Saturday, April 9, 2016

Scaling effects in archery

By Ugo Bardi – 2000

On the right: one of the smallest examples of bows. A replica of an Indian bow demonstrated by Jim Hamm, this diminutive bow (90 cm, 36") is nevertheless not a toy and can be used for hunting.

On the left, a Roman Palintonon catapult according to the reconstruction of M. Viollet le Duc. Among the largest bow-like machines ever built, the bow arms could span up to 3-4 meters and the the weapons could shoot 400 Kg projectiles

Hand held bows are normally made as large as it is practically possible. This large size makes them somewhat unwieldy and difficult to handle. Nevertheless, at the simplest level, it seems logical that a larger bow can shoot a larger a heavier arrow. That may be a good thing for the archer (if not for the prey or enemy). We know that in ancient times bow-like machines much larger than ordinary bows (catapults) have been built and used to shoot very large arrows or projectiles. On the other hand, hunters or people using bows while riding a horse would normally use smaller bows. Indian bows were almost never the full length of a European longbow. Again, there seem to be logical reasons for wanting to use a small bow: it is easier to handle and to carry. Now, what is exactly that makes longbows more effective than short ones, and what are the physical factors that control the characteristics of short and and long bows? In physics you call these effects scaling laws.

Let’s consider first small bows. How small can a bow be made and still be effective for some practical purpose? In "Gulliver’s travels", Jonathan Swift describes how the protagonist is attacked by Lilliputian armies of archers. In the story, Gulliver is almost unaffected by the small arrows shot at him, and his only worry is to avoid to be hit in the eyes. Intuitively, it sounds logical, but let’s see if we can say something more quantitative. Let’s consider a Lilliputian archer and for focussing our thought we may think of it as a something like 20 cm (8") tall. There is a host of scaling factors to be considered about the size and proportions of a biological creature. In general, we can be sure that if evolutionary pressure were to act to make humans smaller ("turn them to pixies"), it is unthinkable that the proportions of the various body parts would be maintained. For instance, if we reduce the linear size of a human to a pixie ten times smaller, the body mass would be reduced not by a factor of 10 but by a factor of 1000, but this should be corrected taking into account a series of further scaling factors. For instance the thorax should be larger to accommodate proportionally larger lungs which have an inner surface proportional to volume. And then, the head should be large enough to accommodate a cortex with a sufficient number of neurons to make the creature capable to use a bow, no small feat. On the other hand, the relative weight and volume of some parts of the body can be reduced. For instance, the legs have to carry a proportionally lower weight, and so can be thinner. On the whole, if created by real biological forces, a Lilliputian would not look so much like a small human being, more like a mouse with a large head, and it would not be necessarily pretty.

But we are not interested so much here about our small archer being pretty or not, and our initial question can be perhaps reformulated as "can a mouse us a bow?". Why not? The little beasties are known to be smart, and you never know what they could do. So, the main point about archery is to shoot arrows with the highest possible kinetic energy, so that they’ll go as far away as possible and penetrate the target as deeply as possible. In bows, this energy is given simply by the very definition that you can find in textbooks: a force that moves its application point. In a bow, the force is the "pulling force", the elongation is what is called "draw". The larger these two factors, the higher the energy. Very obviously there are limits to both: human force is limited, and the length of one’s arms determines the length of the draw. A smaller archer is limited both in draw length and in force. We said that we would consider an archer of 1/10 size of a human being and that means that the draw is reduced of the same factor. Regarding force, this is roughly proportional to the area of the section of the muscles involved (this is why people engaging in weight lifting or arm wrestling have such thick arms). In proportion, the force would then be reduced of a factor of 100 (this assumes that the thickness of the arms of the elf is proportionally the same as in a normal sized human beings, we said that evolution would make this is unlikely, but it is not impossible either). So, on the whole the energy involved with a "toy bow" about 10 cm length that could be operated by a mouse (or an elf, or a pixie, or a Lilliputian, or whatever) is a factor of 1000 smaller than that of a human sized bow.

This factor of 1000, impressive as it is, is not such a great handicap if you consider that the arrow, too, is smaller and lighter. Again, being everything linearly reduced of a factor of 10, the arrow weight is 1/1000th of that of a human-sized arrow. So, things even out, and the speed at which a Lilliputian arrow can be shot is just the same as that of the human one. Now, 10 cm (4") bows are not so common around, but this result fits with the evidence for what we know about short bows. Even today some archers (human ones) like to use short bows, and will try to convince you that their bow is as god as a long bow by measuring the initial speed of the arrow and showing you that it is the same as (or even higher than ) that of a longbow. Yes, a short bow - actually any bow independently of size - can shoot a light arrow at a remarkably high initial speed, but there is a problem with small bows that we’ll see now: aerodynamics.

Aerodynamics is the key factor governing the flight of an arrow. We know that in general an object moving through a viscous fluid experiences a resistance force proportional to speed (v) and to the cross section (A): F=KAv. The accelleration (actually a deceleration) caused by this force is simpy this force divided by the object mass (Newton’s law: F=ma). So we have an equation of motion to solve as

Speed (v) is a function of time and solving this equation we can see how the arrow speed varies after having been shot. This is a simple equation to solve: for a function (v) to be proportional to its first derivative it has to be some kind of exponential, that is the basis of the natural logarithms, e, elevated to something. The result of the integration is vo is the initial velocity, the speed of the arrow just as it leaves the bow. We see that for very large t the exponential tends to zero and this speed tends to zero, too. Also, there is a time for which the speed of the arrow becomes half of the initial speed. This "halving" time is given by

We can’t calculate this time exactly since we don’t know the values of the constant K, but we see that a heavier arrow will, in principle, maintain its speed for a longer time and hence fly farther (for the same initial energy). This may seem to contradict the archers’ common habit of choosing arrows as light as possible, but there is a reason for that, too. A heavy arrow will start slower than a light arrow, but it will maintain its speed for a long time and eventually overtake it. But both arrows will feel the effects of gravity and fall down at the same speed. Being slower at the beginning, the heavier arrow will lose more height and as a consequence has to be shot with a more arched trajectory. Archers (as all shooters) much prefer flat trajectories which make it much easier to aim, hence the preference for lighter projectiles.

Now, back to our Lilliputian extra small arrow, we said that its mass is 1/1000th of that of a human arrow. We can also say that its cross section (A) goes with the square of the length and is by necessity 1/100th. And, alas, when combined in the formula, we see that t1/2 for the small arrow is ten times smaller than for a normal, human-sized, arrow. Here is the effect of aerodynamics: small projectiles are always at a great disadvantage with respect to large ones. An Elvish arrow would start very fast, but rapidly lose speed. Since the distance covered is proportional to speed, its range would likely be also 1/10 of that of the human sized one.

The effects of aerodynamics on small arrows do not end here. The arrow flies in air but at some moment, hopefully (for the archer) it will hit the target and do its effects. The arrow penetration into living tissue can be approximately described by the same aerodynamical equations written above. Just, the constant "K" will be much larger for a denser medium. Again, at the same speed, the Lilliputian arrow would penetrate approximately 1/10 less depth than the human arrow. You see that Swift correctly described Gulliver’s situation even though he probably didn’t make any calculations.

In the end, all this reasoning has led us to discover that what is called in physics the "scaling factor" for bows is just one. A bow half size is – very roughly – just as deadly at half the distance, but there may well be conditions in which range is not the most important factor: hunting for instance. The maximum range of a longbow is a few hundred meters, but the practical range (the range at which you have reasonable chances to hit anything smaller than a mammoth) is only of a few tens of meters.

So, a small bow may have its practical uses, too, and there are interesting experimental data confirming this point. Jim Hamm reports in his book on the archery of American Indians that in many cases the Indian archers neither pulled their bows to the maximum possible elongation, nor to the maximum force. Their "pizzicato" style of archery (holding the arrow between index and thumb) made it impossible to use all the force that a European archer can muster by using three fingers to pull the string. In practice these Indians let the arrow fly much before the maximum force was reached, and their bows were very small, too (though not Lilliputian!). There is plenty of logic in this style. As we have seen, using light arrows you can shoot at a very high speed, and the pizzicato style makes sure that you shoot fast – anybody who has tried intuitive archery knows that thinking too much is the sure way to miss. In hunting you have no time to aim, you just let fly, speed is everything. As a shortcoming, these Indian arrows do not have a very long range but, again, what would be the benefit of shooting at hundreds of meters? That’s not the way to get a turkey for dinner. Incidentally, penetration here is often not an issue either: when hunting small game Indians would often use "blunt" arrows (i.e. without metal or stone tips) designed to stun the target rather than killing it immediately. The perhaps surprising conclusion is that maybe, for hunting, humans are somewhat oversized for the purposes of using a bow and that hunting bows could be profitaby used by smaller creatures, if not Pixie-sized perhaps at least Hobbit-sized.

A different story is bows used in war. In that case, precision becomes a lesser concern and range becomes fundamental. In war you are not aiming at a single enemy, it is a mass of archers shooting at another mass of troops some distance away. Here, if you can shoot from a larger range you have a definite advantage, and if the enemy is charging at you can start shooting earlier and shower them with more arrows. The shape and size of military bows follow from these considerations: maximum range obtained by large bows pulled to the limits of the strength of the archer. Also, arrows are large with heavy tips, true mini-javelins designed to maintain their speed for a long time and to fall on the enemy with full force from above. It may be that for military archery humans are actually undersized, and this explains the development in history of all sort of bow-like devices (crossbows and catapults) designed to cram more energy into the arrow.

So, our initial question was: "could a very small creature (Elf, mouse or Lilliputian) profitably use a bow?" The answer is yes, provided that we take into account range and size of the target. If our Elf archer can get within a couple of meters of his prey (say, a mouse), then a well-placed arrow to the creature’s chest will likely penetrate at a sufficient depth to cut the inner organs and snuff it. Whether this would be practical or not it is difficult to say, but in fantasy novels and games we can at least imagine the existence of armies of pixie archers……...