Relevance to Available Data on Paintballs and Equipment
Well, I have shown a lot of calculations, but can we put some real world data to the test. Unfortunately, there is very little hard reliable data for comparison. Although I have some reservations about the available data, I will try and relate it to the calculations.
For our first case, we will examine the study by TRicKOuT (!?) on testing how "good" some paintballs are over others. Quite a few different paintballs were used for the tests. The test consisted of dropping ten balls of each type from various heights onto concrete and determining at what height they broke. This is a nice straightforward test, and the author is to be commended for the work. However, the published information lacks rigid control. First, how old was the paint? How was it stored? What was the temperature and humidity during the test? Were the balls dropped with the seam toward the ground or the pole? Does it matter? (It might.) What was the criteria for breaking? Did all have to break, or one? In my own just as uncontrolled tests I had quite different results on two paints. Brass Eagle did not break at 6 inches, more like 4 ft. RP Scherer Premium purchased a month before I tried the dropping tests, did not break even at eight feet, while the above study found the balls broke at about 6' 10". Of course, the problems could even be different lots of paint in addition to all the aforementioned factors.
Despite the limited data and need for more precise controls, there is some very useful information here. It would appear that most paintballs will break at distances somewhere between 6 feet and 10 feet. What are the speeds associated with these heights? Well the trajectory program won't allow a zero velocity, but if we make the number very small and use an angle of 0 degrees, we can obtain the speeds that the paintball will reach just as it hits the ground. From 4 ft, 15.8 ft/s; 6 ft, 19.3 ft/s, 8ft, 22.1 ft/s; 10 ft, 24.6 ft/s. So the speeds are not very high. Remember, we calculated that even a maximum trajectory would have a speed of 60 ft/s as it hit the ground. However, a big factor here is the type of surface it is hitting.
The above ball breakage data would appear to conflict with a comment in another study - the "Lapco Barrel Challenge" by Richard Allen on barrels. This study was concerned with trying to answer what is the best barrel. In the document, it was mentioned that some balls were bouncing off a plywood target at 150'. I calculated the speed that the balls should have at this distance and found it to be around 110 ft/s. This is quite a bit different than the ~24 ft/s found in the previous results. The difference between hitting a plywood surface and a concrete floor should be only slight, so it is not clear why the large difference. One possibility is that in this study the balls were hitting the target at a glancing angle rather than perpendicular to the surface as in the drop tests.
The latter study may have one other useful bit of comparison information. The results are difficult to fully interpret for our purposes because some important information is unclear or the test procedures could lead to inaccuracies. For example, the guns were not in a vise, which could introduce a measure of dispersion at the target. We also don't know high the guns were mounted, and what was the angle of the shots. Although the velocity of each ball for one series of tests was recorded, the velocity was apparently not recorded for the other two series. It is not clear whether the velocity of each ball for the clocked series was compared to its impact on the target. (It would be interesting to see if there were any correlations.) However, one finding that seems to be true is that if the bore of the barrel was oversize compared to the paintball. (by several thousandths) the balls tended to be more erratic. At distances of 150 ft, the variation was on the order of ~36". Going back to the last figure for different spins and orientations we see that this is roughly in the range we would expect for random spins around 4K rpm at 125 ft. We can do a little better. A calculation with the velocity information, a starting height of 5 ft and assuming our 4 ft target point, I found the range of variation is from 30" at 2K rpm to 58" at 4K rpm. The consistency with the observed group range is quite good.
This raises an interesting question as to what is the mechanism that gives rise to the inaccuracy when there is a large mismatch between the barrel and paintball. If it was a seam or misshapen ball problem, then why would the variation show up as a function of barrel size. My guess is that we have unbalanced pressure variations in such cases that cause not ball distortion, but spinning.