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Figure of Merit of a Bow

Starting again with Eq. (1) and assigning zero value to m, we obtain

 v=(2rW/K)½. (2)

This equation establishes a figure of merit of a bow from the standpoint of cast. It represents the limiting velocity that the bow could impart to an arrow approaching zero mass. Obviously no arrow shot with this bow could achieve higher velocity than that given by Eq. (2). In the light of what has been said about the fact that a certain amount of available energy in the limbs implies a certain minimum value of K, Eq. (2) shows that increasing the "weight" of a bow by increasing the width of its limbs does not appreciably change the figure of merit. We are assuming here that no other dimensions are changed, since, if the bow is properly made, we may neither increase its thickness, nor decrease its length, without risk of breakage. Consequently, doubling the width without changing any other dimensions would also approximately double the value of K, and hence would leave the figure of merit unchanged. This does not mean that there is no advantage in doubling the weight of a bow, as is quickly demonstrated by computing velocities of arrows of several masses shot with two bows, one of which has double the "weight" and double the virtual mass of the other. The result is shown in Table I. This table, which is based on assumed values of m, K and w that are well within the ranges of those used in practice, shows clearly why doubling the weight will not increase the velocity by more than a small fraction of the factor of increase in "weight."

 Mass of  Arrow Bow A: 45 lb; Bow B: 90 lb; Difference in  Velocities, K1, = 0.035 lb; K2 = 0.070 lb; Available Energy, Available Energy, 1200 ft pdl 2400 ft pdl m m + K1 v1² v1 m + K2 v2² v2 v2 - v1 (lb) (lb) (ft²/sec²) (ft/sec) (lb) (ft²/sec²) (ft/sec) (%) 0.050 0.085 28300 168 0.120 40000 200 19 .060 .095 25300 159 .130 37000 192 21 .070 .105 22800 151 .140 34300 185 23 .080 .115 20800 144 .150 32000 179 24 .090 .125 19250 139 .160 30000 173 25

TABLE I. Effect of doubling the "weight" and virtual mass of a bow on the velocities of arrows of different masses.

An approach to the problem of the limiting velocity of a given bow with m approaching zero, is to consider the symmetrical limbs of a bow from the standpoint of the free period of vibration of the limbs when the handle of the bow is solidly clamped in a vise. For a given set of dimensions of the bow limbs there is a certain period of vibration. If it be assumed that the dimensions of the limb represent limiting values below which we may not safely go, the only way to increase the stiffness of the limb is to increase its width. Increasing the width, however, does not further decrease the period, a conclusion that bears out the reasoning regarding the figure of merit of two similar bows, one of which has double the "weight" of the other.