The mass of the limbs and cams makes little difference to arrow speed

The summary today

Nobody has measured it: the question has only been put to models of the bow, and the models say something narrower than the statement seems to say. In the 1937 calculation for a yew bow, at the loose the push on the arrow is lower than the force on the fingers but falls more slowly, because the limb tips, now moving, hand their kinetic energy back to the arrow: this is why massive tips take little speed away. In the 2009 compound model the final speed does not change at all with the effective mass of the limbs or the inertia of the cams; only the way the arrow accelerates changes, with more limb mass giving a higher, sharper and later peak in the push, and more cam inertia starting the push later. That model, though, fixes the energy reaching the arrow through a virtual mass derived from speeds measured on the bow, so it says that how the mass is distributed matters little, not that heavier limbs never slow the arrow. And the 1981 model with an elastic string shows that the idea of energy being handed back in full cannot be taken for granted: when the arrow leaves, 11 per cent of the stored energy is still in the moving limbs.

Also phrased as: "light limbs and arrow speed"; "virtual mass of the bow"; "limb weight and arrow speed"; "cam inertia"; "heavy limb tips"

Evidence

• 1937
  For Model Published later

  Dynamic model of a yew bow with an inextensible string: the dynamic force on the arrow starts lower than the static force on the fingers but falls more slowly, thanks to the kinetic energy of the limb tips. For this reason, the author writes, massive tips do not reduce arrow speed by very much: the energy spent setting them in motion returns to the arrow later. Curves only, no tabulated speed values.

  The Dynamics of A Bow and Arrow · C. N. Hickman

• 1981
  Context Model Published later

  The 1937 longbow model redone with an elastic string: when the arrow leaves, 11 per cent of the stored energy is still kinetic energy of the limbs and 9 per cent of the string, and efficiency falls from 92 to 78 per cent. It does not vary limb mass, but it shows that the energy spent moving the limbs does not all return to the arrow, which was the assumption of the 1937 calculation.

  Bow and arrow dynamics · W. C. Marlow

• 2009
  For Model Published later

  Dynamic model of a single-cam AR34 compound: the arrow's final speed does not change with the effective mass of the limbs or the inertia of the wheel and cam; only the acceleration profile changes. With more limb mass the peak force on the arrow becomes higher, sharper and later; with more cam inertia the main push starts later (comparisons in graphs only, reference inertia 0.00014 kg m²). The energy reaching the arrow is set by the force-draw curve and by a virtual mass derived from the speeds of arrows of different mass.

  A compound archery bow dynamic model, suggesting modifications to improve accuracy · J L Park

Status history

1. 1937 Transferred The first dynamic model of the bow, worked out for a yew bow with the string treated as inextensible, finds that at the loose the push on the arrow is lower than the force felt on the fingers, but then falls more slowly than the static force because the limb tips, now moving, hand their kinetic energy back to the arrow. The author concludes that massive tips do not reduce arrow speed by very much, and adds that the late push may not help accuracy. It is a calculation, with curves and no tabulated speed values: transferred.
2. 1981 Transferred A model that redoes the 1937 one with an elastic string does not vary limb mass, but it casts doubt on the assumption behind that calculation: with a string that stretches, the arrow leaves while the limbs and string are still moving, and at that moment 11 per cent of the stored energy is in the limbs and 9 per cent in the string. That energy never reaches the arrow, and the calculated efficiency falls from 92 to 78 per cent. The picture is still made of calculations alone: transferred.
3. 2009 Transferred A dynamic model of a single-cam compound, the AR34, built with almost every parameter measured on the bow, reaches a sharper conclusion: the arrow's final speed does not change with the effective mass of the limbs or the inertia of the cams; only the acceleration profile changes. More limb mass gives a higher, sharper and later peak in the push; more cam inertia starts the push later, and the string and arrow stay near the face for longer. The model, though, fixes the energy reaching the arrow through a virtual mass derived from measured speeds, so it says that how the mass is distributed matters little, not how much of it there is. Two models in favour, no measurement: transferred.

Related

Related articles

• 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
• 12 studiesCompound mechanics: cams, limbs and symmetry

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

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