**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:

*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.

My point was more about thrown items; I was also trying to point out that you can't assume these people will somehow throw or shoot within an inch of the ceiling perfectly. The tendency to hit the ceiling or overcompensate in trying to not hit the ceiling will have an effect, even if the shot is within range. When you also consider the objects are not points in space but actual physical objects, the length of the axe for the period of the time it is travelling perpendicular to the ground will reduce the height you can throw. When you combine the overhand motion and the length of the axe and some cushion space, there is not much height left to a 10’ ceiling.

ReplyDeleteYou wouldn't be trying not to hit the ceiling, you'd be throwing it as hard as you could at the target - you'd purely be trying not to aim it too high a part of their body. Mostly hitting the ceiling would be aiming too high and would simply be missing what you aimed at.

ReplyDeleteYou *could* increase your height by one foot when you're throwing an axe to compensate for its length, but not unless you're also making a distinction between people who are 5'5'' and 6'5''. It doesn't affect other thrown items like your example of a ball. The overhand motion is why I've assumed you're throwing from your height (it's slightly lower for bows).

During my run this morning I was wondering about an axe-specific effect for the length of the axe reducing head height and hence max range. If you threw entirely flat, a throwing angle of zero, then max range would decrease a lot – thrown from it would go from 80’ to 27’. But even a slight angle of 10 degrees, rising only 1 foot, would go 41ft. This would be enough to drop the range one category. However I then recalled that the axe (like the knife) at max range is thrown at a far flatter angle than 45 degrees, and hence its velocity for a particular max range is much higher than other projectiles, which would lead to the opposite effect – a lessening of the effect of low ceilings. When I ran the calculations this raises the range back to 50ft, cancelling out the first effect!

ReplyDeleteJust been playing the Saxophone and thinking some more about this. I'm modelling the range as being able to hit the middle of your opponent - that means I'm saying at the extremes of the range you can't hit the top half of your opponent (you'd hit the ceiling). So perhaps at the furthest extent of range (in this case 40'-50') you should have an extra -3 penalty to hit. If this causes you to miss you'd say it hit the ceiling, so it adds flavour.

ReplyDeleteI like that you included formulas. Did you do that for the determination of difficulty for ranges?

ReplyDelete"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)."

Hi Joe, isn't 7ft+6ft = 13ft which would mean a six foot tall person couldn't throw an axe 50' with a 10' ceiling?

The maximal height for the axe is going to be equal to the height at release plus the rise of a ballistic (not straight line) throw.

The height at release is a simplification, really we need the height of the center of momentum at release. For a bow that is where the arrow is on the draw which is about shoulder height. For an axe I think it would be a bit higher than a bow, maybe a bit below head height. And for a javelin higher still, above head height, I imagine. It should be possible to check those initial heights by measuring an actual throw. So rather than 4' as we need to go higher. In addition the axe rotates around it's center of mass which is, I imagine, closer to the head. So you need to add in something over half the length to the axe to the distance as well. So I think for axes and javelins we are looking at adding 6' or more to the h of the throw itself.

Also note that a big person/creature still needs not to hit the ceiling with the top of the axe or javelin while swinging above their head. That means an 8' tall critter will hit the ceiling with their axe or javelin, maybe even with their hand without a weapon in it.

I like the fact that shorter creatures like dwarves will have a range advantage underground over taller creatures like humans.

"I like that you included formulas. Did you do that for the determination of difficulty for ranges?"

ReplyDeleteThanks. Yes I used formula for the difficulty for ranges, but not in the blog post.

"Isn't 7ft+6ft = 13ft which would mean a six foot tall person couldn't throw an axe 50' with a 10' ceiling?"

In this case yes: the thrower's max range is 60' and the ceiling is 10', which from the table reduces the Max Range to 40'. But it depends upon the max range for that thrower - another thrower with a max range of 80' would not hit the ceiling.

"The height at release is a simplification"

Yes, I could model it more closely as you outline, but I think the loss of head room with an axe is roughly cancelled out by the fact that the max range was from a flatter angle than 45 degrees. That is, the axe with a max range of 80' has a higher initial velocity than a bow with a max range of 80'.

"It should be possible to check those initial heights by measuring an actual throw."

You could do some experiments, but they'd be tricky to do as you'd need to measure initial velocity and angle and max height and range, and you'd need someone capable of throwing the axe very fast!

"So rather than 4' as we need to go higher. "

Note it's not 4' release height in the table, it's 4'+ and all heights up to 8'. The values are actually for 6'.

If you want more accurate results, then the area it falls down most is not having a column for a 15' ceiling. I tried that but the table becomes massive and unwieldy.