[Johnn Schroeder]

Kevin,

Not sure what "arrow constant (c)" is, but all things that are propelled through the air must follow the laws of Ballistics. The one thing that has a dramatic effect on such flight is Ballistic Coefficient, or BC.

In ballistics, the ballistic coefficient (BC) of a body is a measure of its ability to overcome air resistance in flight. BC is inversely proportional to the deceleration of the projectile, (a higher number indicates a lower deceleration.) BC is a function of mass, diameter, and drag coefficient. It is given by the mass of the object divided by the diameter squared, so that it is presented to the airflow divided by a dimensionless constant i which relates to the aerodynamics of its shape. Ballistic coefficient has units of lb/in² or kg/m².

The formula for calculating the ballistic coefficient for a body:
(This is not the same thing as the BC used by bullet manufacturers.)

This is the BC as defined by and used in Physics and Engineering. Although it would not be incorrect to use this equation on bullets, the BC obtained from this equation would not gave same value as the BC from a bullet manufacturer because their value is a comparison to the G1 bullet model. Projectile length is important in arrow (and bolt) flight, and this formula allows that to be factored in.

BC = M / (Cd X A) = (p X l) / Cd

where:
• BC(physics) = ballistic coefficient as used in physics and engineering
• M = mass (grams)
• A = cross-sectional area
• Cd = drag coefficient for a streamlined body (use 0.04)
• ρ (rho) = average density (select density of arrow or bolt material)
• l = body length (In inches)
In fluid dynamics, the drag coefficient (commonly denoted as Cd, Cx or Cw) is a dimensionless quantity which is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area. The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic drag: skin friction and form drag.
The density of a material is defined as its mass per unit volume. The symbol of density is ρ (the Greek letter rho).

Mathematically:

P = m / V

where:
ρ (rho) is the density,
m is the mass,
V is the volume.
Different materials usually have different densities.

In this manner you can calculate the likelihood of an arrow or a bolt’s ability to retain energy and travel farther and or faster if the BC is high. By comparing the mass and BC of any projectile, you can quickly see how they will respond, even before you fire it for the first time. The math is easy enough, and the information easy enough to get on-line, like shaft material density, Drag Coefficient for different tip profiles and the like.
One thing here, the length of an arrow or bolt affects flight because of the effects of boundary layer drag. The friction of the slower moving air flowing down the length of the shaft actually slows the arrow in flight. A simple trick here is to put a couple of windings of monofilament around the shaft right behind the head area, bonded with simple super glue and trimmed neatly. What this does is induce a small amount of turbulence along the shaft, stripping away the boundary layer drag to some degree and reducing the arrow's tendency to lose energy to boundary drag.

[Kevin S.]

one more question.

When you measure how much you draw, do you measure the complete draw-length?

Or do you measure the complete draw-length minus the brace-length?

[Johnn Schroeder]

As far as the mechanical system is concerned, I would consider only the length of the mechanical travel that is used from the first point the string contacts the bolt, to that point at which the rearward travel stops (trigger mechanism lockup), to that point at which the string finally leaves the nock of the bolt with the bolt in forward travel. This last point will be a bit passed the point at which the string/nock were first mated, because the string will move farther forward as the bow limbs flex forward. (Check the wear pattern on the frame to see where the string stops rubbing and you're close.) This is the distance at which the bolt is subjected to any force from the string. Take the point at trigger lockup (trigger set) to that point at which the string is no longer thrusting the bolt forward, and this is the effective length you have for energy to be imparted.

I once worked on a small prototype crossbow in which the limbs of the bow were mounted on cam plates, and the bow's spring force was further enhanced by a small leaf spring arrangement that allowed the additional spring force to be transferred through the cam plates and thus to the bow limbs. The small prototype was fun to play with, and the addition of rollers with integral bearings and adjustable wear plates, made for a very powerful bow in a small space. While the crossbow was half again the size of a 'pistol' version of a crossbow (to keep the model small enough to work on at a bench) the calculated draw weight was just over 490 pounds to trigger lock up point. In reality, the actual measured draw weight was 447 pounds, the loss being from mechanical friction, and other small losses as was to be expected from such a simple prototype. I used a secondary cable to the bow tips to prevent the arms moving so far forward that the bow string would be shock loaded in recoil. All in all, I wished I could have completed the work on this one, it had promise of getting interesting!