I've heard that heavy arrows with more momentum (like from a war bow) were more effective against armor than lighter faster arrows.
This video shows some penetration testing with different weights of arrows.
http://www.youtube.com/watch?v=VAfK0sBsZBw
All the arrows used are light in comparison to English war bow arrows, but you can see the lighter arrow penetrated more than the heavier arrows.
I've also seen other tests where heavier arrows penetrated more than lighter ones.
Does anyone have any ideas on how this would translate to arrows against armor?

(Here we have a longbow arrow completely penetrating a phone book, though it's with a needle bodkin fired from a much heavier bow than in the first test
http://www.youtube.com/watch?v=hvEJZKB1kpE )

Light arrows tend to fly faster and have a flatter trajectory. Heavy arrows are better for punching through armour. I think it was King Charles who bemoaned the fact that nobody could shoot a quarter-pound arrow anymore.

A heavier arrow from the same bow always[1] has more energy and more momentum. Somewhat more energy, and a lot more momentum. The lighter arrow always has more speed. To check this, one ideally uses a chronograph, rather than indirect tests like penetration. Some tests showing variation in speed with arrow weight: http://www.atarn.org/islamic/akarpowicz/turkish_bow_tests.htm ; note that the "efficiency" is the fraction of stored energy transferred to the arrow. From a given bow, the efficiency tells you the kinetic energy (or you can calculate it from the weights and speeds).

For penetration into soft tissue, penetration is limited by viscous drag (mostly). In this case, more momentum gives more penetration.

Where penetration is limited by friction (like a lot of archery targets), then more kinetic energy gives more penetration. Assuming that the frictional force is the same for different arrow speeds. Which it won't necessarily be, since the target deforms. Perhaps this is responsible for some of the test results?

For penetration through armour (metal or rawhide hard armour, at least; not sure about textile and fibre armours), more energy gives more chance of penetration. To put an arrowhead through a steel plate, you need to cut through the steel, and curl the edges of the cut out of the way. This takes a certain amount of energy. Only by exceeding that energy can you go through. This is supported by plenty of testing; a summary is given by Atkins in The Science and Engineering of Cutting.

Manually shooting arrows from a bow is a poor way to test - too many ways to vary the draw and get the result you want, even if unconsciously. Karpowicz's shooting machine in the tests linked above is much better. Other good test methods include pneumatic launchers or drop-weights (which eliminate (or at least greatly reduce) effects of arrow spine). A manual-shooting test would ideally have the arrows shot through a chronograph.

[1] Where "always" means that the draw and release are identical, and we're ignoring energy that goes into flexing the arrow.

Most excellent answers so far.

I'll just add an explanation of how an arrow like a quarter Pound or even heavier could be made.

The weight of the Arrow depends on a number of facotrs:

-The arrowhead shape, overall size and of course material used will affect weight. We'd expect iron or better yet tempered steel if one is to peirce metal based armour though. But there were Arrows made with hardwood tips in medieval times. Some are enbedded in an old fortified log house from 14th Century at the Skansen heritage outdoors museum here in Stockholm, Sweden, showing plenty of penetration even with such crude arrowheads.

- The type of wood used, oak would be far heavier than say the same size ash but also more liable to crack rather than bend.

- The shape of the Arrow shaft for example conical with a wider back end than at the point which also helps vs splitting while being shot with a really Heavy bow. Lighter flight arrows could be made straight or close to it just like modern target arrows.

One must also remember the length of the Arrow used, most modern made Arrows are considerably shorter than say for an historical warbow which is drawn further. One way to make and Arrow heavier would be to simply make it longer, though at some point it would become unweildy and impossible to shoot.

Some other things to bear in mind

1: bows don't scale linearly with draw weight, diminishing returns set in quickly
2: heavier arrows are required to get benefit from a heavy bow, so if you are shooting a 120lb war bow a light arrow is not an option the arrow would most likely shatter or even shoot slower then in a lighter bow
3: even a relatively inefficient bow like an English Longbow packs more then enough energy to kill anything without armor at pretty low draw weights (say 70lbs or thereabouts) so the only reason I can think of for massive bows in punching through some degree of armor
4: the only way to really tell what would happen when you shoot armor is to shoot armor. Similarly the best way to tell what happens when you shoot flesh is to shoot dead animals , which is what a lot of the modern day testers do
5: even modern recurve and reflex/deflex "longbows" are significantly more powerful then ancient bows. This is true even for bows that are made mostly of traditional material, the only real equivalent to an English Longbow is something very close to the original design

I'm not sure there is a direct, general, relationship between the weight of the bow and the maximum range of the arrow

Relatively light bows can shoot a long long way, the distance an arrow flies is a factor of both weight of the arrow and energy delivered to it.

for instance this example a < 35lb recurve shot 500 yards

http://www.worldrecordacademy.com/sports/long...101831.htm

There might be a relationship between accuracy at that range and the weight of the arrow, or a relationship between armor piercing ability. Could also be there is some peculiarity of English longbows that drive toward heavier bows giving longer shots

Please read the Ashby report for details on arrow effectiveness on animals;
http://www.alaskabowhunting.com/Dr.-Ed-Ashby-W26.aspx
Heavier is better in pentrating flesh and bone with arrow heads strikingly similar to those used throughout pre-history. 50lbs draw weight is reckoned to be enough to kill any animal in North America including Elk (alces alces) and Bear (ursus arctos). I would have thought the idea behind the heavy draw weights (estimates aren't they?) of some historical bows was to maintain effectiveness at long ranges. Effective and lethal not being the same thing.

I could see the high mass of an arrow making it more deadly when fired on an extreme parabolic arc

it probably would not shoot farther but might hit harder and be more accurate, less effected by windage

It doesn't appear that heavy (say, 80g, or 1000gr, or wherever we want to draw the line, and up) arrows are needed to be effective at long range, against horses and unarmoured soldiers. Ming and earlier Chinese, Korean, Indian, Turkish, and Mongol use of arrows in the 20-40g range, out to long range, effectively, shows this.

In East Asia, we have Manchu/Qing bows, and Japanese bows sacrificing range and accuracy for energy, and accepting very short archery battle ranges. Very heavy arrows, often 100g and more. To do so in an attempt to defeat armour makes sense. It certainly wasn't for long range performance.

The hunting information is interesting, but I don't know how applicable it would be. There's a big difference between shooting from an ambush at point targets 30 yards away or less and shooting at 300 yards at massed troops/cavalry who are wearing armor and wounding them is just as good as killing.

Certainly, for plunging fire, a heavier arrow is going to retain much more energy and have a higher terminal velocity than a lighter arrow. Trajectories are never parabolic because the arrow slows down as it travels. At extreme range the arrow will be falling close to vertical.

I don't know that many arrows would even be used at direct fire ranges (<70 yards) - at least under the English system. As mentioned, the ability to disrupt cavalry at long ranges was the killer app.

Apparently the Manchus felt differently, which is interesting.

Here is Hardy's summary of his tests from The Great Warbow (p. 18): Warbows of 143-165 lbs shooting arrows weighing 3.5-4.0 oz. flying as far as 240 yards and losing 15-30% of their initial velocity at the point of impact. Lighter arrows lose a lot more of their initial velocity than this.

I have limited knowledge and experience with archery but wouldn't it be reasonable to assume that factors affecting the armour penetration of arrows would be similar to those of other projectiles? If so, my assumption would be that the most important attributes affecting armour penetration would be the shape of the arrowhead, density of the material used in the arrowhead versus that of the armour, and speed. I don't think weight would necessarily be as important other than its effect on the speed of the arrowhead and as a result of its density. The efficiency of the arrowhead in preserving its energy as it strikes the target would probably be what determines its effectiveness so a heavy arrowhead that decelerates too quickly wouldn't work well.

Looking at modern day projectiles, a 45 caliber pistol round is heavy compared to other pistol rounds but has relatively poor performance due to its lower velocity and shape. An armour piercing 9mm round is smaller and lighter but has higher velocity and is made out of denser materials which makes it better at punching through armour but may possibly compromise its effectiveness against "soft" targets.