Archery and mathematics
Archery is the skill of using a bow to propel arrows, historically significant for hunting and warfare, and currently practiced as both a sport and recreational activity. The earliest known bows date back to the Mesolithic period and archaeological evidence suggests that archery has existed for tens of thousands of years. In various cultures, it has served as a vital tool in military strategy, with notable historical depictions highlighting its importance in battles. The sport of archery has evolved over the centuries and was included in the modern Olympic Games starting in 1900 for men and 1904 for women.
Mathematics plays a crucial role in understanding the mechanics of archery, particularly in the design and performance of bows. Mathematical modeling has advanced significantly since the 1930s, allowing for a better comprehension of how bows function. Key concepts include the Bernoulli-Euler equation, which describes how bending moments affect the curvature of a bow's limbs, thus aiding in the calculation of energy stored and released during shooting. This blending of archery and mathematics not only enhances the performance of modern bows but also provides insights into the characteristics of historical bows, which can be difficult to assess due to material degradation over time.
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Archery and mathematics
Summary: Mathematics is essential in modeling and predicting a bow’s performance.
Archery is the practice of propelling an arrow with a bow for the purpose of hunting, warfare, or sport. A bow is a pair of elastic limbs connected at the tips by a string. A bow acts as a spring and stores in the limbs the energy applied by the archer. As the archer releases the string, the arrow is propelled with a force proportional to the tension on the string. The path of the arrow is a parabola whose shape is determined in part by the angle of release from the bow, measured with reference to the ground.
The origins of archery are lost in the beginning of civilization and probably will never be determined with precision. The earliest bows known today were found in the Holmegaard area of Denmark and were made of elm and yew. The Holmegaard bows date form the Mesolithic period (10,000–3000 b.c.e.); however, there is archaeological evidence of projectile wounds—possibly caused by bows—from the Upper Paleolithic (40,000–10,000 b.c.e.) in all continents. It is speculated that archery was first used for hunting and, later, for warfare as social structures became increasingly complex. By the twelfth century b.c.e., archery was a decisive branch of military power. For example, the wall of the Theban temple of Ramses III depicts the Aegean fugitive fleet—driven from Crete by the Greek immigration—engaging in and losing a battle against the Egyptian fleet, whose primary weapon is shown to be archery. Archery remained the weapon of choice in the West for distance combat until the introduction of gunpowder toward the fourteenth century c.e. Archers in the medieval era would fire in a high arc, achieving accuracy by volume rather than deliberate aim. Today, archery is practiced as a precision sport and for hunting. Men’s archery was one of the events of the second modern Olympics in 1900. The first Olympic archery event for women was held in 1904.
Mathematical Modeling of Bows
Since the 1930s, engineers and scientists have studied the design of bows. In 1947, C. N. Hickman made the first accurate mathematical model for flat bows, consisting of an idealized representation of two linear elastic hinges and rigid limbs with point mass (an idealized representation of a body used to simplify calculations) at the tip. More recent modeling efforts by B. W. Kooi and C. A. Bergman consider the limbs as beams that store elastic energy by bending.
The Bernoulli-Euler equation , named for Daniel Bernoulli and Leonhard Euler, describes the change in the curvature of a beam as a function of the “bending moment” (tendency to rotate about an axis) and is used to estimate the force in the string. When the archer draws the bow, the force exerted at the middle of the string causes an increase in the bending of the limbs, thus increasing the momentum and storing more energy for the shot. The elasticity modulus of the bow’s material—the proportionality constant that relates limb deformation versus energy stored—determines the force with which the limbs recover their original shape after being deformed.
Mathematical modeling is a viable alternative for the evaluation of the performance of old bow models. As time passes, environmental conditions and natural processes cause considerable degradation within the cell structure of the wood used in ancient bows, which prevents a realistic assessment of the original density of the material and precludes direct testing.
Bibliography
Grayson, Charles E. Traditional Archery From Six Continents. Columbia: University of Missouri Press, 2007.
Kooi, B. W., and C. A. Bergman. “An Approach to the Study of Ancient Archery Using Mathematical Modelling.” Antiquity 71, no. 271 (1971).
Miller, Frederic P., Agnes F. Vandome, and John McBrewster, eds. History of Archery. Beau Bassin, Mauritius: Alphascript Publishing, 2010.
Slater, Steven. “The Physics of a Wooden Bow: In Traditional Archery, Not All Bows Are Equal.” Suite 101 (July 6, 2009). http://www.suite101.com/content/the-physics-of-a-wooden-bow-a130234.
Soar, Hugh D. H. The Crooked Stick: A History of the Longbow. Yardley, PA: Westholme Publishing, 2009.