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The Dynamics of a Bow and Arrow
Part 1 of 5

While contributing the articles on the static strains and stresses in bows. Dr. Hickman was working on analytical methods of computing the dynamics of bows and arrows. He was not satisfied with his treatments of this more difficult problem for several years. Even after working out a treatment that was satisfactory, the results were not published until June 1937. This paper on the Dynamics of a Bow and Arrow appeared in the Journal of Applied Physics.

Making use of Fig. 1 and the formulas given in the article "General Formulas For Static Strains And Stresses in Drawing A Bow", Dr. Hickman gives the following dynamical treatment of this problem:

Dynamical Treatment

Let x represent the distance traveled by the arrow in t seconds. Then: dx/dt=V where V is the instantaneous velocity of the arrow. But x=D'-D where D' is the value of D for the fully drawn position. (Prime letters will be used to represent the value of the variables for the fully drawn position. Prime letters will therefore be constants.)

dx/dt=d(D'-D)/dt= -dD/dt=V.

In like manner: -dN/dt=v equal velocity of bow tip. But: dD/dN=(dD/dt)/(dN/dt)=V/v=R where R is the ratio of the two velocities. Also,

dD/dN =(dD/dA)(dA/dN).

But D=H+ P. Therefore, substituting the values given for D=H+P and differentiating

dD/dA = (3B1/4) cos A+ (3B1Y/4P) sin A.

But A = 4N/3B1 and dA/dN =4/3B1. Therefore:

dD/dN=V/v=R=cos A+ (Y/P) sin A.

Let M equal mass of arrow, m/2 equal effective mass of each limb located at tip and W equal potential energy of both limbs. Since N is the displacement of each bow tip: W'-W= C(A'N'-AN) is the loss in potential energy of both limbs from position of full draw to any other position. The author with a shooting machine which he designed for use in testing arrows. This machine was used in conjunction with a chronograph for measuring the velocity and acceleration of arrows. It is equipped with pneumatic release. Click for a larger image