I always like a good discussion. Takes me out of my comfort zone, and forces me to dig deeper. In the end I always feel I’ve learned something.

Is drag additive relative to air density? Increase air density by 10% and drag will increase by 10%.

In a world without friction and drag, an arrow would should a symmetrical, parabolic curve. The only force countering movement is gravity. With drag, however, the horizontal distance during the downward path is shorter than the upward horizontal distance, because drag slows the arrow down from the moment it leaves the bow.

Since drag is quadratically related to velocity (arrow speed), this asymmetry becomes more and more exaggerated with faster and faster launch speeds. Arrows being shot very fast will fly further, but will also fall out of the sky in a more vertical manner (landing at much more than 45° relative to the ground) since drag reduced the initial velocity more. I’m sure the experienced flight shooters have witnessed this.

Under exceptional circumstances, Alan’s theory about beneficial drag in a mildly rotating arrow might be valid (with a change in drag behavior due to different arrow movement through the air beyond the apex), but I don’t think when I shoot it matters a lot ;-). In general, the less drag, the more the arrow follows the drag-less path, and the further it will shoot. Drag that slows the descent also slows down the horizontal speed. With no horizontal speed, the object falls vertically.

How did I calculate the distances? I didn’t do complex calculus. I modified the spreadsheet presented in the technical archery page I linked to. Through the spreadsheet I track the vertical distance, Y (column AC). The horizontal distance is read at the moment the value in column AC reaches zero again.

I didn’t have input for drag coefficients (which differ for each arrow). For the calculations this wasn’t necessary either: I just needed to see how an increase in drag of 10, 15 and 23 % influenced a regular shot. Still, I used Steve’s rule of thumb’s relation between 10 gpp arrow speed and distance: 170 fps will yield c. 200 yards (183 m), 205 fps will yield c. 300 yards (274 m). From this, you can calculate the expected drag coefficient. If you know initial arrow speed (chronograph) and distance, you can calculate drag.

If you then increase the drag coefficient by 10% (effect of 5°C versus 25°C), 15% (effect of altitude) or both (23.5%) you can track the max distance in the spreadsheet. This teaches me that for a typical 200 m shot, the effect of altitude or temperature isn’t negligible at all.