In considering the possibility of improving the efficiency of a bow—aside from the obvious necessity of using the best possible materials from the standpoint of hysteresis loss—a most useful concept is that of virtual mass.  This I define as a mass which, if it were moving with the speed of the arrow at the instant the latter leaves the string, would have precisely the kinetic energy of the limbs and the string at that instant. Letting K represent the virtual mass of a bow, we may write
where m is the mass of the arrow and v is the corresponding velocity. If K is independent of the velocity of the arrow as it leaves the string, then Eq. (1) enables us to determine the curve of cast for a particular bow—namely, the curve of v versus m—provided the available energy rW is known. That the virtual mass K is in fact a constant, has been determined in many measurements with a large number of bows. These measurements consisted of determining by means of a high speed spark chronograph, the velocities imparted by the bow to six arrows ranging in mass from 250 to 700 grains (Fig. 5), in steps of 75 grains. . This range includes practically all masses of arrows used with target or hunting bows, as well as most flight arrows. If we measure two velocities v corresponding
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Fig 5. Velocity of an arrow as a function of mass. k=190 gr.
to two masses m, differing preferably by several hundred grains, we can eliminate K from Eq. (1) and solve for rW, thus obtaining a check on the value of rW, determined as previously described. The values of rW thus found are usually higher than those obtained by either the photographic or the step-by-step method, indicating that the available energy in actual shooting is somewhat higher than that determined by the slower methods.