Bows and arrows, like horses and tents, played a very important part in the lives of the ancient Turks on the steppes of Asia. These nomadic people who hunted on horseback armed with bows and arrows were a nation of archers, as were the Scythians, Avars, Mongols and the Tartars. In the early Ottoman period bows and arrows were the most effective weapons used by the army, but as firearms became increasingly reliable, arrows were gradually superseded by muskets and rifles from the middle of the 16th century. However the archery was not abandoned and lived as an important sport among the Turks until the beginning of the 20th century. Ottoman composite bows, which achieved their final form at the end of the 15th century, are reflex bows, which mean that when unstrung the limbs bend in the opposite direction. Ottoman arrows are the shortest known to have been used by any people. They were also slender, and this made them light, so they remained longer in the air and traveled extremely long distances. When the string of a bow is stretched, the archer’s muscle energy is stored in the form of potential energy on the bend limb of a bow. And secondly when the string is released the stored energy is transformed to the arrow in the form of kinetic energy. The bow producer’s object is to create the greatest energy by reaching the upper endurance limit of the used material. In Ottoman bows the reversal bending created an additional impulse, so that the initial acceleration of the arrow was extremely high. In this paper the construction and use of the Ottoman reflex bow is reviewed. A mathematical model for the shooting including the air friction is formulated, so the conditions and parameters for the optimum distance archery derived. The mathematical results are compared with real historical records.