What the bow does not give the arrow stays in its moving parts: recoil and nock separation count for next to nothing

The summary today

The full account exists for a single bow, a compound measured in the laboratory and published in 2015 as a preprint. The bow makes 71.0 joules available and the 20.6 gram arrow carries off 57.1, or 80 per cent. The author attributes the rest, almost 14 joules, to the parts of the bow that move with the string: he does not measure it on its own but obtains it by difference, after showing that the other items are tiny. The bow's recoil, measured with an accelerometer, comes to 0.23 joules, pulling the nock off the string about 0.05, arrow oscillation and string shortening a few tenths of a joule each. Expressed as a virtual mass, it is as if the arrow dragged about 5 grams along with it, and half of those grams are in the string, the serving, the D-loop and the peep sight. The formula that follows stands up to the chronograph, since the speeds of three arrows of different mass fall on the predicted curve; two practical consequences, that a heavier arrow makes the bow more efficient and that a lighter string sends the arrow off faster, are deductions from the formula rather than separate measurements. The models reach the same picture by another route: in 1981 a calculation with an elastic string gave the arrow 78 per cent of the energy and left almost all the rest in the motion of limbs and string, with air resistance and hysteresis under 2 per cent, and in 1997 a model put the arrow's lateral motion at 0.05 of the drawn bow's 30.2 joules. There is one limitation, and it is a large one: one bow, one laboratory, one author, and the registry holds no matching measurement on a recurve.

Also phrased as: "bow efficiency"; "virtual mass of the bow"; "bow recoil energy"; "light string and arrow speed"; "where the bow's energy goes"

Evidence

• 1969
  Context Model Published later

  An analytical model of the recurve that divides the energy between arrow, limbs and string. According to the author it explains why heavy limbs give such a strong recoil that the bow may fly from the hand, and why a dry fire often breaks the bow or the string: the energy has not left the system through the arrow. It is a qualitative remark, with no figures.

  Ballistics of the Modern-Working Recurve Bow and Arrow · Burton G. Schuster

• 1981
  For Model Published later

  A model with an elastic string, applied to the longbow described by Hickman: at arrow exit 78.1 per cent of the stored energy is in the arrow, 11.0 in the motion of the limbs and 9.8 in that of the string, while the potential energy left in string and limbs is 1.9 per cent. Air resistance and hysteresis come, by the author's estimates, to under 2 per cent. The inextensible-string model predicted an efficiency of 92 per cent instead, against measured values of 70 to 85 cited by the author.

  Bow and arrow dynamics · W. C. Marlow

• 1997
  Context Model Published later

  A model of the archer's paradox: the energy of the arrow's lateral motion reaches at most 0.05 Nm, against 30.2 Nm available in the fully drawn bow, about 0.2 per cent. The authors use it to justify calculating the lateral motion separately.

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

• 2015
  For Equipment tests Published later

  A sample compound with a 20.62 gram arrow: available energy 71.0 ± 0.5 joules, arrow kinetic energy 57.1 ± 0.8, efficiency 0.80. The bow's virtual mass derived from those two figures is about 5 grams, of which the string accounts for 1.1 (linear density 3.3 grams per metre), the serving 0.7, and the D-loop and peep sight together 0.7: half the virtual mass is in the string and what is attached to it. Chronograph speeds for three arrows of different mass follow the resulting formula, so efficiency rises with a heavier arrow.

  Applications of Physics to Archery · H. O. Meyer

• 2015
  For Equipment tests Published later

  Recoil is tiny: with the 6.5 kilogram bow, conservation of momentum predicts about 0.18 joules; a wireless accelerometer on the bow records a peak of about 37 metres per second squared for 6 to 8 milliseconds, a recoil speed of 0.26 ± 0.04 metres per second and so 0.23 ± 0.06 joules, against 71.0 available. Nock separation costs even less: hanging weights from a nocked arrow shows that about 10 newtons are needed to pull it off the string; from the bow's force curve that force is reached about a centimetre past brace height, and the work spent in the snap is about 0.05 joules.

  Applications of Physics to Archery · H. O. Meyer

• 2015
  Context Equipment tests Published later

  The arrow stays on the string for about 13 milliseconds, confirmed by the 12 to 14 of string noise in the audio recording, with a peak force of about 200 newtons and an acceleration of about a thousand times gravity. The same account also puts arrow oscillation energy, about 0.43 joules at a 2 centimetre amplitude, and the energy lost as the string shortens, 0.2 to 0.3 joules, among the negligible items.

  Applications of Physics to Archery · H. O. Meyer

Status history

1. 1969 Not studied An analytical model of the recurve divides the bow's energy between arrow, limbs and string, and explains qualitatively why heavy limbs give a strong recoil and why a dry fire breaks the bow: the energy that does not go to the arrow stays in the system. No figure says how much stays or where: the question is still not studied.
2. 1981 Transferred A model with an elastic string makes the first count: at arrow exit 78 per cent of the energy is in the arrow, 11 in the motion of the limbs, almost 10 in that of the string and under 2 remains as potential energy, while air resistance and hysteresis come to under 2 per cent. The author notes that the measurements of the time gave efficiencies between 70 and 85 per cent, whereas the inextensible-string model predicted 92. What does not go to the arrow stays in the moving parts, but it is a calculation: transferred.
3. 1997 Transferred A model of the archer's paradox calculates that the energy of the arrow's lateral motion reaches at most 0.05 of the 30.2 joules available in the drawn bow, about 0.2 per cent. It confirms that the arrow's oscillation counts for little in the balance, but it is still a model: transferred.
4. 2015 Reported The only complete measurement arrives, on a compound: the 20.6 gram arrow carries off 57.1 of the 71.0 joules available, an efficiency of 0.80, and the rest goes into the moving parts of the bow, a virtual mass of about 5 grams, half of it string and accessories. The measured recoil comes to 0.23 joules and nock separation to about 0.05. The measured efficiency is close to the 78 per cent of the 1981 model. One bow, one laboratory, in a preprint: reported.

Related

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• How much energy actually reaches the arrow, and where the rest goesHow much of the drawing work reaches the arrow: models say about three quarters, while the few measurements show where the rest goes and what the arrow changes.

Research lines

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

Other claims in this domain

• CorroboratedAlong its flight the arrow loses an appreciable part of its launch speed12 studiesStatus since 2011
• CorroboratedAs it leaves the bow, the arrow flexes and snakes around the riser instead of leaving straight19 studies · 47 archers measuredStatus since 2012
• 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

See all claims on equipment and tuning