Analysis of Archer’s Paradox
Due to the relatively high force of the bow string acting on the arrow, it rapidly accelerates and a high-speed camera is required to capture the motion. Also, because this large force is acting in the direction of the arrow shaft, it begins to oscillate (fishtail) back and forth. This is the natural vibration the arrow experiences when subjected to a brief but large force. It’s similar to hitting something with a hammer and it vibrating as a result. The force acting on the arrow is equal to the draw force as the archer pulls the string back to the release position. This force can be in the range of 30-50 lb which is very large compared to the mass of the arrow, which is why it accelerates so rapidly (due to Newton’s second law, F = ma).
The term archer’s paradox refers to the phenomenon of an arrow traveling in the direction it is pointed at at full draw, when it seems that the arrow would need to pass through the starting position it was in before being drawn, where it was pointed to the side of the target. The bending of the arrow (around the bow), when released, is the explanation of the paradox and should not be confused with the paradox itself. Flexing of the arrow when shot from a modern ‘centre shot’ bow is still present and is caused by a variety of factors, mainly the way the string is deflected from the fingers as the arrow is loosed.
The term was first used by E.J. Rendtroff in 1913, and as understanding was gained about the arrow flexing out of the way of the bow as it is shot, and then experiencing oscillating back-and-forth bending as it travels toward the target, this dynamic flexing has incorrectly become a common usage of the term, causing a good deal of misunderstanding by those only familiar with modern target bows, which being ‘centre shot’ do not actually show any paradoxical behaviour as the arrow is always pointing visually along its line of flight.
Details
In order to be accurate, an arrow must have the correct stiffness, or “dynamic spine”, to flex out of the way of the bow and return to the correct path as it leaves the bow. Incorrect dynamic spine results in unpredictable contact between the arrow and the bow, therefore unpredictable forces on the arrow as it leaves the bow, and therefore reduced accuracy. Additionally, if an archer shoots several arrows with different dynamic spines, as they clear the bow they will be deflected on launch by different amounts and so will strike in different places. Competition archers therefore strive not only for arrows that have a spine within a suitable range for their bow, but also for highly consistent spine within sets of arrows. This is done using a static spine tester.
Choice of bow and spine
Less powerful bows require arrows with less dynamic spine. Less powerful bows have less effect in deforming the arrow as it is accelerated from the bow and the arrow must be “easier” to flex around the riser of the bow before settling to its path. Conversely, powerful bows need stiffer arrows with more spine, as the bow will have a much greater bending effect on the arrow as it is accelerated. An arrow with too much dynamic spine for the bow will not flex and as the string comes closer to the bow stave, the arrow will be forced off to the side. Too little dynamic spine will result in the arrow deforming too much and being propelled off to the other side of the target. In extreme cases, the arrow may break before it can accelerate, which can be a safety hazard.
Calibration
Dynamic spine is largely determined by shaft length, head weight, and static spine. Static spine is the stiffness of the center portion of the shaft under static conditions. The Archery Trade Association (ATA) (formerly the Archery Manufacturers and Merchants Organization (AMO)) static spine test method hangs a 2 pounds (0.91 kg) weight from the center of a 26 inches (0.66 m) suspended section of the arrow shaft. The American Society for Testing and Materials (ASTM) F2031-05 (“Standard Test Method for Measurement of Arrow Shaft Static Spine (Stiffness)”) hangs a 880 grams (1.94 lb) weight from the center of a 28 inches (0.71 m) suspended section of the arrow shaft. The (obsolete) British Grand National Archery Society (GNAS) system used a 1.5 pounds (0.68 kg) weight and a variable length with the arrow supported just behind the head and just in front of the nock. Because of this, GNAS cannot be directly converted to ATA or ASTM.
The primary unit of measurement for spine is deflection in thousandths of an inch (a deflection of 500 equals 0.500 inches) Deflection is sometimes converted to pounds of bow weight by dividing 26 by the deflection in inches. (26 divided by 0.500″ equals a spine of 52 pounds.)
