Bow Ballistics And Scale

I like to think that the roots of our game (Chainmail, et. al.) are well-thought out with a good grounding in historical reality, insofar as scales, movement, and ranges go. They're not entirely fantastical (and inasmuch as that is so, we can continue the tradition of leveraging connections to real-life research to enrich our games). Let's think about longbows for a bit, and at the end we'll consider whether our game uses a good scale for representing them or not.

A few specific resources I'll use here. One: I tend to use Wikipedia as an acceptable source of raw data, especially in the rough context of a game (colleagues pointing to specific, better research is fair game, but people simply crying, "no, because it's Wikipedia!" is a pretty weak excuse). Two: Google does instant unit conversions (did you know that)? Three: Green Man Longbows page on arrow-speed tests taken by chronograph. Four: Stephen R. Schmitt's Ballistic Trajectory Calculator using JavaScript (illustration above taken from Stephen's site).

Okay. It seems like we can identify two different classes of longbow. The first I'll call the normal longbow, which basically matches the historical hunting longbow used globally, and also the modern longbow, of about 50 pounds draw weight. The second I'll call the warbow longbow, equivalent to the African elephant bow or the famed English longbow (which took a lifetime of practice and literally bent the bones of the users), of about 150 pounds draw weight.

According to Green Man Longbows, an estimate can be made by "adding 100 to the bow's draw weight and using this number as the expected arrow speed for that bow" (and note that all of their practical tests are done on the "normal" type, averaging around 40 pounds draw weight).

Let's calculate the maximum range of the normal longbow, assuming simple ballistic flight (tail feathers will modify this, but it's unknown by how much). We expect the launch velocity to be 50+100 = 150 ft/sec (46 m/sec), max range at 45 degrees (known from ballistics, or else experiment and recognize that any other angle lowers the range), and a launch height of a man's shoulder at 5 ft (1.5 m). Pumping this through our ballistics calculator, we see a maximum range of R=217 m, i.e., 237 yards.

Obviously this is an abstract estimate, so how does to it compare to other sources? Wikipedia says "Modern longbows have a useful range up to 180 m (200 yd)". Our games of Chainmail and D&D give the longbow a maximum range of 210 yards. So, our result seems compatible with a wealth of different sources -- somewhere in the low 200-yard range.

Let's do the same for what I'm calling the warbow style longbow. Here we expect a launch velocity of 150+100 = 250 ft/sec (76 m/sec), again at a 45 degree angle and a 1.5 m launch height. For this the ballistics calculator reports a maximum range of R=590 m, i.e., 645 yards.

How does this compare to other reports? Not quite so well; Wikipedia documents tested ranges of around 360 yards, and that "A flight arrow of a professional archer of Edward III's time would reach 400yds". So we're off by about 50%, but not by a whole order of magnitude. Furthermore, we can conclude that D&D longbows are better interpreted in terms of the "normal" hunting-style longbow of around 50 pounds, usable by any fighter-type, without requiring exceptional strength or a lifetime of training.

So let's consider our findings so far. Are D&D (Chainmail) longbow ranges accurate? Yes, they are in fact very accurate as given by a variety of sources and calculations -- assuming we mean the "normal" type longbow of around 50 pounds draw weight. A bit more than 200 yards is an excellent simulation of its capacity.

Now let's do something a bit different. The preceding assumes use of a bow outdoors, i.e., with no ceiling overhead to interrupt the flight path of the arrow. If we instead assume a limited ceiling height, how far is our maximum shot then? Here's how we'll do that: go to our ballistics calculator and by trial-and-error input different shot angles until the maximum height h is less than our ceiling height (say, a 10' tall corridor, i.e., 3 meters). The resulting range R will be our maximum indoor shot.

Do this for the normal longbow. Start velocity is 46 m/sec, best angle is then 6.7 degrees with a 1.5m start height. Resulting max range is R = 61m = 200 feet. Also do this for the warbow style weapon. Start velocity is 76 m/sec, best angle is then 4.0 degrees with 1.5m start height. Resulting max range is R = 100m = 328 feet.

Now here's the interesting thing, looking at the first type, which is apparently the D&D longbow. I tend to ridicule Gygax a bit for shifting the dungeon game scale in D&D from "yards to feet" as a rather short-sighted cop-out. But it turns out that the normal hunting longbow (50 lb draw weight) does have a max outdoor range of about 200 yards, and it does have a max indoor range (limited by 10' ceiling) of precisely 200 feet. So we see that, to my immense personal surprise, the yards-to-feet conversion for dungeons is rather freakishly accurate, at least for the limited domain of ballistic missile ranges.

I'll even go so far as to say that I didn't really want this to be the case; I prefer a game scale of 1"=5 feet (matching miniature scale), but this finding is certainly easier to apply for a scale of 1"=10 yards outdoors, and 1"=10 feet indoors (thereby matching D&D ranges given in inches). Another interesting implication is that while outdoor bowfire is very much "indirectly arced" at the maximum range, indoor bowfire is always basically a directly aimed shot (only 4 or 6 degrees angle at most). We could then conclude that dungeon bowfire is always effectively at "short" range, and should simply not be subject to any range penalties at all (which is actually the converse of the rules given in Chainmail, where mass fire outdoors has no range penalty, but man-to-man fire does).

Addendum: Let's add an assumption that you need access to the target's head for a normal D&D to-hit roll (i.e., not just their foot, but potentially at least 2m off the ground at the end of the shot). Then by my calculation the maximum indoor range would be R = 45m = 148 feet, still on the same general order of magnitude.