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The Dynamics of a Bow and Arrow
Part 2 of 5

Neglecting the dissipation in heat and sound, the gain in kinetic energy is equal to the loss in potential energy. Therefore:

MV2/2 +mv2/2=C(A'N'-AN).

But V=Rv. Therefore:

v = [2C(A'N'-AN)/(MR2+m)]1/2

But N=3B1A/4. Therefore:

V = [3B1C(A'2-A2)/2(MR2 + m)]1/2.

And

V=Rv=R[3B1C(A'2-A2)/2(MR2 + m)]1/2.

Let a equal the acceleration of the bow tip and A the acceleration of the arrow. (Bold face type will be used to indicate dynamic values of variables.)

a=dv/dt=(dv/dA) (dA/dt) = (dv/dA) (dA/dN) (dN/dt).

But dA/dN=4/3B1 and dN/dt= -v. Therefore:

a=-(4v/3B1) (dv/dA). Therefore by differentiation:

a=2CA/(MR2 + m)-2MRC(A'2-A2)Z/(MR2 + m)2,

where

Z= sin A-(Y/P) cos A+ (3B1S2/4P3) sin2 A.

Also A=dV/dt=d(Rv)/dt=vdR/dt+Rdv/dt.

dR/dt= (dR/dA) (dA/dt)

= (dR/dA) (dA/dN) (dN/dt) = - (4v/B1)dR/dt.

Therefore

A=2CAR/(MR2+m)+2mC(A'2-A2)Z/(MR2+m)2.

The dynamical force f on the tip is: f=ma/2.

The dynamical force F on the arrow is:

F=MA.

T=F/2 sin E=SF/2P=SMA/2P= (f-f) /sin (A + E).

If the angle A is expressed in radians, the masses in pounds, the lengths in feet and the forces in poundals, the velocity and acceleration will be obtained in feet per second and feet per second per second, respectively.