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':
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:
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.