The physics of bow and arrow: Meyer 2015 study

A physicist at Indiana University takes a compound bow and an arrow and measures everything that can be measured with garage tools: he weighs the parts, bends the shaft, makes the arrow vibrate with a magnet, draws the string in small steps photographing every step, times the launch, and records the noise of the shot with two microphones to find out how much the arrow slows down.

The arrow is an Easton LSR 500: 0.696 metres long, 20.62 grams, shaft radius 3.63 millimetres with a wall of 0.50, centre of mass 0.414 metres from the nock, FOC 0.093. Measuring its stiffness (5.17 newtons per square metre) he derives its spine according to the Archery Trade Association definition: 492, close to the 500 stated by the manufacturer. Ten arrows of the same type, rolled on two supports, deviated from a straight line by between 10 and 100 micrometres, 55 on average. Set vibrating, the arrow resonates at 84.8 hertz, with nodes at 0.125 and 0.630 metres and a quality factor of 400: the oscillation dies away in about 1.5 seconds, longer than the flight time in most target shooting, so when the arrow reaches the target it is still oscillating.

The bow is a Hoyt Contender, with a brace height of 0.185 metres, a draw of 0.481, a peak force around 200 newtons and a let-off of about 65 per cent. The archer hands it 73.7 joules, the bow gives back 71.0 (the rest goes into internal friction and the cams) and the arrow receives 57.1: efficiency is 0.80, and the difference corresponds to a virtual bow mass of about 5 grams, of which the string and its serving account for nearly half. The acceleration lasts about 13 thousandths of a second and amounts to about a thousand times gravity; the axial load squeezing the arrow over that stretch, about 60 newtons, is a little over half the critical buckling load computed from Euler's formula, about 100 newtons.

The arrow leaves at 74.5 metres per second and loses speed at a rate of 2.31 thousandths per metre, which corresponds to a drag coefficient of 1.94. From here comes the result of most interest to anyone competing: at 70 metres, an error of 1 metre in the distance moves the point of impact by about 7 centimetres vertically, 1 metre per second more launch speed raises it by about 13 centimetres, and a 20 per cent change in aerodynamic drag moves it by about 7 centimetres.

Every measurement is described so that it can be repeated. The stiffness of the shaft is obtained by resting it on two knife edges 0.550 metres apart, hanging known weights at the centre and reading the deflection with a dial indicator: the displacement grows in proportion to the force, as beam theory predicts, and the stiffness comes out of the slope. The Archery Trade Association spine is the same number in other units, and the author gives the conversion: 2544 divided by the stiffness in newtons per square metre.

The transverse oscillation is measured by supporting the arrow at its nodes and driving it with a two-hundred-turn coil on an iron core, set five millimetres from the steel point and fed with alternating current whose frequency is controlled to a hundredth of a hertz; the amplitude is read by lowering a 25 micrometre steel reed on a micrometer stage until contact, detected electrically. The resonance curve thus obtained was compared with a finite-element model (COMSOL) and with the Euler-Bernoulli beam formula.

The draw force of the bow was measured by pulling the string in small steps, with a piezoelectric load cell, photographing every step with a 200 millimetre lens at ten metres to reduce parallax: some fifteen exposures on the way out and as many on the way back, then digitised. The stored and the released energy are the integrals of the two curves. The string tension at rest comes from the slope of the curve at brace height, and the tension at full draw from the balance of the three vectors at the nocking point. The launch velocity is the mean of ten measurements with an optical chronograph two metres from the bow.

For the deceleration a chronograph could not be used at long range, so the author measures the time between two noises, that of the string and that of the impact, with two microphones fixed at 4.6 and 36.6 metres and a free audio analysis programme: 25 shots at 7 distances between 27.4 and 82.3 metres, with a shooting machine in place of an archer. The final check on the trajectory was made by moving the sight until the gold was hit at various distances and converting the sight positions into launch angles, then comparing them with the numerical integration of the ballistic equations.

One bow and one arrow, not a series: every number here applies to a Hoyt Contender compound with an Easton LSR 500, shot with a mechanical release. Nothing in this work concerns the finger release, bow torque, the limbs of a recurve or the variability of an archer, because in the long-range tests there is no archer: a machine shoots.

Several quantities are not measured but derived. The virtual bow mass is obtained by assuming that all the available energy ends up as kinetic energy of bow and arrow, and the author only checks that the other contributions are small (0.05 joules in pulling the nock off, 0.2 to 0.3 in the string, 0.43 in the arrow's oscillation, 0.23 in the recoil, measured with an accelerometer). The axial load that might make the arrow unstable is estimated as 30 per cent of the peak draw force, that is, from the ratio of point mass to total mass, and from there comes the prediction that arrows with spine above 800 would fail: that is the extrapolation of a formula, not a breakage test. The absence of aerodynamic lift is not measured directly either: it is inferred from the sight positions agreeing with calculations that leave it out.

Where model and measurement meet, they do not always agree: the finite-element model gives 85.4 hertz against the 84.8 measured, which is good, but the uniform beam formula gives 97.2 hertz, 15 per cent higher, and the author acknowledges this while still using the formula for the ratios between quantities. In the archer's paradox calculation, moreover, values other than the measured ones are used (0.6 metres of length and 71 metres per second instead of 0.696 and 74.5), chosen for convenience. The drag coefficient is assumed constant along the flight, while a real arrow rotates and oscillates, something the author himself lists as an open point.

The document is an arXiv preprint from November 2015, written avowedly for scientists with an interest in archery, and has no peer review. The author closes by listing the things which in his view are still discussed without data: dynamic spine, the relation between FOC and lift, and the effect of the non-straightness of an arrow that rotates and oscillates. On FOC he notes that the range between 0.07 and 0.17, accepted in the archery world, has no quantitative evidence behind it.

The most useful part is in the closing pages, where the author puts into a formula how far the point of impact moves when one thing alone changes. At 70 metres an error of 1 metre in the estimated distance is worth about 7 centimetres vertically; 1 metre per second more launch speed raises the impact by about 13 centimetres; a 20 per cent change in aerodynamic drag, for instance from different vanes or from the air on another day, is worth about 7 centimetres again. These are calculations on one particular compound, but they show how little it takes to lose a centre.

Two numbers change the way one thinks about equipment. The first is efficiency: with this arrow and this bow, 20 per cent of the available energy stays in the bow, and the formula shows why a heavier arrow is more efficient and a lighter string gives more speed. The second is the damping time of the oscillation, about 1.5 seconds: longer than the flight, so the arrow enters the target still bent, and its straightening out in flight cannot be taken for granted.

On spine the study offers something concrete: the Archery Trade Association definition is a measure of stiffness, and the author gives the conversion, stiffness and spine being the same figure in two sets of units. On this shaft the measured spine (492) almost coincides with the manufacturer's (500). On the rest the tone stays cautious: dynamic spine, for the author, is a concept discussed without measurements, and the FOC range the archery world takes as good has no quantitative evidence behind it.

The experiments are within reach of an archer with a chronograph, a scale accurate to a hundredth of a gram and an audio recorder: the two-microphone method, in particular, measures the mean speed over the whole distance without putting the chronograph at risk. None of this concerns anyone's technique or score.