 films: contact with the button is not continuous, it drops to nothing for an instant just after release, resumes and ends before the arrow leaves the string, and the nock crosses back over the bow's median plane at about the moment the string releases it, so the shaft snakes around the riser instead of striking it with its tail. The author also notes that allowing for longitudinal compression predicts more flexing than models which ignore it. It is a simulation, with no new measurements on archers: the experimental curves are those of a single shot, read off a 1990 figure, and the comparison between the old model and the new one is made by superimposing the drawings, with no measure of the discrepancy. The new model also uses one free parameter fewer, since it does away with the nock's initial offset, and the fit is not corrected for that difference.

  Bow-arrow interaction in archery · B. W. Kooi

• 1998
  Context Archers Published later

  Nine archers from the German national squads, with an accelerometer at the centre of the bow and high-speed films: the arrow separates from the string six to eight milliseconds before the first maximum of the bow's acceleration curve. It locates in time the phase in which the arrow is still on the string, on archers, but it does not measure the shaft's flexing.

  Coordinative Aspects of Archery - An Approach Using Surface-Electromyography · Jürgen Edelmann-Nusser, Albert Gollhofer

• 2001
  Context Model Published later

  A model applied to a 202 N bow with two 660 mm arrows: the stiffer one stays in contact for 16.2 ms against a vibration period of 15.6 ms, leaves in the first quarter of the next cycle and passes cleanly; the softer one is in contact for 15.5 ms against a period of 18.9 ms, leaves after three quarters of a cycle and strikes the riser with its tail. The paradox works if the arrow completes a full cycle while still on the string, and that depends on spine. The practical point is that the static buckling test, with the rule of a draw weight two and a half times above the critical force, picks the other arrow: 2.5 times 84.3 makes 211 newtons against the bow's 202, while the stiff one would give 290. The two methods point to different shafts, because in the static test spine does not depend on mass and in the shot it does. In the bow and arrow system, moreover, the critical dimensionless force is 14.79 against the static test's 9.87, so the test underestimates the threshold by a third. All of this is calculation: no arrow was shot.

  Lateral deflection of archery arrows · Zanevskyy

• 2007
  Context Model Published later

  A book chapter in which the author summarises his own work on bow mechanics. It sets the arrow shapes predicted by the 1997 model against those filmed on the Polish archer once more, every 2 ms up to 12 ms, that is until the nock passes the grip, and the agreement is still graphical. It brings no new measurements of the flexing.

  Design and materials in archery · B.W. Kooi

• 2009
  Context Model Published later

  A bow and arrow model compared with a single figure: the predicted period of the two-node bending mode is about 18.4 ms, against the 18 to 19 ms of joint bow and arrow motion read off a 1992 demonstration film by an accessory maker, on a 712 mm, 14.3 g shaft. That film is not a study, and it is the work's only experimental support. It gives the same quantity, about one full cycle before the arrow leaves, and estimates the deformation amplitude at around 5% of the arrow's length.

  Modeling of the Archery Bow and Arrow Vibrations · I. Zaniewski

• 2011
  Context Model Published later

  The same approach applied to a compound bow with ProTour 470 arrows: peak flexing before the arrow leaves is a few millimetres, against the 10 to 15 mm expected with a recurve bow, and the arrow leaves the rest at 14 ms, the string at 17 ms and the bow at 20 ms. The stroboscopic photographs also show that the second harmonic is clearly present and can give the shaft an S shape. It marks the limit of the claim, which concerns the finger release. Worth flagging, because it is what the other models promise and do not do: the dominance of the nocking point's path over the longitudinal force is shown by elimination, holding the force constant and seeing that nothing changes, and separation from the blade is predicted at 13.83 ms against the 14 measured photographically. The agreement between calculation and measurement, however, is declared excellent below 14 ms and less good after that, which is precisely on the maximum flexing and the exit condition, and in the horizontal plane between 10 and 14 ms no usable camera position was available.

  The Behaviour of An Arrow Shot from a Compound Archery Bow · J L Park

• 2012
  For Archers Published later

  16 world-level recurve archers (8 men and 8 women) filmed at 4,000 frames per second at the London 2011 test event: at release the string drives the nock an average of 13.1 mm towards the body (range 9 to 20 mm, standard deviation 2.8 mm) at 5.4 ms, the shaft is straight again at 8.6 ms and leaves the string at 14.6 ms, while contact with the pressure button is over within 8 ms. While the string runs off the fingers, string and nock move sideways by 4.7 mm on average, from 3 to 6 mm. It is the largest sample in the bibliography and comes from a group independent of the 1990 one. The times are those in the paper's table; in the text two of them appear a tenth of a millisecond higher.

  High-speed video analysis of arrow behaviour during the power stroke of a recurve archery bow · James L Park, Oliver Logan

• 2012
  For Model Published later

  The same author as the films, this time with a recurve bow in a shooting machine checked against calculation: with the sight centred on a target at 20 m, the arrow at full draw is aligned about 220 ± 25 mm to the right, against 226 mm predicted, because its centre of mass picks up a small sideways velocity to the left (for a right-handed archer) as it leaves the bow. In the model the shaft's first bending mode is at 74 Hz. It confirms that the arrow does not leave in the direction it is pointing, but without archers.

  Arrow behaviour in the lateral plane during and immediately following the power stroke of a recurve archery bow · James L Park

• 2012
  Context Equipment tests Published later

  Three types of carbon shaft, three specimens of each, modelled and measured under free-free conditions: in the first bending mode the shaft has two still points, at about 139 to 145 mm from the nock and at 636 to 647 mm on a 710 mm shaft, with models and measurements agreeing. It says where the bending nodes seen in the films sit, with no bow and no archer.

  Modelling the three-dimensional vibration of composite archery arrows under free–free boundary conditions · Marianne Rieckmann, James L Park, John Codrington, Ben Cazzolato

• 2013
  Context Model Published later

  The same flexible arrow model already used in the horizontal plane, applied to the vertical plane of a recurve: the vertical flexing is about ten times smaller than the horizontal. In the paper test with a shooting machine, from 0.8 to 12 m, point and vanes pass less than 1 mm apart vertically; raising the nock by 10 mm makes the tear at 2 m reach 40 mm. It confirms that the flexing that matters for the paradox is the lateral one.

  Arrow behaviour in the vertical plane during and immediately following the power stroke of a recurve bow · James L Park

• 2014
  Context Equipment tests Published later

  Four arrow types shot from a bow fixed in a machine, six shots each, flexing read from frames set from 80 cm onwards at about 300 km/h: the shaft flexes sideways by between 4.63 and 5.53 mm. With no fingers and no archer the flexing is still there, so it does not arise from an imperfect release; the arrows, however, are described neither by spine nor by weight.

  Measurement of Archer's Paradox Size using Multiple Frames · Jonggeun Kim, Yeongsang Jeong, Moonjae Song, Sungshin Kim

• 2015
  Context Equipment tests Published later

  A robot with mechanical fingers compared with an international-level archer: the amplitude of the flexing as the arrow leaves is 6.94 ± 0.23 mm with the sliding release against 6.86 ± 0.34 mm for the archer, a difference of 0.08 mm. A device that reproduces human flexing that closely is good evidence that the measurement is sound, not a study of archers.

  2A2-T08 Development of Archery Robot : Development of finger mechanism for controllable the bending vibration · Jun Wakatabe, Chisato Kanamori, Ryu Ando, Takeshi Miyazaki

• 2015
  Context Equipment tests Published later

  A calculation, not a measurement. Using beam theory and values chosen for the example (initial speed 71 m/s, a 0.6 m shaft, a frequency of 85 Hz), the author shows that with a finger release the nock and the middle of the arrow wind around the bow with room to spare. His measurements, by contrast, concern a single compound shot with a mechanical release and by a machine, with 74.5 m/s and a 0.696 m shaft measured: for that bow, he writes, the paradox matters little, because with a release aid the sideways oscillation is smaller.

  Applications of Physics to Archery · H. O. Meyer

• 2019
  For Review Published later

  A review of target archery: from a well-tuned recurve bow the arrow leaves with large lateral flexing and small vertical flexing, and the behaviour depends mostly on the choice of arrow and on the way the string leaves the fingers. It also adds the limit: among compound bows, design differences are such that no general rule can be given for predicting flexing and departure angles.

  The impact of the atmosphere on target archery · James L Park

• 2022
  Context Equipment tests Published later

  Thirty-five X10 arrows shot by a robot and tracked with high-speed cameras and automatic image analysis: the shaft's apparent length comes out up to about two centimetres shorter than its real length, with a periodic pattern, from which a bending frequency of around seventy-five hertz is estimated. The flexing does not end as the arrow leaves the bow, it carries on in flight.

  Automatic image analysis of an archery arrow in free flight using high-speed video cameras · Riku Murakami, Hiroya Mamori, Takeshi Miyazaki, Naoya Takahashi

Status history

1. 1987 Reported The first direct measurement does not look at the arrow, it looks at the bow. An accelerometer mounted on the bow of one Italian national-team archer records the lateral acceleration at 2,500 Hz, and three phases can be read from the signal: over the first twenty-two milliseconds the flexing shaft pushes the button to the end of its travel, between twenty-two and twenty-six the button pushes back, and between twenty-six and thirty the arrow has already left the riser while still on the string. It is the sequence of the paradox read from the bow's side, on a single archer: reported.
2. 1990 Reported High-speed films of a Polish national-team archer and the model that reproduces them show the phenomenon from the arrow's side, including the case where it does not come off, since a shaft that is too weak ends up striking the bow with its fletching after separation. The status does not rise: each study still has a single archer.
3. 1997 Reported A model treating the shaft as a vibrating beam, pushed by the string and held by the button, reproduces the shapes filmed on the Polish archer, and at ten and twelve milliseconds reproduces them better than the model that came with them: the arrow bends three times before the nock passes the button, and the last bend helps the vanes clear the bow. It leaves the string just as the nock crosses back over the plane of the bow, and a weaker shaft ends up against the button. The comparison with the films is by eye only, and the measurement is still that of one archer: reported.
4. 1998 Reported A fuller model of the interaction between bow and arrow reconstructs the whole sequence: contact with the button is not continuous, it drops to nothing for an instant just after release, resumes and ends before the arrow leaves the string, and the nock crosses back over the bow's median plane at about the moment the string releases it, that is before it passes the handle. The same year, on archers from the German national squads, films place separation from the string six to eight milliseconds before the bow's first maximum deflection. The mechanism closes, but the measurement on more than one archer is still missing: reported.
5. 2001 Reported An independent model puts the condition into figures: the arrow snakes around the riser if it completes one full cycle of vibration while still on the string. In the worked example, the tuned arrow stays in contact for 16.2 milliseconds against a period of 15.6 and passes cleanly, while the untuned one leaves after three quarters of a cycle and brushes the handle with its tail. The picture gets stronger but is still made of models and of measurements on one archer at a time: the status holds.
6. 2007 Reported A review chapter by the author of the 1997 model sets the calculated flexing against the flexing filmed in 1990 once more, up to the moment the nock passes the grip, and the agreement is still graphical. No new measurements arrive, and nobody has yet measured the flexing on more than one archer: reported.
7. 2009 Reported The quantitative description of the flexing arrives. A model checked against high-speed footage gives almost one full cycle of two-node bending while the arrow is on the string, with an amplitude of around five per cent of the shaft's length, over the eighteen or nineteen milliseconds in which bow and arrow move together. It is the how, not a new measurement on archers: the status does not change.
8. 2011 Reported Stroboscopic photographs show that the arrow does not flex in the fundamental mode alone: the second harmonic is clearly present and can give the shaft an S shape. It adds detail about the how and does not shift the status.
9. 2012 Corroborated The measurement on more than one archer arrives. In sixteen world-level recurve archers the string drives the nock 13.1 mm towards the body at 5.4 ms, with values from 9 to 20 mm from one archer to the next, and the shaft is straight again at 8.6 ms, before it passes the bow. On the bench, with a recurve bow in a shooting machine and the sight centred on a target at twenty metres, the arrow at full draw is aligned 220 mm to the right, against 226 mm calculated, because its centre of mass picks up a small sideways velocity as it leaves the bow; and vibrometry places the two nodes of the first bending mode at 140 and 640 mm from the nock. Two independent research groups have now measured the flexing on archers, the tests on the equipment point the same way, and no study in the bibliography points the other way: corroborated.
10. 2013 Corroborated The flexible arrow model is applied to the vertical plane and finds flexing about ten times smaller than in the horizontal; in the paper test with a shooting machine, from 0.8 to 12 metres, point and vanes pass less than a millimetre apart vertically. It confirms that the paradox is played out in the horizontal plane, the plane of the riser, and the status stays corroborated.
11. 2014 Corroborated After the shooting machine of 1990, further evidence arrives that the flexing does not come from the fingers: with the bow fixed in a machine, with no human release, four types of arrow still flex sideways, by between four and a half and five and a half millimetres, over the first few metres. The status stays corroborated, and the idea that the flexing is a fault of the release loses further ground.
12. 2015 Corroborated Two supporting confirmations. The calculation with beam theory, using values chosen for the example, shows the nock and the middle of the shaft clearing the bow with room to spare when the release is with the fingers, while with a mechanical release the oscillation is smaller. And a robot with mechanical fingers reproduces the flexing amplitude of an international-level archer to within eight hundredths of a millimetre, which is to say the measurement is sound. The status does not change.
13. 2019 Corroborated A review adds the shape of the phenomenon and its limits: from a well-tuned recurve bow the arrow leaves with large lateral flexing and small vertical flexing, and it depends mostly on the choice of arrow and on the way the string leaves the fingers; among compound bows, on the other hand, design differences are such that no general rule can be given. The status does not change.
14. 2022 Corroborated Automatic analysis of high-speed images puts a figure on the flexing that survives into flight, about seventy-five hertz on X10 shafts shot by a robot. Context that confirms, without shifting the status.

Related

Related articles

• Fishtailing and porpoising: what arrow flight saysDoes your arrow wag sideways or wave up and down? What causes fishtailing and porpoising, which to fix first, and how to confirm it with a bare shaft test.
• The archer's paradox: the physics of the flexing arrowAt full draw the arrow points to one side and still reaches the centre: how it flexes and clears the riser in the first twenty milliseconds, and how spine matches it to the bow.
• Stabilisers in archery: what they actually do, beyond the folkloreWhat research has measured about stabilisers: how the bow moves at release, muscle work, vibration, barebow weights and accuracy.

Research lines

• 29 studiesThe archer's paradox in research: from 1987 to today
• 22 studiesHow much energy passes from bow to arrow: models and measurements since 1937

Other claims in this domain

• CorroboratedAlong its flight the arrow loses an appreciable part of its launch speed12 studiesStatus since 2011
• ReportedA camera-based scoring system misreads some arrows: the ones barely sticking out and the ones that shift the target face7 studies · 2 archers measuredStatus since 2017
• ReportedA chest band with fibre optic sensors measures breathing and mean heart rate with small errors, and an archer can wear it while shooting2 studiesStatus since 2019
• ReportedA compound bow stores between 70 and 100 joules and sheds about two thirds of the force at full draw4 studiesStatus since 2015
• ReportedA damper tuned to the tremor frequency reduces the bow's vibration while aiming1 studyStatus since 2017

See all claims on equipment and tuning