In a few years, Genghis Kahn and his sons conquered the largest contiguous empire in history.

It is hard to quite explain the dramatic Mongol victories of the 13th century (the 1200s) over far larger armies of China, India, Russia, Bagdad, Germany, and Poland. Over 50 years or so, Genghis Kahn, an illiterate horseman with few troops built the largest contiguous empire in history, defeating far more advanced armies. The British empire was larger, but not contiguous, and they mostly fought less-trained armies than their own.

Most attempts to explain the victories talk about Mongol unity, an issue I treat elsewhere, but include comments to the effect that the Mongol bow worked better and sent an arrow further. Armed with better bows, the books and articles maintain, the Mongols could kill from a distance without fear of being killed. There is also a curious note by Weatherford (Genghis Khan and the Making of the Modern World) that the Mongol arrows could not be nocked in the Russian bows, but that Russian arrows were easily nocked in Mongol bows. At first glance these two sound contradictory, and bizarre. Let me try an explanation — one that seems to fit the physics — but first allow me to provide some background and reject some (all) the explanations I’d seen:

Force vs pull distance for several medieval bows, and a modern compound bow (blue). The longbow was better than the Mongol

There are several reasons to doubt that Mongol bows were better than the European, Moslem, and Chinese bows of the time. Most significantly, each Mongol made his own bow and his own arrows, or stole them, and that is a recipe for some very crude bows and some very un-straight arrows. Most countries had professional shops of bow makers and professional families of arrow makers. One person or guild made only arrow heads; others grew straight wood for arrows and bows;  yet-others made finished bows, or added the feathers (fletching) to arrows. It is highly unlikely that a soldier could do anywhere near as well as this team of professionals. While it is possible that some arrows were better, it is far more likely that they were worse and that all were different, besides.

Books and web-sites also like to claim that the Mongol’s highly recurve (bent forward) bow was the key advantage. Recurve helps, but not much, and recurve also appears in the bows of the Moslems and Chinese. As the force-distance curve above shows, recurve bows provide, at most, 10% more energy than a straight bow other things being equal – arrow energy is related the area under the curve. This is far less advantage that with a modern compound bow (one with wheels– the blue curve above), and no 13th century culture had the modern compound bow. Recurve or not, the force-distance curve of a non-compound bow is a virtual straight line. Unless the Mongols had longer arms or stronger pulls, their arrows would go the same distance as everyone else’s, or would fall shorter because of lousy craftsmanship.

2. The nock issue. There are two ways this might work: a) that the Mongol bowstrings were narrower, e.g. silk instead of flax, or b) that the Mongol arrows were shorter. I’m willing to accept both answers. Either answer would explain how the Mongols could use European arrows while the Europeans were not able to nock the Mongol arrows. But these answers just make the range issue thornier. A silk string does not give you extra distance; a short arrow would decrease the shot-length. The energy given to the arrow is the integral. In a perfect world,

E =  ∫fdx

E is the energy of the arrow, f is the force produced by the bow, and x is distance. Thus assuming no energy goes to the twanging of the bow, the energy of the arrow is equals the area under the force-distance curve, above. if the bow is unchanged, and the arrow is shortened, the draw distance will be shortened, and the range will be reduced.

The real reason that Mongol arrows went further, I suspect, has to do with the fact that the Mongols like to fire their arrows from horse-back, and I like  to imagine from horse running towards the enemy at 25 to 35 mph. Now lets compare two archers. One, a standing archer (Russian, lets say) carries a 70 lb bow shooting a 100 gm arrow (2/9 lb) at 50% energy efficiency. I call this a 70 lb bow because the maximum force at the end of the pull is 70 lbs — more force than we use currently, but quite reasonable for medieval war where shot distance was the key to survival. Modern bows are approximately 80% efficient, but I take 50% efficiency just because wood (or bone for Mongols) isn’t as good as fiberglass in this regard, and leather bow strings are not as good as nylon.

I could solve the integral above for the arrow velocity and distance, using Euler’s approximation, but for now I’ll use a simpler approximation. For the above case, the average force is seen to be 35 lb (the force goes from 0 to 70 lbs), or 156 Nt. The energy in the arrow is then 1/2 156 x, where x is the pull distance, the draw. If we assume the draw is 1/2 m. the energy deposited is 39 joules and maximum speed of the arrow turns out to be 28 m/s, 63 mph.

v = √ (2)(.5)(39)/(.100) = 28 m/s.

The Mongol army was almost entirely cavalry archers.

The above suggests a flight time of about 8 seconds and a maximum range of 28 x 8 =  224 meters. This is about what you find with European bows. Now consider firing that same arrow from the back of a horse galloping at 32 mph, 19 m/s: The range for the same 8 second hang-time is now about 50% more, about 336 meters or 380 yards. If the riding Mongol released his arrows while going 32 mph, he would easily hit a standing archer 350 yards away, while the archer could not reach the Mongol. The only trick is the Mongol has to pull up before he reaches the 225 yard mark. Having a moving horse did more for the Mongol than having a better bow, better arrows, or a longer pull.

Robert Buxbaum, September 24, 2017 I’d mentioned Genghis Kahn in an essay about military statues. and felt compelled to come up with an explanation of how he won so much land, so fast, with so few men. Similar to my treatment of Mongol bows, I discuss aircraft carrier catapults.