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.