Ranged weapon physics

Unfortunately, trying to extract parameters useful for an actual assessment of a historical weapon from descriptions is hard. One problem is that the wrong parameters are reported - while it is not difficult to learn about draw weight and range, what would be of much more use is kinetic energy and projectile mass. Another problem is that descriptions are frequently at odds with each other as far as range is concerned - and there is of course a reason, because 'range' isn't a property of a bow or crossbow in the first place. So let's take some time to understand what is what.

From draw weight to kinetic energy

The name draw weight indicates the force with which the string needs to be held when the weapon is fully drawn. This isn't usually a mean force - most weapons work in an approximately linear regime in which the force grows with the length that is drawn, so if you draw only a bit you're quite far from experiencing the draw weight - that comes only if you draw to full length. But it isn't even necessary - a modern compound bow has a relation between drawn distance and force that is more constant.

If we assume that force is proportional to length - which is mostly approximately true for historical weapons - then the elastic energy stored in a bow (or crossbow, or torsion catapult) is proportional to the draw weight times the full draw length squared. That means that even if they have the same draw weight, the tall archer who can draw 70 cm has nearly 15% more energy stored in his weapon than the small guy who can only draw 65 cm. It is also relevant to understand the difference between a bow and a crossbow - crossbows frequently reach draw weights that by far exceed those of any bow - but they also come with a much shorter draw length, and so the actual energy stored becomes comparable again.

It is tempting to identify the energy stored in the bow with the energy of the arrow (or other projectile) - but alas, this too is simplistic.

When the string is released, the bow snaps back into its normal configuration --- energy stored in the deformation of wood or a torsion spring is converted to kinetic energy. All parts of the bow receive the energy - that means its arms as well as the string that accelerates the arrow. The transfer function determines what fraction of the total ends up with the arrow.

Its general characteristics are such that the less kinetic energy is in the bow, the more goes to the arrow. In other words, when the bow can't relax to its configuration fast because it accelerates a heavy arrow, the transfer is good, if the bow can relax fast because the arrow is light, then lots of energy is dissipated when the string finally stops the process (this is very evident from firing 'empty' - don't try it, it is dangerous!).

So to get most of the energy to the arrow, we want to use heavy arrows - but they are slower than light arrows for the same kinetic energy precisely because they are heavy.

From kinetic energy to range

In a frictionless world, the range of a projectile is proportional to its squared velocity. If the kinetic energy is given, that means that cutting the mass of the projectile in half doubles the range. And that in turn means that whenever we discuss the range of a weapon, we need to ask: What is the projectile mass?

For the real world, it's not quite as simple. First of all, kinetic energy is not constant, the transfer function says that if we make arrows lighter and lighter, they will receive less and less kinetic energy. Second, light and fast objects will also experience a lot more drag. So, there's a projectile weight for which the range is optimal - if we make the arrow any lighter, the transfer and drag will kill range, if we make it heavier it won't have enough velocity. If some modern replica of a weapon is tested for maximum range, this might be the range that is found.

However, the maximum range arrow is almost certainly not what is of military use. Even at somewhat less than maximum range, a heavier arrow will do nicely - and it will have more energy upon impact (better transfer, less drag) - which means more armor penetrating power. Of course that too has limits - if we make arrows too heavy they won't reach enough - so the weight with which archers shoot is a compromise between achieving a decent range and a high penetration power. In short, a longbow is better operated with 100 g arrows and 200 m range than with 25 g arrows that reach 400 m but don't penetrate armor unless the target is closer than 25 m.

So a likely scenario is that often military archers would not use more powerful bows to shoot the same arrows farther, but to shoot heavier arrows the same range. That isn't necesssarily the agenda of modern archers who don't benefit so much from high energy impacts which tear their straw targets apart in a hurry. Understanding this, we need not be troubled by vastly different accounts from historical sources and modern replica tests.

All of this means that kinetic energy (unfortunately) can only really be estimated if we can discover a combination of projectile mass and range - only one of them will not do.

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