Recurve bow and arrow ballistics: Schuster 1969 study

Schuster, of the National Center for Atmospheric Research in Boulder, builds a mathematical model of the modern working recurve, the kind in which the whole limb works and unrolls during the draw, and derives the equations of motion of bow and arrow, which he then integrates by computer. In the idealised model, with no friction, no stray vibration and a string that does not stretch, all the stored energy goes into the arrow except the part needed to move the string: efficiency is 100 divided by (1 plus a third of the ratio of string mass to arrow mass). This is the same practical result, already known from measurements, that the effect of the string is equivalent to adding a third of its mass to the arrow. At the moment the arrow leaves, the limbs are at rest.

The simulations, with a 350-grain arrow, show that during the power stroke the limbs absorb part of the energy and then release it all at once, producing a spike in the arrow's acceleration. With limbs ten times lighter the acceleration rises smoothly and follows the draw curve, while the arrow's velocity and displacement change little. Over the first few inches a simple approximation follows the measured draw curve very closely, up to 6.5 inches; the full model's curve comes close enough to that of a commercial recurve to be used in the calculations.

For flight after the bow, for the same energy the drop of the trajectory grows in proportion to arrow mass: a light arrow flies flatter, but according to the author the archer pays for that with a more abrupt transfer of energy, which can disturb the aim. For hunting, penetration depends on momentum, which favours heavier arrows.

It is a theoretical paper. The working section of the limb is described as a circular arc that unrolls along its initial tangent, with the string peeling away from the limb as the bow is drawn. For the stored energy Schuster uses an idealised limb made of two layers of fibre held apart by a core, of constant thickness and with a width that tapers linearly towards the tip; the model is normalised to a point measured at 11 inches of draw. The kinetic energies of limbs, string and arrow enter an energy balance from which acceleration, velocity, displacement and energy partition over time are derived, calculated for two values of limb mass. The section on free flight is a first-order estimate with no air drag; the one on penetration treats the target as uniform tissue with resistance proportional to speed and depth.

The near-100% efficiency is a consequence of the assumptions, not a measurement: the model leaves out hysteresis losses in the materials, vibration and string stretch, all of which matter in real bows. The only comparison with real data is the draw curve of a commercial recurve, which the author himself says resembles the model only in part; calculated velocities and accelerations are not checked against measurements. The arrow is treated as a point moving along its own line, with no flex and no lateral effects, and free flight takes place without air. It is also a paper from 1969, on bows with glass-fibre and wood limbs: the mass ratios have changed with modern materials. The archive PDF is a scan and the text was read through optical character recognition: not every formula is legible, but the prose is.

For anyone shooting today its value is mainly historical and a matter of principle. The model explains why a lighter string performs better (it counts for a third of its mass, as if attached to the arrow) and why light arrows and heavy limbs make for a harsher shot: with aluminium in place of wood, the author notes, the arrow goes faster but the bow kicks more, and a bow dry-fired without an arrow risks breaking because the energy has nowhere to go. Choosing arrow weight remains a trade-off between a flatter trajectory and a more manageable shot, which the author leaves to each archer's own abilities.