588 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1962 



curve for the longbow approximates a straight line. For a short, 

 straight-limbed bow it curves slightly downward from that for the 

 longbow. Backwardly curved tips produce convexity upward, as do 

 strongly reflexed limbs of the kind employed in the Turkish bow, 

 and some modern bows which employ modifications of such limbs. 

 Hence the energy at full draw is relatively lower in bows with 

 straight limbs than in reflexed bows, or bows with backwardly cur- 

 ving tips or ears. It would appear, therefore, that with substantially 

 higher energy content for the same maximum force, the bows which 

 follow a modified Turkish design, or use backwardly curving tips, 

 are to be preferred to bows with straight limbs. This is indeed true 

 if they are as effective in transferring energy to the arrow. 



V eloGity-mass relations in arrows shot from a given how. — ^An arch- 

 er, as he gains experience with his bow, will notice — provided he 

 shoots arrows of different weights — that the light arrow takes off with 

 higher velocity than the heavier. This raises questions. First, can one 

 establish a systematic relation between the mass of an arrow and the 

 velocity imparted to it by a given bow ? Second, can one find a rela- 

 tion between the mass of an arrow and the energy wliich is trans- 

 ferred to it from the bow ? These two questions, and the search for 

 their answers, lead to some other considerations of interest. 



Figure 9 reproduces a mass-velocity curve obtained by plotting 

 measured velocities of arrows of different masses against the mass 

 values, all shot from a bow with given energy at full draw. The solid 

 line is plotted from computed values of v and m, using the relation 



my- Kv^ 



E= + » 



2 2 



simplified to ^=i/^(m+Z')'y^ 



The equation says that the energy E in the drawn bow is accounted 

 for by the kinetic energy in the arrow, 14 ??iy-, and another energy 

 term, 14 iTv^, which represents the part of E which failed to be trans- 

 ferred from the bow to the arrow ; it is that part of the energy which is 

 left behind when the arrow leaves the string. This term employs the 

 same velocity v as that of the arrow, and a quantity K which has the 

 dimensions of mass. This I have called the virtual mass of the bow. 



A physical picture of virtual mass may be drawn as follows. 

 Imagine the bow and string to have no mass, and imagine a mass 

 K to ride "piggyback" on the arrow until the instant the arrow leaves 

 the string. The velocity v is that which corresponds to kinetic energy 

 equal to the energy E^ transferred to a mass of m + K. The energy 

 carried off by the arrow is i/^ mv^^ which means that the energy left 

 behind is i/^ Kv^. The virtual mass K has been found by many experi- 



