Arrow mass and ballistics: Kuch 2019 study

The French federation wanted to know how much the Easton X10 shafts reaching its archers actually weigh, and how much the weight difference between one shaft and the next matters on the target. The authors weighed 26 dozen bare, unbuilt shafts in seven spines, and not one dozen stayed within the manufacturer's stated tolerance of 0.5 grains (about 0.032 grams) between two shafts from the same pack. When shafts of the same spine from different dozens were pooled, as archers do, the gap between the lightest and the heaviest ran from 1.96 to 3.69 grains, and reached 11.64 grains for the 380s.

They then used a simplified flight model to work out where a lighter or heavier arrow lands at 70 metres, compared with a reference arrow, for the same bow energy. Half a grain more lowers the impact by 0.87 cm; half a grain less raises it by 0.94 cm. A difference of 1.96 grains moves the impact 3.59 cm up or 3.52 cm down, 3.69 grains moves it 6.76 or 6.67 cm, and 11.64 grains moves it 21.3 or 20.93 cm. According to the model, shafts of one spine would need to fall between 3.35 grains lighter and 3.37 grains heavier than the reference to stay in the 10, and only the 325, 410, 500 and 600 shafts did. With mixed 380 shafts, weight alone would spread the shots between the 7 and the 10; with the 350 and 550 shafts, every arrow would still score at least 9.

The shafts were Easton X10s of the kind used at the Olympic Games: three dozen each in spines 325, 350, 380 and 410, five dozen each in 500 and 550, and four dozen in 600. They were weighed bare, without point, fletchings or nock, on an A&D FZ-200i scale reading to a thousandth of a gram; the shaft makes up roughly two thirds of the finished arrow's mass. The spread between the lightest and heaviest shaft was worked out for each dozen, and then across all shafts of the same spine.

The ballistic part is a calculation, not a shooting test. All the elastic energy of the bow, set at 45 J, is assumed to become kinetic energy in the arrow, with no friction losses; the arrow then flies in the vertical plane under its own weight and air resistance, with a drag coefficient of 1.5 taken from a Japanese study of arrow aerodynamics. The reference arrow weighs 23.361 g (360.5 grains), leaves from a height of 1.7 m at an angle of 5.1 degrees and hits the centre of a target at 70 metres, set about 130 cm off the ground. The equations of motion are solved in Matlab.

This is a three-page preliminary conference paper, and the model says so: it considers mass alone. Spine, the arrow flexing around the bow, diameter, balance point and wind are all left out, and the energy delivered to the arrow is assumed to be the same for arrows of different weight, whereas a heavier arrow usually takes slightly more energy from the bow. The calculated shifts should therefore be read as an order of magnitude: no arrow was shot, and the step from weight to score assumes shots that are identical in every other respect.

The shafts are those supplied in France at the time, with no information on production batches, and the figures for each dozen appear only in a chart. The bare shaft is what was weighed, while a finished arrow adds a point, fletchings and a nock, each with its own weight tolerance.

For a recurve archer the useful figure is the order of magnitude: according to this model, at 70 metres one grain of difference between two arrows is worth almost 2 cm vertically, and the differences found within a single supply can exceed the radius of the 10 ring, which is 6.1 cm. The study does not say how much of this shows up in practice, where spine, point and release also play their part, but it explains why some archers weigh their shafts and sort them by weight before building arrows. It does not set a tolerance to aim for: it estimates how much weight matters on its own.