Aerodynamics of the arrow: Higgins 1933 study

Higgins, an aeronautical engineer, applies to the arrow the methods that aeronautics then used to calculate drag. He starts from an observation: the feathers give the arrow far more stability than it needs, so it always stays aligned with its own flight path and air resistance comes almost entirely from friction over the surface, not from the wake. On that basis he writes the equations of flight and replaces them with approximate formulas that can be solved by hand.

For an average arrow (300 grains, 5/16 inch in diameter, 28 inches long, flight feathers) launched at 180 feet per second, about 55 m/s, the calculation gives a maximum range of 652 feet, about 199 metres, reached in 6.87 seconds; the arrow lands at 127 feet per second at an angle of 53 degrees. Without air the same arrow would reach 1,007 feet. The best angle is between 43 and 44 degrees, and moving 10 degrees away from it costs only about 30 feet, 4.5% of the range. A 10% change in diameter, length, friction coefficient or feather area shifts the range by about 30 feet; 5 feet per second more or less at launch, a difference the author thinks can occur from one shot to the next, shifts it by 26 feet, or 4%.

The work is purely theoretical. Drag is calculated with the skin friction formula used in aeronautics, with a coefficient read from a curve as a function of the arrow's speed and length (about 0.005 in the example) and with an area equal to that of the shaft multiplied by a factor for the feathers: from 1.02 for the celluloid vanes of flight arrows to 1.12 to 1.27 for target arrows, and up to 1.25 to 1.50 for hunting arrows. Drag is taken as proportional to the square of speed.

The equations of motion, split into horizontal and vertical components, cannot be integrated by hand because they contain the arrow's angle, which changes in flight. Higgins solved them graphically and then looked for simpler formulas that would reproduce those curves; the graphs show the two solutions staying close along the whole flight path and matching at full range. For the effects of launch speed and arrow weight he uses a bow efficiency of 0.60. The paper does not say which bow, nor does it report measured shots: the comparison with experimental data rests on one sentence and a reference to another paper.

Everything rests on the assumption that the arrow flies perfectly aligned and that the wake is negligible. For arrows with small, well-fitted feathers this may be reasonable; for hunting arrows or large feathers much less so, and the table of factors itself shows how much the surface changes. The first metres of flight, when the arrow is still flexing and oscillating, are left out, and so is wind.

The approximate formulas are fitted to the reference arrow and to speeds around 180 feet per second: outside that range their agreement with the full solution is not shown. The effect of arrow weight depends on how weight changes launch speed, which the author takes from another paper, and the graph shows the advantage of a heavier arrow reversing beyond a certain point without the text saying where. The experimental check is stated, not shown.

For a modern archer, especially one shooting target at competition distances, the study says little directly but offers some useful orders of magnitude. The first is that air resistance weighs heavily: without air the reference arrow would fly about half as far again (1,007 feet against 652). The second is that launch speed matters more than many construction details: 5 feet per second, about 1.5 m/s, shifts the range by 26 feet, almost as much as a 10% change in the arrow's diameter or length. The table of factors also gives an idea of what feathers add: on a target arrow they increase the area exposed to the air by 12 to 27% compared with the bare shaft.

These numbers are calculated for maximum range, at around 45 degrees, in other words for flight shooting, not for the point of impact at 70 metres on a flat trajectory. They do not tell anyone how much a score would change.