No they aren't. Longbows are very large (nearly as tall as an average man), have much bigger arrows, and take a lot longer to pull back. They also shoot to a a much farther distance and can penetrate most armor with ease.
Short bows a re small, lightweight and low strength bow used for closer range and mobile shooting. Kind of a close range forest hunting sort of deal.
Uhm, no. That thing about penetration, simply wrong: http://en.wikipedia.org/wiki/English_Longbow#Range_and_penetration
There are also videos on youtube that shows this in a test bench - penetration is very rare occurance, but would barely scratch the wearer.
You need crossbows and heavier bolts to penetrate. Alternatively a very special bow (you use your feet) with very long and heavy arrows, but it's not a medevial thing.
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So if we have five different bow types and sizes, like
Japanese Yumi (huge) - http://medieval2.heavengames.com/m2tw/history/miscellaneous_history_folder/japanese_weapons_folder/kyudo.jpg
English Longbow (large) - http://poorfolkbows.com/elb7.htm
English Warbow (medium) - http://farm4.static.flickr.com/3284/2800778643_0f2f4e26c1.jpg
Mongol Composite Bow/Recurve Bow (small) - http://www.centenaryarchers.gil.com.au/images/niya_recon.jpg
Indian Short Bow (tiny) - http://1.bp.blogspot.com/_VyekL5IHloE/TOAGE56nwZI/AAAAAAAABak/X1hmF2r8Wl8/s400/Indian-Holding-Bow-Arrow.jpg
then how would we deal with them?
The Yumi seems like asking for clipping problems, so lets just take that one out. The indian short bow does shows slight evidence of recurve although not as extreme as the mongol version, as does any pictures of short bows. So we can relabel the two smaller ones to short bow (recurve is part of making it small), suitable for mounted combat and small prey hunting. So as I see it, we have three possible bow types, and I suggest these features on them (more gameplay related than realism related):
* Long Bow - You need to be standing still, but it can penetrate armor (penalty 0). Long draw speed. Does weight up for missing crossbow (which *is* problematic to include).
* War Bow - Can be fired on the move, but not from mounted and does not penetrate armor (penalty -2). Fast draw speed, but still reduces stamina while aiming.
* Short Bow - Can be fired from mounted but does not penetrate armor (penalty -4). Fast draw speed, but no stamina penalty.
With these three variants we can add three "styles" reflecting English/Imperial (no recurve), Indian/Elvish (medium recurve) and Mongol/Dwarven (heavy recurve). Imperial design sacrifices draw speeds for power, Dwarven sacrifices power for draw speed, Elvish being the all rounder. These designs can well be culturally chosen, so they appear more frequent in some cultures than in others.
We can also have at least two different build styles: simple and laminated. The simple one doesn't offer as much power, but will need less repairs. The laminate one is somewhat more powerful (armor penalty +1), but will break down more quickly.
That's 3 types, 3 designs, and 2 builds giving us 18 different bows for different purposes. I'd like to forget completely about silver and ebony bows, that stuff hardly makes sense, but belongs more with the arrow designs. As for armor penetration, the secret lies in perks and arrow material. Without any perks, you'd need a daedric arrow to defeat a daedric armor with a Long Bow (penalty 0). If you get four ranks of perks, you can defeat daedric armor with a short bow (penalty -4 + 4 perks = 0), but you still have to use a laminate bow.
Finally, arrows degrade. You can craft basic arrows in the wild consisting of wood and stone - the most basic one - in three qualities; used, normal, and fine, depending on your crafting skill. For more exotic arrows, you have to visit a smith. You can choose to carry a bunch of arrows and not worry about degration at all, or you can carry less and choose to maintain them. Magic arrows (other than unique ones) loose their enchantment when used and degrade to a normal arrow of that material, but still in fine condition until you use it next time.
Does any of this make any sense?