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Originally Posted by Maerlande
Excellent point and one I did not address. Easiest way to model it mathematically is by an efficiency factor. I don't know the details sufficiently to make an analysis of this effect but I would be very interested if you can find some references or can supply some more information. Along with that same point there is a serious problem with the arrow flexing. It is my understanding that this is why modern hollow shaft metal and composite arrows are much more effective. They fly faster and straighter. But frankly, modern materials are not relevant to this discussion. We are talking about medieval bows and crossbows.
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Ignoring all "complicated" stuff like friction or speed-dependent forces it would still be damn hard to model, and flexible objects can only be really analysed via computerprograms afaik. However there are two simple points to add:
The energy put into the bow ( - losses) doesn't equal the energy of the arrow but the energy of the arrow and all the moving bowparts at the moment of launch. I can't calculate how a bow would actually behave but for a simple balista with rigid arms and torsion springs in the middle this should come to
E(bow) / E(arrow) = mass(bow) / 16 * mass(arrow)
With E(arrow) = (1/2) * mass(arrow) * speed²(arrow)
So speed²(arrow) = 2 * E(input) / (mass(bow)/8 + mass(arrow))
So if the bow weights 8 times as much as the arrow the catapult has only an efficiency of 50% (and that's not even counting entropy) a catapult which weights 24 times as much only 25%.
(While every real bowshape should be more energy efficient this still applies with different factors)
Now, so making the bow heavier (or changing its shape) makes it less efficient, why would we do so anyway? For one thing because we want additional draw weight.
And that makes two factors that limit the crossbows we can build. It's undoubtedly true that the crossbowman can put a lot more energy into the crossbow, but the question is could the medieval people build crossbows that could handle that extra energy while putting enough of it into the bolt to make a difference.