# Item statistics

## Projectile velocity and distance

Note: for now only throwing weapons considered, no launchers nor firearms. TODO: test in the field.

The game context for the numbers is tile size of 1 m and standard game actors moving every 0.5 s. So the actors walk at 7.2 km/h, which is a very fast walk or a slow run.

### Sources

Lots of averaging and rounding below. The speeds listed there are initial projectile speeds. We give them in km/h for comparison to car speeds.

http://en.wikipedia.org/wiki/Projectile#Typical_projectile_speeds

• thrown club (expert thrower, weight conjectured, range computed) 1 kg 150 km/h 170 m
• spear-thrower dart --- the same stats conjectured

Sports data, sorted by weight; range or velocity computed, if necessary:

• sling (tennis or golf ball) 100 g 200 km/h 300 m
• baseball fastball pitch (range computed) 150 g 150km/h 170 m
• Olympic javelin (a long run needed) 1 kg 110 km/h 100 m
• Olympic discus (uses air resistance to fly further) 2 kg 80 km/h 70 m
• Olympic shot put (velocity computed) 7 kg 50 km/h 20 m
• Olympic hammer (with a long wire, and multiple spins) 7 kg 120 km/h 80 m
• weight throw (a short handle, velocity computed) 15 kg 60 km/h 25 m

http://w4.ub.uni-konstanz.de/cpa/article/view/181 (fit individuals, but not sportsmen, velocity computed from v = sqrt (g * d), g = 127000 km/h*h):

• tennis ball 60 g 70 km/h 40 m
• tennis ball 180 g 80 km/h 48 m
• tennis ball 360 g 70 km/h 40 m

http://www.jssm.org/vol9/n3/24/v9n3-24text.php (two-handed overhead throw (a major factor!), fit individuals, but not sportsmen, range computed from d = v*v / g):

• medicine ball 500 g 40 km/h 12 m
• medicine ball 1 kg 32 km/h 8 m
• medicine ball 3 kg 25 km/h 5 m
• medicine ball 5 kg 20 km/h 3 m

assorted from Wikipedia:

• pilum (possibly requires a short run, e.g., during a charge) 3 kg 60 km/h 30 m
• throwing axe 500 g 45 km/h 15 m

For throwing, in general, comparing with baseball and short put, fit individuals seem to reach half the speeds of sportsmen. There's not enough data to say how much better object with wires or spear-throwers are; it may be around 50%. If we disregard object shape, the speed seems to be affected as much by density as by weight. High density causes less air resistance and enables one hand throwing with fully extended swinging arm. A handle can similarly help with throwing, even with low density or very high weight.

### Simplified velocity formula

We only consider quick methods of launching projectiles, which rules out Olympic hammer and Olympic javelin and renders heavy crossbows one-shot weapons. We disregard rotation and elasticity of the projectile (but not of the launcher). We assume fit individuals with basic training. The velocity represents the average speed of the projectile throughout its flight, which is lower, but not much lower than the initial velocity, unless the object's shape causes a lot of air resistance. Measure units are m for distance, m/s for speed, g (gram) for weight.

We assume thrown objects of up to 250 g do not require an extraordinary strength to throw them at full speed, so can travel at the average speed of 20 m/s (72 km/h). Objects between 250 g to 1.5 kg are either dense enough or have a handle good enough for one-handed throw, though their weight restricts throwing effectiveness and higher air resistance is probable, resulting in 10 m/s (36 km/h) at 1.5 kg. Objects above 1.5 kg can be thrown effectively only two-handed with, e.g., 5 m/s (18 km/h) for 6.5 kg.

• thrown object 250 g and less: 20 m/s, from 250 g to 1500 g: 20 * 1250 / (weight + 1000) m/s, 1500 g and more: max 0 (11500 - weight / 1000) m/s
• object with a wire (but without multiple spinning) or thrown with a spear-thrower or with a sling: 50% better
• particularly dense or well-balanced object with a handle: 25% better
• object not suitable for throwing: varying maluses
• any object light enough to be lifted can be thrown a token distance (just dumped on a neighbouring tile)

### Simplified range formula

We compute range from velocity, but clip it due to the assumption that the combat takes place in interiors with a ceiling at a few meters, close to the height of combatants. We don't take into account excessive air resistance due to weird object shapes, but we conjecture that objects worth throwing, but heavier than 500 g, are rare and relatively expensive and so can't be selected or modified as easily to have lower air resistance. However, instead of reducing range, we reduce velocity to account for that (so the velocity is not the initial velocity, but an average).

The formula for range, given velocity in m/s, is taken from

http://en.wikipedia.org/wiki/Range_of_a_projectile#Uneven_ground

We simplify the calculations by setting the elevation angle to 0, since the ceiling is close to the initial height of the projectile. To compensate for the lowered angle, we set the initial height of the projectile to 4.5 m, while 2 m would be more likely for two-handed overhead throw and 2.25 for a single-handed throw. We get

d = (v / g) * sqrt (2 * g * h), where h = 4.5 m

which simplifies to

d = v * 3 / sqrt g

which rounds up to

d = v

The consequence of this simplification is that each projectile flies for exactly one second and then drops to the ground. In fact, that's just the correct physics of throwing things from 4.5 meters height, strictly horizontally, with no air resistance.

With gravity g / 2 we can assume lower ceilings and get the same result with h around 2.5. For outdoors we could use the simplified formula d = v*v / g, but it does not mix well with the formula above, in particular for small speeds the distances become shorter than in reality. OTOH, if we insisted on completely correct general formula, in essence a weighted sum of the two simplified terms, it would be complex, with lots of trigonometry and so not easy to derive from player observations, especially sampled with tiles and turns.

### Simplified accuracy formula

This is only partially implemented ATM.

The tiles are small (1 m), but the projectiles are assumed to always travel through the intended tiles. However, they can still miss their target and continue, up to their maximum range. Simplifying, we assume the size of the target (and the missile) is directly proportional to its weight (in grams), so let the chance of hitting the target be (weight of target + weight of projectile) / 1000 percent. A 100 kg actor will always be hit, two 50 kg tables thrown against each other will always collide, but two chairs much less often, and two bullets almost never. This is modified by any percentage bonuses applying to projectiles and targets (possibly the same bonus as for throwing them) and the attacking actors (the skill and the launcher, if any).

There is a problem with projectiles that miss their target. If they are to continue their flight, they should momentarily occupy the same tile as the missed target. Since projectiles are actors, we have to actors in the same tile, which is wrong from both the UI and simplicity of game rules perspective.

There is no problem due to inability to determine who hits whom first, if two actors move and collide, because even if one of the actors is descroyed (HP <= 0), its body remains in the arena for enough time, to cause damage to the other actor.