Dynamics of bow and arrow: Hickman 1937 study

Hickman, one of the physicists who brought scientific method into archery in the 1930s, sets out a way of calculating on paper what happens between the release and the arrow's exit: the forces, speeds and accelerations of the arrow and of the limb tips. The calculation is meant to replace high-speed filming (one to three thousand frames per second), which the author had already done and describes as slow and expensive.

Applied to a six-foot yew bow with a 6-inch brace height, a 27.5-inch draw and a 350-grain arrow, the model describes a precise sequence. At the release almost all the energy goes to the arrow, and only about five pounds of force is left to accelerate the limb tips; this force falls to zero at around 15.5 inches of draw and then becomes a braking force. The tips reach their top speed at around 15 inches and stop when the string returns to brace height. In the first instants the force pushing the arrow is lower than the static draw weight, because part of it goes into setting the limbs moving; later it is higher, because the limbs, as they slow, hand back to the arrow the energy they received. For this reason, according to Hickman, heavy limb tips cost less speed than one might think, but they increase string tension. In the calculation the bow is 100% efficient; an Osage bow built by Klopsteg on the same pattern reached 93%.

The model starts from statics. The bow is made of a rigid central section and two limbs that, at every point of the draw, bend in an arc of a circle; Hickman shows that the limb tip then moves almost exactly along a circle whose radius is three quarters of the working limb, accurate to better than two parts per thousand. The force at the tip is proportional to its displacement. From this he derives string tension and draw force at every point of the draw, then requires the energy lost by the limbs to become kinetic energy of the arrow and the tips, ignoring heat and sound, and obtains speeds and accelerations.

The example uses an 8-inch central section, 32 inches of working limb on each side, a constant thickness of 0.6 inch and a width that grows steadily from the tip to 1.5 inches near the handle, so that every section works at the same stress. The wood is yew, with a density of 43 pounds per cubic foot and an elastic modulus of 1.46 × 10⁶ in the paper's imperial units. Each limb's mass is equivalent to 334.4 grains concentrated at the tip; to the 350-grain arrow Hickman adds a third of the weight of the 105-grain string, for an effective mass of 385 grains. There are no new measurements: the results are calculated curves.

It is an idealised model, and the author says so: 100% efficiency, no losses in the wood or the air, a string that does not stretch, limbs that always bend in an arc of a circle. Hickman himself warns that, to come close to measured values, the final speed should be read about 2 inches before brace height, because in practice not all the tips' energy reaches the arrow. The comparison with measurements is asserted but not shown: there are no experimental data alongside the curves.

The bow is a wooden longbow with straight limbs and a heavy string; a modern recurve, with composite limbs, working tips and light strings, has a geometry this scheme does not describe. The arrow is treated as a mass moving in a straight line, without flexing: the archer's paradox, which according to the introduction the films had settled, does not come into it. The remarks on accuracy (short bows, heavily reflexed bows, heavy tips or a low brace height making the arrow accelerate sharply at the end) are the author's reasoning from the curves, not measurements.

For archers today the study matters mainly as history: it is one of the first attempts to describe the bow as a machine, and some of its ideas have lasted. The most useful is that the force on the arrow does not follow the static force-draw curve: at first it is lower, because some of the work goes into the limbs, and at the end it is higher, because the limbs give energy back. The same idea returns, with far more complete models, in later work on modern recurves.

The remarks about equipment should be read for what they are: heavy limb tips cost little speed but raise string tension and can make the final push more abrupt; a heavy string makes the tips spring back after the arrow has gone. These are deductions from a model of a wooden longbow, and they do not say how much accuracy changes.