Archery
Physics

When a projectile is thrown by hand, the speed of the projectile is determined by the kinetic energy imparted by the thrower's muscles performing work. However, the energy must be imparted over a limited distance (determined by arm length) and therefore (because the projectile is accelerating) over a limited time, so the limiting factor is not work but rather power, which determines how much energy can be added in the limited time available. Power generated by muscles, however, is limited by force–velocity relationship, and even at the optimal contraction speed for power production, total work by the muscle is less than half of what it would be if the muscle contracted over the same distance at slow speeds, resulting in less than 1/4 the projectile launch velocity possible without the limitations of the force–velocity relationship.

When a bow is used, the muscles are able to perform work much more slowly, resulting in greater force and greater work done. This work is stored in the bow as elastic potential energy, and when the bowstring is released, this stored energy is imparted to the arrow much more quickly than can be delivered by the muscles, resulting in much higher velocity and, hence, greater distance. This same process is employed by frogs, which use elastic tendons to increase jumping distance. In archery, some energy dissipates through elastic hysteresis, reducing the overall amount released when the bow is shot. Of the remaining energy, some is dampened both by the limbs of the bow and the bowstring. Depending on the arrow's elasticity, some of the energy is also absorbed by compressing the arrow, primarily because the release of the bowstring is rarely in line with the arrow shaft, causing it to flex out to one side. This is because the bowstring accelerates faster than the archer's fingers can open, and consequently some sideways motion is imparted to the string, and hence arrow nock, as the power and speed of the bow pulls the string off the opening fingers.

Even with a release aid mechanism some of this effect is usually experienced, since the string always accelerates faster than the retaining part of the mechanism. This makes the arrow oscillate in flight—its center flexing to one side and then the other repeatedly, gradually reducing as the arrow's flight proceeds. This is clearly visible in high-speed photography of arrows at discharge. A direct effect of these energy transfers can clearly be seen when dry firing. Dry firing refers to releasing the bowstring without a nocked arrow. Because there is no arrow to receive the stored potential energy, almost all the energy stays in the bow. Some have suggested that dry firing may cause physical damage to the bow, such as cracks and fractures—and because most bows are not specifically made to handle the high amounts of energy dry firing produces, should never be done.

Modern arrows are made to a specified 'spine', or stiffness rating, to maintain matched flexing and hence accuracy of aim. This flexing can be a desirable feature, since, when the spine of the shaft is matched to the acceleration of the bow(string), the arrow bends or flexes around the bow and any arrow-rest, and consequently the arrow, and fletchings, have an un-impeded flight. This feature is known as the archer's paradox. It maintains accuracy, for if part of the arrow struck a glancing blow on discharge, some inconsistency would be present, and the excellent accuracy of modern equipment would not be achieved.

The accurate flight of an arrow depends on its fletchings. The arrow's manufacturer (a "fletcher") can arrange fletching to cause the arrow to rotate along its axis. This improves accuracy by evening pressure buildups that would otherwise cause the arrow to "plane" on the air in a random direction after shooting. Even with a carefully made arrow, the slightest imperfection or air movement causes some unbalanced turbulence in air flow. Consequently, rotation creates an equalization of such turbulence, which, overall, maintains the intended direction of flight i.e. accuracy. This rotation is not to be confused with the rapid gyroscopic rotation of a rifle bullet. Fletching that is not arranged to induce rotation still improves accuracy by causing a restoring drag any time the arrow tilts from its intended direction of travel.

The innovative aspect of the invention of the bow and arrow was the amount of power delivered to an extremely small area by the arrow. The huge ratio of length vs. cross sectional area, coupled with velocity, made the arrow more powerful than any other hand held weapon until firearms were invented. Arrows can spread or concentrate force, depending on the application. Practice arrows, for instance, have a blunt tip that spreads the force over a wider area to reduce the risk of injury or limit penetration. Arrows designed to pierce armor in the Middle Ages used a very narrow and sharp tip ("bodkinhead") to concentrate the force. Arrows used for hunting used a narrow tip ("broadhead") that widens further, to facilitate both penetration and a large wound.
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