What was the range of medieval bombards?
score:4
First, let's discuss some basic properties of guns: muzzle-loading launchers of solid shot, usually round, at sub-sonic muzzle speeds, with no internal mechanism for absorbing recoil. The absence of any recoil-absorbing mechanism dictates that launch elevations are restricted to just a few degrees, else the carriage is rapidly destroyed by shot recoil.
The solid shot being fired does damage solely by impact, unlike the explosive shells fired by howitzers and mortars at high elevations. Combined with the low muzzle velocity (typically about Mach 0.9, or ~900 fps), low firing elevations (less than 3 degrees), and high-drag of spherical shot, ranges were quite low. Further, there were two firing modes with different ranges:
Anti-personnel mode involved skipping the shot over the ground to sweep a path of destruction before the battery. The intent was to simultaneously attempt to keep the shot under about 5 or 6 feet of height through its entire travel, and maximize the number of skips. The shot was considered spent at a range when it began rolling and hopping instead of skipping. (However spent shot remained dangerous to the unwary for an additional few hundred yards of range - General de Division St. Hilaire and Marechal Lannes were killed and fatally wounded respectively within minutes of each other at Aspern-Essling by spent shot.)
In this mode range is limited by the hardness and smoothness of the ground (ie by the number of skips possible). Assuming a muzzle height of 3 feet and 3 skips at 0 degrees elevation, range would be a bit less than 7 * t * 900 feet where t is the time for the shot to drop 3 feet, or sqrt(3 / 16) ~ 0.4. Then range is a bit less than 7 * 0.4 * 900 = 2520 feet = 840 yds = 750 m. This is the effective range of the gun. There is some variance on caliber due to the cube-square law, favouring heavier guns, but in effect this means that heavier guns get one or two more skips because aerodynamic drag (proportional to the square of the shot diameter) is less effective against the shot's momentum (proportional to the mass, ie the cube of the shot diameter).
Beyond effective range the spent shot becomes dodgable (for those not in close formation) and is more likely to wound than immediately kill. There are apocryphal tales of British officers attempting to trap spent shot with their foot as they would a cricket ball - but 12 or even 8 pound iron shot is a far cry from a 5 oz cork ball, and they typically lost their leg in consequence.
Range could be extended slightly (see hybrid mode below) by firing at slight elevation, though fewer skips would be obtained, and there would be dead ground in front of the battery where the shot was travelling above any targets.
Bombardment mode is the employment of elevations above level (up to about 3 degrees only) in the attempt to hit a target, undoubtedly stationary, on the fly. Here the limitation on the range is the maximum angle of elevation, about 3 degrees, at which the gun can be repeatedly fired without destroying its carriage from the recoil. For a maximum elevation angle of 3 degrees the time to land for a shot is about 2 * 900 * sin(3) / 16 =~ 6 seconds, giving a maximum range of about 6 * 900 = 5400 feet = 1800 yds = 1600 m. At these elevation angles there is no skipping of the shot as it impacts the ground with a downwards velocity component of ~48 fps.
Here again the cube-square law advantages heavier guns relative to lighter guns, with the calculated ideal maximum range above being reduced more by drag in the case of lighter guns than for heavier guns. I have not attempted to calculate those effects in the rough calculations above. It is also possible that the heavier carriage of a heavier gun might be able to tolerate a slightly greater maximum elevation, for again a slight increase in maximum range; however I would probably only credit this to post-Gribeauval designs without hard experimental evidence.
Hybrid mode was sometimes attempted at very low elevation angles of about 1 degree or less, as at Waterloo by Napoleon's grand battery. Here an attempt was made to achieve one skip of the shot in order to target Wellington's troops behind the ridge; but it is uncertain how effective this was. It certainly would required very hard ground for success.
Notes
I am unsure precisely why (nearly) all guns before the early-19th century are subsonic, but I suspect it is a casting limitation on the barrels. A second possibility though is that achieving higher velocities was inefficient in powder, gaining just a little in range but dramatically reducing ability to maintain fire rate and duration.
Guns prior to the early 19th century absorbed recoil solely through friction of their own weight against the ground. With no internal springs, it was necessary that the gun recoil almost straight back so as to not over-stress the wooden carriage. This is the reason for the very low elevation ranges for guns. Howitzers and mortars were able to launch at higher elevation ranges by using a very much lower powder charge. This is the reason why guns and howitzers are always distinct weapons until development of the Napoleon 12-pounder (named for Napoleon III, not his more famous uncle) in the 1850's.
The Napoleon 12-pounder well known through the American Civil War is a very different weapon than the Gribeauval 12-pounder so favoured by Napoleon I. The latter was a true gun designed in the late 18th century, while the former is a hybrid gun-howitzer designed for napoleon III in the 1850's.
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