Low Ceilings Revised

I've had some helpful feedback on my Effect of Low Ceilings on Range post, and this has lead me to revisit the subject and revise the rule.

Q: Gaston’s Hat contrasted my max range for an of Axe 80ft with competitions over very short distances e.g. IKTHOF  (3/4/6/7/9m) and AKTA (13/15/21/30ft).
A: Although you can throw an axe 80ft, this is close to the record (and in Explore you could only achieve this with a high Strength and Athletics skill). Also it is quite hard to be accurate at 80ft with an Axe (in Explore you would get -9 on to hit for range, and axes are harder to aim in the first place). Both of these facts would imply that you'd have competitions at much lower distances than max range, just as you do in archery, so this range for competitions is as expected.

Q: Gaston’s Hat observed that in this video the axes are clearly being thrown quite high in the air to hit a target only 60ft away.
A: To achieve max range, which it looks like those throwers are at, you have to throw at 45 degrees. In this case the projectile reaches a height of one quarter the range, which would be 15ft in that video (plus 6ft for the thrower’s height). Reduce the distance only slightly, and if you throw with the same velocity, you can reduce the angle massively, and hence the maximum height. At a range of 50ft the max height is only 7ft (+6ft).

A picture is worth a thousand words, so here's a diagram of the trajectory of a projectile with max range 80' launched at targets at ranges 10' to 80'. As you can see, the max height drops rapidly as the distance to the target reduces.

Q: Thiles Targon commented that throwing a ball only 20-25ft he was often hitting a 9ft ceiling.
A: Evidently if you shoot someone with a gun at this range, you can ignore the 9ft ceiling, similarly with a bow. The effect of a low ceiling depends upon the velocity of the projectile, not the distance to the target. Hence, if you throw the ball slowly, or mis-throw it, or throw it over someone, then you’ll likely hit the ceiling. If you only throw the ball fast enough to go 20ft, in a 45 degree arc, then the ball would go 5ft+6ft = 11ft high and bounce off the ceiling. On the other hand, if you’re world class you’d have a range of 480' and hence the ball would go in almost a straight line, and only rise by 1.3 inches.

Here's a diagram of the trajectory of a projectile with max range 20' at a target 20' away, and then thrown at the same target with sufficient velocity for max range 40' and 80':

Revised Rule
Note that in these answers I have invoked the height of the thrower - in my original results I stated that I was ignoring this. I thought the effect wasn't large enough to worry about, but on second look it is, as these discussions show.
In addition, I was always rounding to the nearest category - but this is often rounding up quite a lot. I'd say no reduction when it was actually 60' reduced to 51'.
Time for a re-examination of the figures....

Firstly by introducing half-way values I stop the rounding up issue (so 51' is now 50').
Secondly I noted that the effect of introducing the height of the thrower is almost the same as reducing the height of the ceiling by the height of the thrower.
The solution has to be a compromise; I've tried several approaches but in the end settled on a single table for all races, but you use different columns depending upon your height:

For example, with a 10' ceiling and 160' max range, a 6' human uses the middle (white) line for ceiling height, so has the max range reduced to 70', whereas a 3'6'' halfling or kobold uses the bottom row, hence uses the column one to the left and has it only reduced to 100', giving them a big advantage in missile combats in corridors.

The heights are 2'+/4'+/8'+ rather than 2.5'/5'/10' as this is what gives the closest fit.

Performing the Calculations
My calculations were initially done with a computer program – but as I'm ignoring wind resistance I should have just solved the equations! When ignoring the height of the thrower the equations aren't too complicated, and the result is:

Let R be the max range, H the height of the roof, r the reduced range (due to the low roof) we can derive the following results:

So it is a quarter-ellipse!

For example, with max range R=120, height in the X axis, r in the Y axis:

From this you can calculate my results above, but not the effects of the height of the thrower. Unfortunately when you try and calculate the range of a projectile thrown at 6' high, with a 10' roof, aiming at a target 3' off the floor, the maths becomes rather complex and I reverted to my simulation program.

The Effect of Elevation on Range

In my recent series of posts on missile weapons, I’ve been promising an analysis of the effect of the archer being at a different elevation to the target. This was a bold statement as my initial investigations had yielded nothing and I was hoping for inspiration to strike. Fortunately it has, and I'm very surprised at the result.

Note that as the Referee I'd do these range calculations, and just tell the players if they’re in range or not and what the relevant bonuses are!

Imagine you have max range R and are shooting at a target horiz distance X away, height H below you.

For example:  throwing an Axe with max range 80' from a 20' high wall at something 75' away.

Here's a diagram to accompany the example:

Firstly, guesstimate the direct distance D by adding half the height H to the horiz distance X. If H is bigger than X then do the reverse - add half the horiz distance X to the height H. From this direct distance you get your to-hit penalty for range.

In our example it is 75'+10'=85' direct distance, so -9 to hit.

Secondly we check that it is within range: direct distance must be less than Range + Height.

In our example range + height is 80'+20'=100' which is bigger than 85' so it is within range.

Lastly you calculate the damage bonus/penalty: Take Range + 2*Height and see how many categories that shifts the range, you get +1/-1 for each shift.

In our example Range + 2*height  = 100'+20'=120' and this is +1 range category so +1 damage.

Apart from the approximation for the direct distance, this is a 100% correct model (ignoring wind resistance) - no simplification needed. These are the actual equations when targeting a missile at something with a different elevation!

As a second example, consider throwing an Axe with max range 80' up at the top of a 20' high wall, 75' away. The direct distance is 75'+10'=85' as before. However 80'-20'=60' so it's out of range.

I've added one extra "limiting rule" (to mimic terminal velocity) –

You can add at most 960' to the height when calculating the damage.

This is unlikely to ever be needed!

As a third example, consider throwing a javelin with 240' max range off a 2000' cliff at a target 400’ away. It is clearly within range. Damage is capped at 960'+240' = 1200' which is +4. The direct distance is 2000'+200' = 2200', which means you'd be severely unlikely to hit anything, as the to hit penalty is now -36!

Example #4: Throwing the Axe (max range 80') from the top of a 100' high cliff at a target 100' away. Direct distance = 150' which is -12 to hit. This must be less than 80'+100'=180', so it's in range. Damage is @80'+100'+100'=280' which is +3.

Finally we have two observations – with these rules both of the following happen to be true:

The max rise is half the range.
The horiz distance is capped at twice the range.

This second one is a happy accident, which I discuss at the bottom of the post.

Example #5: Throwing the Axe (max range 80') from the top of a 200' high cliff at something 160' away. Direct distance is 200'+80' = 280' which is -18 to hit. This actually the max, which is 80'+200'=280'. Damage is @80'+200'+200'=480' which is +4.

We'll apply these rules to a common in-game situation:

Attacks by Flying creatures
For simplicity we’ll assume diving attacks are always at 45 degrees. This means the diving speed of a bird should (for simplicity) be given in stat blocks as the speed direct / speed horizontal & vertical (which is two-thirds).

For example, a Roc dives at speed 440'/rnd, which is 2/3*440' = 293'/rnd both vertically and horizontally. Hence we'd specify Roc Dive:440'(290').

When attacked you’d shoot at point blank range, and then one round before that, and one round after that, so distances are always a simple multiple of the movement speed.

Continuing our example, we have a bow (range 960') to protect us from the Roc. Is it in range at 440'? The max is 960'-290'=670', so it is in range (just). To hit penalty is -24. Damage is @670'-290'=380', so -2. In actual play the Roc swooped in, all shots at range missed, the next round (at point blank range) it successfully grabbed Elanor but got badly wounded, the subsequent round it got shot out of the sky due to the bonus to hit from it being wounded – Elanor was extremely grateful for having earlier polymorphed to grow a pair of wings!

Explanation
So how did I arrive at these rules? This involves a bit of maths – sorry – but it’s purely by way of explanation; it’s completely irrelevant when playing the game and certainly will not appear in the rulebook!

My initial thoughts were that if:
1) shooting a distance R requires an initial KE of mg x 0.5R
And
2) shooting up height H requires kinetic energy of mg x H
Then shooting at range R and height H would impact with the KE of (1) minus (2), i.e. mg x (0.5R-H), i.e. the same KE as shooting a distance of R-2H. Hence we arrive at the damage reduction rule.

I had hoped that the max horizontal range would be X-2H, but results from my simulation program showed the max range was a straight line close to X-H, but above a 30 degree angle or with big drops the line curved away and it was difficult to have a simplification without wacky edge cases. By complete accident I added in an extra row to my results showing the direct distance, and found it was an exact straight line which seemed rather too good to be coincidence. Going back to this standard formula, I calculated the max direct distance for a given height and range...

and was astonished that everything cancels! Gosh, I haven’t written out an equation using Latex since I finished my thesis in 1997…

An Accidental Range Limit

One consequence of using an approximation for the direct distance isn't clear at first: the horiz range maxes out at double. With a correct calculation for direct distance you would have the rate of increase slowing down but never stopping, there would be no such maximum, but in reality you’d also have wind resistance, so we want a cut-off, so it is a happy accident.

For example with 100' range, 200' drop, we have max horiz range as 200', but it should be 224’, which is still very close. But if we increase the drop to 400’ our equation still gives 200' when it should have increased to 300'.

The Effect of Low Ceilings on Missile Fire

I've been busy creating a new adventure locale for my group (and the adventure went in an unexpected direction, which is good) so for the blog I've chosen an easier topic to resolve: Does the height of the ceiling significantly affect the range of missile weapons indoors?

Now in D&D Gary Gygax had this odd “distances are in yards outdoors, feet indoors” rule (AD&D DMG p39), which reduces all ranges and areas of effect by a factor of three in dungeons. To me this seemed firmly based upon reconciling the awesome power of Fireball in Chainmail with the desire to reign in the power of Magic Users for dungeon crawls, but it also applies to missile fire with the justification of low ceilings. Is this drastic universal range reduction justified? In particular it has a big effect on short range missile weapons like axes. The reduction on range for bows seems plausible (apart from his odd statement that Crossbow bolts don't follow a parabolic flight), but the 3" for throwing axes and spears from Chainmail (p10) means 30 yards outside, but only 30 feet indoors. The ranges are carried forward with minor changes through to AD&D where it's become 30ft for Axes, 60ft for Javelins (PHb p38).

I’ve run some simulations for missiles of different maximum ranges over various maximum ceiling heights, and calculated what the maximum range is reduced to. For my purposes I’ve made several simplifications: I’ve lumped all missiles with the same range together (ignoring different trajectories for different types of projectile due to wind resistance – it has little effect). I’ve ignored the height the missile was thrown at (which should be about 5 foot – it has little effect, and I ignored it when calculating ranges in the first place). I’ve omitted ceiling heights under 10’ high as that’s unlikely to be relevant, though note that it follows the same pattern.

The results are as follows: (all heights and ranges as per my standard "double every other category")

I’ve shaded the ranges which are reduced. I then simplify this table (below) by altering the reduced range to be the nearest category, allowing myself latitude in a couple of cases (marked red) where I round up for some results just less than halfway between to make the results fit a regular pattern.

Now the pattern in that table is easy to see but difficult to extrapolate from, but it's probably an acceptable simplification to drop every other column (who ever heard of a 15' ceiling anyway) and the resultant cut down table is much simpler, especially if we reverse the order of the columns:

So we note that unless you’re aiming at something more than 80’ away, the low ceiling has zero effect on range. Furthermore, if you’re shooting at more than 80’ you get a -12 range penalty on to hit. So range restrictions would only ever come into play if the target you’re aiming at is so far away that you’re unlikely to hit it in the first place!

Summary

So in summary, the range reduction can be quite easily reduced to a simple pattern which can be summarised in one short table. However, I think that it is extremely unlikely that this range reduction will ever actually have an impact on play (since it only ever takes effect over longer distances than are usual). Note that this result is absolutely nothing like “reduce the range threefold” except for long ranges with a 10' ceiling, and in fact ignoring ceiling height altogether would be a more accurate simplification for most situations!

Comparison with Delta's Results 

Delta did some similar analysis using D&D’s fixed ranges,  but in Explore the range varies according to the archer so these results are not much use to us except for comparison purposes to check the results.

Delta shows thrown weapons (range 90') having a reduced range only with a 10’ ceiling – when it is reduced to 63'. The nearest equivalent 80’ in Explore would only be reduced to 69’, which in this case I rounded up not down, hence in our case there’s no range reduction at all.

Similarly the three bows have fixed ranges 150/180/210/240 yards or 450/540/630/720 ft. In Explore we don’t have this level of granularity in ranges, but we get similar results: approx. one third range with a 10' ceiling, one half with a 20' ceiling, two-thirds with a 40' ceiling.

Mass Targets

I've just come back from a week's holiday in the Lake District with no Internet Connection, so to satisfy my cravings I'm going to address Leland's questions in the comments on archery versus mass targets. Here's the first:

I concur with Delta's analysis about treating range as reducing target size, but I think this falls apart somewhat when firing at long-range dispersed targets. I think of the dispersed target as a big area that the archer is just trying to drop an arrow into somewhere. The target is quite large, so it's not hard to hit the area. But what's important here is, how much of the target area actually has a target in it? And also, what's the effect of angle of incidence -- what you really want is the projection of the actual target area into a plane orthogonal to the arrow's direction of flight (also allowing for potential overlap/shadowing of individual targets). If (to choose a simple example) the target area is 50% real target and 50% empty space, then it's simple in your system: just apply an additional -3. So I think you're baking in some assumptions about density of the massed troops. Personally I think 50% density is a bit high, but that's probably because we have different mental images of the targets.

Firstly I'll recap with the help of some illustrations. If you shoot at a target which is twice as far away, then that is the same apparent size as a target which is half the size (which is a quarter of the area):

The further warrior is twice as far away as the near warrior and the warriors are about the same area as the targets. The further target (twice as distant) is the same apparent size as the nearer central green section, which is only 4 studs instead of 8. Hence hitting more distant targets is simply like shooting a smaller target.

The distribution of shots is a bivariate normal distribution (i.e. both x and y are a bell curve), so 1000 arrows shot into a target would look like this:

As you can see, once you get into the four central squares, the distribution is even. If you hit one of these four central squares, whereabouts you hit is completely random. If we highlight the best 50% of shots you can see that the distribution of these is completely random:

For shooting at massed target shots there is the chance of hitting the area with the target in, and then the chance of hitting something within that area. If you have less than a 50% chance of hitting the target area, then all you need to know for hitting an actual target is what proportion of the area is filled with target and add in the appropriate penalty. My eldest son described it as being like "shooting at a target with holes in it". If half the area is empty, then that's a -3 penalty.

I don't have any armies to hand, so I was going to take photos of miniatures (Lego figures are not proportionate) but it's tricky to see as my figures are all on painted bases, so I've taken this approximation of people as x 9.5'' x 1'6'' x 5'9'' and made stacks of Lego bricks with the same proportions. I've given a gap of 2 studs - 1'6'' - which seems pretty close when you're holding a sword. According to Wikipedia this is the spacing of a tight phalanx.

Viewed head on, only the front rank can be seen:

However, from a distance, arrows will come in at an angle, so present a target something like this:

For a maximum range shot, you're shooting at approx 42 degrees, and the arrows are coming down at approx 48 degrees, and from this angle the troops look like this:

At this angle a 5'9'' man only presents the same size target as a 4'5'' man, but as the angle has increased the amount of overlap has reduced to almost zero. The area to hit has gone up from 95 sq ft to 131 sq ft. Half range would be shooting at 12.5 degrees, coming down at 14 degrees, and at this angle the area would be 114 sq ft. Hence shooting at range has actually increased the target area.

In contrast let's consider the ultra tight synaspismos formation the phalanx adopted when under "extra pressure, intense missile volleys or frontal cavalry charges". This is no spacing between figures, and only one stud between rows. At 48 degrees there is now only 112 sq ft presented to the arrows of the enemy, this might result in a small -1 to hit penalty, but overlapping shields would result in a more significant advantage.

Now a phalanx is tighter than you'd expect - if the spacing between troops was doubled as per a loose phalanx, then the troops would occupy four times the area, and when the arrow comes down onto the phalanx three quarters of the area would be empty. This would apparently give a -6 penalty to the archer, but on the other hand the troops would cover a much larger area.

For comparison, Here's loose, tight, and ultra-tight formations together in a single photo:

My assumption is that the area the archer is shooting into is either in loose order, so the whole of the target area is occupied by troops, but only one quarter of the area has a person in it (from the arrow's perspective), or it is in tight order and the troops cover only a the central quarter of the target area.

That is we either have loose order:

Or we have tight order:

And either way, assuming the archer has less than 50% chance to hit, the troops cover a quarter of his target circle, and hence -6 is a good penalty.

Mass Targets or Individual Targets?
Leland also asked:

I guess one question is, where do you transition from shooting at an individual target, vs. at an area that is n% full of targets?

Due to the logic behind how I derived mass target shooting, I'm limiting archers to a minimum of 11  needed to hit mass targets (i.e. minimum of 50% chance to hit). Hence at close range it can become better to switch to individual shooting.

The chance to hit is the best of individual shooting (with range penalty) and mass shooting (-6 fixed penalty, min 11).

For example, at 120' range a +11 archer only needs 7 to mass hit troops with a parry of 12. This is below the lower limit for mass target shooting (11), so they need an 11. For individual shooting it would be a -12 distance penalty (-3 per range category), so a 13 to hit. Hence they stick with the 11 target.

If the troops were 80' away the archer only needs a 10, so they can now switch to individual shooting. 

Hitting Important Targets
A third question was:

Another thought that occurs -- you could have archers shoot at one individual target in a mass, and if they miss by, um, not very much (depending on the density and other modifiers) you could rule that they hit SOMEBODY, just not the guy they were aiming for. 

Now imagine that one of the targets is a special target, an important NPC perhaps. Determine the number of troops around him and roll to see whether he is targeted. E.g. if he is one of 20, a 1 on a d20 would pick him. I'd say one in a hundred maximum. If he is selected then we can see that the rest of the troops are now irrelevant and you simply make your attack roll against him. If it misses him it misses everyone. It he is not selected then you make the attack roll against the normal troops.

This same method can be used for mounted troops - fifty fifty chance of targeting the mount (depending upon the size of the mount).

Field Experiments
Now back to considering Crossbows. My youngest is keen to to do some field experiments with his toy Crossbow...

We're only shooting an apple juice carton, and the biggest danger is loosing the bolt in the hedge!

On Throwing

Today I'm turning from Archery to a far easier topic - Thrown weapons – and I’ll start with the easiest - throwing rocks.

The aim, as before, is to derive some simple rules for range and damage for different missile weapons. The core motivation comes from me not being able to decide the rules for Halflings and Giants, and realising that I was making entirely arbitrary decisions with no firm basis.

Throwing Rocks
I found a very useful study into throwing objects of differing masses. The conclusion is that for weights much heavier than your (hand + forearm) kinetic energy is constant, for weights much lighter than your (hand + forearm), velocity is constant.

The rationale given is that there is a maximum velocity that you can achieve with a throwing motion, and at the point of throwing only your forearm and hand are in motion, so if the object is much lighter than this it doesn't slow down this motion, whereas if it is much heavier than this it becomes more a question of how much ene
rgy you can put into the throw.

For our purposes we want to simplify this to have an "ideal throwing weight" for your size. Above that weight kinetic energy is constant, so you lose velocity and range (+1 weight = -1 range) and below that weight you have constant velocity but lower weight reduces kinetic energy and damage (-1 weight = -1 damage). Hence you'd always be best off throwing a stone of that ideal weight, and we’ll assume that’s what you do.

We'll further assume that the speed you can throw is independent of size, it's the size of what you can throw that varies.

The world record (unofficial) for throwing a golf ball is 170ys = 510'. A golf ball is fairly light (1.6oz) so we can derive from this the maximum range of a thrown stone. We'll take the max range as the category lower (480’) as the golf ball is dimpled and thus has lift.

This is a world record, but not in a proper sport, so we'll take it that the thrower had a Power of 6 (max STR of +3, level 3 Athletics) and that this gives a range of 480'.  Hence when Power=Size max range is six categories less, which is 60'.

Next we take the world record for the shotput: 23.12m = 76 ft. It weighs 16.01 lb. Now that's a Power 7, Size 0 athlete getting +1 range, so they must be throwing +6 weight compared to the ideal. This puts the ideal weight for size 0 to be 2lb.

With arrows the drag length (the distance over which you get -1 on damage) was very important. The lightest rock being considered is Halflings, which are Size of -3, hence their ideal weight is 3/4lb. The drag length for a 3/4lb rock turns out to be 409m. Hence the drag length for rocks is not significant except in outlying cases, so we'll assume no effect of drag.

This gives us the standard rock (for Size 0, Power 0) is 2lb, range 60'. Each +1/-1 size makes the rock +1/-1 bigger. Each +1/-1 power on top of that makes the range +1/-1 range category. (We do not bother with the size of the stone).

Throwing Spears/Javelins

Javelins are thrown at slightly less than 45 degrees (between 35 and 40) because biomechanics allow you to throw the Javelin faster at that angle, which more than makes up for the range lost (this lower angle only reduces the range by about 5%).
In addition, drag reduces the range of a javelin by about 5%, but this is counteracted by lift. Javelins are thrown pointing slightly above the direction they are thrown in, this causes lift, like an aircraft wing. The lift is sufficient that it not only makes up for the drag, the javelin actually goes further than it would if thrown in a vacuum! The lift also meant that often javelins used to hit the ground almost horizontal, so they altered the centre of gravity to make them stick in the ground and be both safer and easier to measure.

Modern javelins are between 2.6 and 2.7 m (8 ft 6 in and 8 ft 10 in) in length and are 800 g (28 oz) in weight. The world record for a Javelin throw (with the old style Javelin) is 104.8m = 344ft

If a Javelin was made of wood, then approx. 800kg/cubic metre would give a diameter of 2.22cm, giving a drag length of 382m = 1253ft. This is far in excess of the distance you’re throwing a Javelin, so drag can again be ignored.

I'll assume like stones there's a most efficient weight for a Javelin, and I'll take that to be the weight of the modern Javelin, 2lb for Size 0, which is one size less than the stone (it is reasonable to assume there's a different ideal weight for different throwing methods).

World record athlete is power 7, range 320ft, so when Power=Size range is 30', also one less than with a stone.

The ideal length of a Javelin I’ll take to be two sizes longer than you're tall – so races that are short for their weight would have thicker javelins.

This gives us a standard spear to be 8' long, 2lb, range 30’.

Throwing Knives

Knives are difficult to throw – they spin end over end in the air, and the distance must be judged just right else the knife will not hit end first. It is often claimed that thrown knives are not only ineffective due to their weight, but also due to wind resistance. You throw the knife in an overarm action, releasing the knife quite early, when it is perpendicular to the ground.

For throwing knives, the biggest I can see is 12'', 18oz. So I'll assume 1lb is ideal. Data is sparse but here states that initial speed has been measured up to 61kmph = 16.94m/s. With no drag that would be range 29.28m = 96ft. Also on that page is a link to the results from the World Knife Axe Throwing Championship which has a long distance competition. In that competition the max distance is 19.7m = 64ft.

This reduced distance (only 64ft instead of 96ft) could be due to drag, or (more likely) due to the mechanics of knife throwing not lending itself to a high angle throw. Because of how you throw knives, the higher the angle you throw at, the earlier you have to release the knife, so the slower the speed it has attained. Hence, like Javelins, there's an ideal angle to throw at, and it's likely not to be that steep an angle. A throw at 21 degrees would give you a range of 64ft, and it seems reasonable to assume that the lack of distance is due to this being the maximum effective throwing angle.

This maximum angle doesn't actually affect our calculations, only later when calculating the damage for the knife, we have to note it was thrown at a velocity for a range 2 categories higher, so this will give us an extra +2 damage.

We’ll make the same adjustments as before, but assume these records were from power 6 individuals (it's not a high profile Olympic sport). Reducing range 60' by 6 categories is 8'.

This gives us a standard knife to be 1' long, 1lb, range 8'.

Throwing Hand Axes

I'm basing Hand Axes on the Francisca, used by the Anglo Saxons in England in the Middle Ages, and similar to a Viking Axe. It averages 1.5lb weight, 1'6''. This is a very similar axe to the Viking Axe and the Tomahawk. Again we'll assume this is the ideal weight for throwing, which is one category less again than the Javelin, one more than the knife.

The physics of throwing are similar to the knife, hence we'll assume that it's the same rules as a knife - no drag and +2 kill for reduced throwing angle. The long distance competition on the knife throwing website has 27.35m = 90ft, which is +1 range compared to a knife.

This gives us a standard hand axe to be 1'6'' long, 1.5lb, range 10'.

Summary
So now our rule for thrown is: The length, weight and range are taken from the table below. Your size alters the length and weight. Your power minus your size alters your range. The kill is always a fixed amount added to your Power.

For comparison I've included details for the standard Longbow and arrow (Height 0, Power 0), but how that varies with the archer is different so should be on a Bows table

Length	Weight	Range	Kill
8''	0.5oz (Arrow)	3’	0 Knife, Rock
9''	0.75oz	4’	1
10''	1oz	5'	2 Axe, (Bow)
1' Knife	1.5oz	8' Knife	3
1'2''	2oz	10' Axe	4 Spear
1'4''	3oz	16'	5 Lead
1'6'' Axe	4oz	20'
1'8''	6oz	30' Spear
2'	8oz	40'
2'4''	11oz	60' Rock
2'8'' (Arrow)	1lb Knife	80'
3'	1.5lb Axe	120'
3'6''	2lb Rock, Spear (Bow)	160'
4'	3lb	225'
4'6''	4lb	320'
5'	6lb	450'
6'	8lb	640'
7' (Bow)	12lb	900' (Bow)
8' Spear	16lb
9'
10'
12’
14’
16’

Coming up: Slings, Composite Bows, Crossbows, Effect of height, Trebuchets.

On Archery II: Longbows

In my last post I talked about to-hit penalties due to range, but left the subject of damage penalties due to range to this second post as investigations were proving a little more complicated. Well, that was two weeks ago and I thought I'd nearly got it cracked, but it seemed the more I thought about it the more questions I had. The sites on competitive archery are helpful but mostly concerned with accurate target shooting, then there are sites on historical archery which contain lots of fascinating information such as making replicas of the bows and arrows found on The Mary Rose, and then there are the hunting websites (all American) - all I'll say is I'm a Vegetarian and I'm on the side of Cecil the Lion!

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 too-heavy 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 well-made 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 one-third 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 hand-to-hand 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.