The aerodynamics of arrows are simple (once you've got the data) but the mechanics of a bow are tricky, and not many sites consider what would be an ideal bow for a Halfling!
This post is all about Longbows – bows made from a single piece of wood (typically Yew). They are termed Longbows as they are typically taller than the archer.
I’ll start with a summary of the rules my investigations lead to:
 The ideal Longbow is one size longer than the height of the archer, arrows are six sizes shorter.
 The kill bonus with a Longbow is your Power (STR+ATHL+SIZE) plus a fixed +3 for Longbows (as per other weapons).
 Your power affects the draw weight of your bow  how stiff it is (note this doesn't affect the actual weight of your bow).
 The range for all sizes of Longbow is 900’, but...
 If your power is less than 2+HEIGHT you get 1 range per 1 power ( due to over weight arrows).
If you're not interested in long range shots or encumbrance, you don't need to bother with the remaining rules:
 Bows have a "drag length" which is the length of the bow * 64. You get a 1 kill penalty per this amount of range. For example, a 3' bow has 192' drag length gives 1 kill at 192', 2 kill at 384' etc.
 A 7' bow weighs 2lb and shoots 2'8'' arrows weighing 0.5oz. Use the standard sizing tables to vary this.
 If your power is greater than 2+HEIGHT you have +1 weight arrows per +1 power.
Note that most of the rules just go off your height and size categories  for humans either 6'/180lb or 5'/125lb. All 5' humans shoot a 6' longbow, weighing 1.5lb, with a drag length of 384'. Strong humans get heavier arrows, weak humans get a range reduction.
Arrow Aerodynamics
In A level maths I learned about projectiles, and learnt that the furthest range is achieved by launching the projectile at 45 degrees. The projectile loses velocity as it climbs into the air, converting kinetic energy into potential energy, and then reverses this conversion as it falls, striking the ground at the same velocity it was launched with.
This simplification ignores air resistance, which causes drag and slows down the projectile. This means the impact velocity and kinetic energy at impact are reduced, which in game terms means that the projectile does reduced damage. Damage of projectiles is linked to the Kinetic Energy – half the mass times the velocity squared; doubling the velocity of a missile (which quadruples the range) is as effective as quadrupling the weight. In Explore either of these gives you a +4 on kill.
A drag force acts upon the projectile which is proportional to the square of the velocity, and also proportional to the drag coefficient which depends on the shape of the projectile (for example it is 0.47 for a smooth sphere, 2.1 for a brick). If you know the drag coefficient and either the launch velocity or the distance it travels, then you can calculate the motion of the projectile through the air. Conversely, knowing the launch velocity and the distance travelled you can calculate the drag coefficient. (This is just an approximation but it’s good unless you’re near the speed of sound). Although the equations are simple, deriving equations of motion are not  there are several examples on the internet of derivations which are unhelpful over simplifications which assume that drag force is proportional to the velocity instead of the square of the velocity, so I wrote a simple simulation program. Here's a graph for the flight of a projectile with drag launched at 45 degrees:
One consequence of drag is that the furthest distance is achieved at an angle below 45 degrees. For example, if we reduce the angle in the last graph from 45 degrees to 37.6 degrees it goes a little further:
Note though that despite the quite different trajectory, the difference in overall distance isn’t that much!
To apply these principles to longbows requires some real world data. Unfortunately finding data for arrows fired from a longbow proved tricky, until I stumbled across the book "The Great Warbow" by Matthew Strickland and Robert Hardy. The appendix for this contains some great data  the launch velocity of arrows fired at 45 degrees and the length of flight. From this they calculate the drag coefficient and the impact velocity and impact kinetic energy. Of key importance to me is the fact that I can check that with these inputs I get the same calculations from my program!
Here are my calculations with the data, matching the calculations in their appendix almost exactly (It's a replica of bows from The Mary Rose  a 150lb bow with 32inch draw length. I've taken air density to be 1.2, and also note there was a tail wind of 9m/s).
Arrow #

Mass (g)

Initial
Velocity

Range
(m)

Drag
Coefficient

Final
Velocity

Initial KE

Final KE

Final KE
/ Initial KE

1

53.6

64.29

313.8

1.80

49.07

111

64.5

58%

64.65

312.8

1.89

48.76

112

63.7

57%


2

95.9

53.36

234.7

2.09

43.70

137

91.6

67%

52.28

228.6

2.02

43.37

131

90.2

69%


3

74.4

57.48

258.2

1.78

44.95

123

75.2

61%

57.77

258.8

1.82

44.88

124

74.9

60%


58.24

260.3

1.87

44.85

126

74.8

59%


4

57.8

62.25

299.7

1.93

48.26

112

67.3

60%

63.09

301.9

2.04

48.11

115

66.9

58%


5

86.6

53.59

230.6

2.15

42.90

124

79.7

64%

53.52

231.2

2.10

43.04

124

80.2

65%

The arrows weighed between 53.6 and 95.9 grams, the range of the shots was 228.6m to 313.8m, the launch velocities 52.28m/s and 64.65m/s, but the computed drag coefficient for the five different styles of arrow was quite consistent, between 1.8 and 2.1.Now given a fixed drag coefficient and launch velocity we can plot the impact velocity at various distances due to altering the launch angle:
As you can see, raising the angle gradually from zero increases the range, but the impact velocity reduces roughly linearly  until you get to maximum range  then increasing the launch angle further starts reducing the range all the way back to zero (whilst the impact velocity stays flat).
For the first half of this graph (before it starts doubling back on itself) we now plot the kinetic energy at impact as a percentage of the original:
This is reducing exponentially, as is clear when we plot the log of the curve and get a roughly straight line:
Hence we calculate that for the arrow with the furthest range (and fastest initial velocity) in the experiment, kinetic energy is reduced by root 2 approx every 140.6m (461 feet), which in Explore means a 1 on damage per 448 feet (nearest length category) – this is our drag length for the bow.
Now the drag depends upon the arrow, so to proceed we need to know what sort of bow and arrow would be used by smaller or larger, weaker or stronger archers.
Draw Length
When you loose an arrow you should draw the string back to your ear. The distance between that and where you are holding the bow  which is slightly shorter than the length of the arrow  is your "draw length". You should always shoot at your natural draw length. It is approximately your arm span (which is approximately your height) divided by 2.5.
Arrow Weight
If you shoot a slower arrow, then not only is the range reduced, but also you have to fire the arrows higher into the air, thus hitting the target is more difficult. In addition against a moving target you'd want your arrows to be as fast as possible.
Given a bow and a fixed draw length, it fires light arrows at a particular velocity. You can increase the weight of the arrows until the velocity drops. You should fire the heaviest arrow which can be fired at that maximum velocity. Heavier arrows have the same kinetic energy and hence the same damage, but you want arrows at that ideal weight.
Unfortunately you can't make good arrows of the correct length below a certain weight, so you can be forced to fire heavy slow arrows. In Korea they used a bamboo "overdraw" to shoot short lightweight arrows, but we're not doing an Oriental game at the moment.
Draw Weight
The force required to draw a Longbow increases linearly from zero to a maximum at full draw, the Draw Weight.
Bow Efficiency
As the force drawing a bow is linear, the potential energy put into the bow is half the draw weight times the distance the string is pulled back through. Longbows are about 70% efficient, which means that 70% of that potential energy gets converted into kinetic energy for the arrow. The loss is mostly through the weight of the string, which is still moving at the point the arrow is loosed. Note that if you use the draw length to calculate efficiency of a bow (as I did) you'll think that longbows are only 50% efficient, but that's because the bow starts out bent and you only pull it back 80% of the draw length.
Optimum Bow Length
In "Primitive Technology 2: Ancestral Skills" by David Wescott, page 110, he measured the speed of arrows launched for a variety of length bows all with the same draw length and draw weight. The fastest ones were 66  67 inches for a 28 inch draw. He says that this ratio of draw length to bow length should be constant for small bows. Hence the optimum bow length should be roughly 67/28 = 2.4 times the draw length. Note that this together with the draw length calculation would make longbows shorter than the archer, so I'm going to keep with these principles but adjust these ratios slightly to match historical bows.
Arrow Speed
From World Records in Flight Archery we can see that the world record for an English Longbow is 339.65m, which is only a little further than The Great Warbow experiment's best shot (313.8m).
At Greenman Longbows we find the launch velocity of a range of longbows with different draw weights and see an English Longbow of only 47lb draw weight (one third of the one in The Great Warbow experiments) which fires at 177fps = 53.9m/s, which again is only slightly less than The Great Warbow's 64.29m/s, and is likely to be due to tooheavy arrows (see below).
From these two sources we see that increasing the draw weight of the bow has not increased the maximum launch velocity, only the weight of the arrows it can fire at that speed. Hence it is reasonable to assume that the maximum possible velocity for bows is the same for all draw weights, and is with a wellmade bow of the optimum length for your draw length.
Short Archers
Note that smaller archers have shorter arms, so a reduced draw length (shorter arms). If you halve the draw length you’d expect the Potential Energy in the bow to be halved and hence the Kinetic Energy of the arrow to also be halved. This would result in additional penalties for Halflings  in addition to the 3 penalty to strength from their size they'd get an additional 2 penalty. If this was the case, however, you'd find that archers with long arms had a big advantage  and they don't. In fact it is the reverse  people say that long arms are a disadvantage. When you look at how you draw a bow the reason for this is clear  everything is stationary except for your upper arm which you rotate backwards to draw the bow. If you double the length of your arm, then you double the length of the leverage the bow force has on your shoulder  you would only be able to draw half the weight. That happens to exactly counteract the gain you made by doubling the draw length! Hence the potential energy that an archer can put into a bow, and the kinetic energy they get out, is derived from their strength with no further component from their size.
Drag for Lightweight Arrows
If you reduce the weight of an arrow by reducing the diameter, then the drag force reduces proportionally with the mass, and the drag length of the bow remains constant. Hence all arrows fired from the bow retain the same drag length. Below a certain point you cannot reduce the diameter of arrows anymore, so beyond this point you start losing velocity, which means 1 range per 1 strength (but the same drag length).
If you reduce the overall size of the arrow (as per a shorter bow length and draw length) then that increases drag. We'll reduce the size of the arrow as per my standard sizing methods, so a half length arrow is one sixth the weight and root onethird diameter. The increased drag has two effects  firstly it reduces the range slightly, but we'll ignore this as you need 7 weight categories before you get 1 range category. Secondly it reduces the drag length, which is important. According to calculations through my simulation program, each 1 weight category on the arrow loosed reduces the drag length exponentially, so that 5 weight categories has halved the drag length.
Statistics for Longbows
I'm choosing The Great Warbow as a starting point for sizing. It's 32 inch draw length, so that should be a 6'5'' bow, which it appears to be from the photos. We'll assume that it could only be fired by a strong archer (+3) with some strength training (+3 athletics) so power 6. The arrows were 53.6  95.9g so 1.9  3.4oz., but the ideal weight for speed is clearly the 1.9oz arrow. The Greenman longbow tests used similar length arrows with 0.75  1oz arrows. I doubt you can get much smaller then this, I'll assume that 0.5oz arrows are the lightest you can have as that'd be 5mm diameter and the narrowest I can find is one quarter inch, which is size 4. Bows of this length appear to weigh about 2lb. According to the introduction to The Great Warbow, the draw weight of a bow is proportional to the fourth power of the diameter, hence draw weight has very little effect on the overall weight of the bow.
So our base statistics will be: 7' bow weighing 2lb for a 6' archer with 2'8'' arrows, weighing 0.5oz, drag length = 64*bow length =448ft, power 2. Below power 2 the arrows are over weight and get 1 range per 1 power. Above power 2 the arrows get +1 weight per +1 power.
Now let's have a size 5 bow, one half length: 3'6'' bow, weighing 6oz, for a 3' archer with 1'8'' arrows, weighing 1/12th oz, drag length = 64*3.5 = 224ft, power of 2 + size = 3.
Note that you can calculate the draw weight of a bow, and it's the amount your character can lift * 2 / bow length. So a 6' man with power 5 can lift 500lb, so has a bow with draw weight 500*2/6 = 167lb.
Shortbows
Halflings are SIZE 3, but are short and fat so HEIGHT 5, hence they get a 3 on kill and increased drag penalty at range, but on the other hand they only get 3 versus missile attacks due to their size, hence it is not surprising they prefer bows to handtohand combat.
In particular, they shoot 3'6'' Longbows, and I would advise you not to refer to them as Shortbows unless you want an arrow in the back.
Next time I'm going to cover Javelins and Sling shots (both about ten times simpler than Longbows) then move on to Crossbows, Composite Bows and the effect of elevation.
This is interesting stuff (to me, at least). I did some math calculations about a year ago on this topic, and I think I came to similar conclusions but from a different methodology, which is encouraging. I'm on vacation and don't have my notes handy; maybe next week I'll review. I've been meaning to comment on your prior Archery post also but never had both the inclination and the time at the same moment. The gist was that I think that firing at long range at large, somewhat dispersed targets (such as firing at a large enemy formation) is maybe somewhat less likely to succeed than you calculate because of the effects of plunging fire and the different grazing angle. At close range with a flat trajectory, shooting at a mass target is like shooting at a really wide person  the target area is like a rectangle about 6' high and however many files wide. But at long range, the angle of the incoming projectile means that the target crosssection of each individual is reduced; I think that would be sufficient to reduce the effective target area enough to warrant an additional penalty.
ReplyDeleteI'd be interested to hear what your methodology and conclusions were. For firing at long range dispersed targets my thoughts on this need some diagrams, and will probably have to require a simulation, so I'll put them into a separate post. In brief when at short range you can only hit the front rank, whereas at long range the arrows coming down at not as steep an angle as you might think so people are say half as big a target, but you can also now hit the back ranks. So what's of interest is the vertical arc for launching arrows between the foot of the front rank compared to the heads of the back rank, and what proportion in between is target.
ReplyDeleteI've found some of my notes, so let's see if I can reconstruct the thinking. I started by looking for the "muzzle velocity" of an arrow fired from a bow with a given draw weight.
ReplyDeleteI assumed an arrow of mass 50g, with an 0.7m (28") draw distance. Somewhere or other [citation needed] I read that the velocity of an arrow in m/s is approximately 8 * sqrt(bow draw weight in pounds).
The work done on the arrow by the bow (the imparted KE) is 1/2 * Kx^2 where x is the draw distance in meters and K is the force applied by the bow at full draw (treating the bow as a spring, K is the spring constant). If we set this KE equal to the KE of the arrow in flight (= 1/2 * mv^2) we solve for v and get:
v^2 = Kx^2/m
But K = D / x (from the spring equation, F = Kx, F = D = draw force at full draw) so
v^2 = Dx/m, v = sqrt (Dx/m)
where D = draw force in Newtons, x = draw distance in meters, m = projectile mass in kg.
For our arrow, v = sqrt (222.5 * 0.7 / 0.05) = 56 m/s. This appears generally consistent with your data above. [And it matches the V = 8 x sqrt (draw weight in pounds) equation I referenced earlier, for whatever that's worth.]
Then I wondered about a large creature, a giant or ogre or something, twice as tall as a man. The draw distance is doubled. I assumed the projectile mass increases by 4x (we double the length, but only scale up the thickness some). I guesstimate that the draw force/weight of the bow is doubled as well. The arrow masses 0.2 kg with a 1.4m draw distance and a draw force of 445N; the initial velocity works out to be ... 56 m/s again.
A small creature, half the size of a man, halves the draw force and length, and quarters the arrow mass, and once again we get the same initial velocity. I found this interesting initially, but it's not actually that surprising since it stems from my assumptions about how to scale the draw weight and arrow mass. Of course, there's a big difference in projectile momentum, and that would significantly impact (heh) effectiveness as a weapon. This also implies that there are more effective arrow weights for larger or smaller bows (which would have different initial velocities).
Hmm, now I'm going to have to think about this some more...
Thanks Leland  sorry for the late reply  I was on holiday with no internet access.
ReplyDeleteYour calculations work, assuming 100% efficiency (all potential energy is converted to kinetic energy). However they imply that the lighter you make the arrow the faster it will shoot, so (as you said) you have to make assumptions about the weight of the arrow for giants.
When scaling up you assume 4* mass and 4* draw force; I went with 6* mass but also invoked the lever principle to mean the draw force only increased to be 3*. Plugged into your equations this gives the same result  a constant velocity.
We'll always have to make assumptions about how much weight a giant can lift (unless you happen to know one) but if have a consistent set of principles then we're less likely to end up with nonsensical results.
As an aside note that the difference between the effectiveness of the different size bows is due to the difference in kinetic energy, not the difference in momentum. During the impact, momentum is conserved (pushing the target backwards) but most of the energy is lost, which is what causes the damage. If you ever find someone's throwing rocks at you, remember to ask them to double the mass of the rock, but halve the velocity.