Here the theory hits the real world fan. I will discuss the limitations inherent in the calculations, and genuflect before the complexity of reality.
By the time you finish reading this section, you may be saying why bother to predict the trajectory with so many possible variables that can affect the sport of paintball. However, barring any major changes in the drag and lift coefficients, our calculations will predict the ball's trajectory. All the real world scenarios I discuss are secondary to the forces that we used to calculate the trajectories. The calculated trajectories are conservative estimates of the magnitudes of the expected changes. The discussion here is on perturbations which will interact with the dynamics system, but for most shots these effects will not greatly affect the balls path (re. accuracy). Another point to take away with you is that the theory does give us insight into what factors are important, and does predict a number of phenomena that we all have seen paintballs exhibit. The magnitude of the changes may be somewhat different, but the mechanisms are the same.
All readers should realize that this section has some speculation in it. I am trying to make the arguments as logical as possible, but the scenarios could be based on false premises, because there is so little real evidence to go on. The reader should think before he leaps.
Environmental Factors Affecting
First, lets take a look at how the external environment affects our calculations. All the constants which depend on atmospheric conditions, such as viscosity and density of air, will change with both the altitude and air temperature. For instance, the density of dry air varies from 1.205 g/L at 20 C and 760 mm Hg to 1.160 at 30 C and 760 mm Hg. So there is roughly a 4 % difference over a 10 temperature range. The viscosity will also change by roughly the same percentage. Moist air is slightly more dense at the same temperatures and pressure, but the difference at the most is less than 0.5 %. Notice that the air pressure is specified when discussing the density. At 25 C the density of moist air changes from 1.186 g/L at 710 mm Hg to 1.231 g/L at 790 mm Hg. The pressure not only varies with highs and lows going through the area, but also the altitude. On the average, Denver must have a lower pressure than Chicago. On the whole, we expect these effects to be relatively minor problems in predicting ball behavior.
Temperature also has other affects. I discussed some of the problems in terms of the Gas law in the Discussion section, but I did not mention the variation in the balls with temperature. RP Scherer indicates that as temperature rises, a ball will get larger and soften. Although a soft ball may have breakage problems, I wonder if there will be a corresponding better accuracy, because the balls will conform better to the barrel. (Does anyone notice more erratic ball behavior in colder weather?
Humidity is also a problem. Again, RP Scherer suggests that paintballs be stored below 50% humidity, because they will swell otherwise. Swelling will cause more ball breaks in the barrel, and will require more gas to obtain a certain speed.
By the way, local day to day changes in both air pressure, temperature and humidity are also part of the reason your marker doesn't shoot the same velocity from one day to the next.
Winds are another sure way for the calculations to be off. Winds will have a large effect on the trajectory. Although the distances will change, you will hardly notice their affect if the wind direction is directly behind you or in front of you. Cross winds will definitely be easy to see in the trajectory. The effect of wind velocity is not that difficult to add to the equations, but for the purpose of these pages, it did not need to be considered.
Reliability Due to Measurements
We have already seen that once the ball leaves the barrel, there is no more force exerted in the direction of motion. Our perception of the velocity relies on choronograph measurements, and to a smaller extent on our "impression" of the speed. In the latter case, most of us can tell the difference between say 250 fps and 280 fps. But how about 275 and 280. Not really! As we have seen, a change of 5 fps can affect the range by about a couple of inches depending on how far the target is from the marker.
The question is just how accurate is a chronograph? I don't really have a clue at this point. However, I suspect the accuracy is no better than a couple of percent. Remember, just because an instrument gives a nice digital number doesn't mean its accurate. If I am correct, about the percentage error than that translates to a +/- 5 ft/s variation with an muzzle exit velocity of 280 ft/s.
Effects due to Barrel Imperfections
A new barrel may be cut and polished to the highest tolerances, but it will not remain that way forever. Dust and dirt particles will eventually cause scratches in the finish. Thoughtless cleaning and especially the hurried barrel cleanout we do in the heat of a game can degrade the surface. Dirt is a good scouring agent. How many of us wash the ball wiper before we clean out the barrel on the field? The result is that these imperfections can cause even a perfectly spherical paintball to start spinning and consistently damage the ball's soft surface. Such damage will cause changes in the airflow around the ball (see next section.)
Effects due to Paintball Imperfections
This is the bad one. After thinking about paintball accuracy, and while writing this document, I believe much of the inaccuracy in the sport of paintball is in the nature of the paintballs themselves. Just carefully look through a batch of balls from any manufacturer. All those little dimples and seams affect the dynamics inside or outside the marker. Thus, paintball imperfections are the most likely source where good markmanship will go bad, especially in conjunction with barrel imperfections. I hope this section will convince you of that. Now there is a big caveat. That does not mean every time you miss a target its the paintball's fault. Only under absolutely controlled conditions can this be proved true. In some cases, the inaccuracy may be related to quality control in paintball manufacture. In many cases, it is related to how the paintballs are stored, used, and abused. The person behind the marker is also a big factor. We should all be more critical of our "abiliities".
An misshapen paintball's interaction with gas and barrel will result in spin. This can be due to gas blowing past a ball, or nonuniform friction while in the barrel. The section on paintball statistics shows that paintballs are not perfect spheres. However, not all ball imperfections will lead to strange paintball trajectories. It depends on the relative geometry between the paintball asymmetry (imperfection) and the barrel. Look at the Figure: The top drawing is the case where a well formed ball is uniformly sealed in the barrel. The second case is a ball dimpled on the "back" side. This configuration still has a good symmetric seal between the ball and barrel; there likely will be no spin developed. However, when the dimple faces the side of the barrel, as in the third case, the gas can blow past the ball. This leads to a difference in force due to the flowing gas on the top side and increased friction on the bottom; there will be a net rotation in this case. In addition, the gas leakage causes a decrease in linear velocity. So not only will the ball spin, but it will leave the barrel at a lower speed. I showed in the Results Section that the lift force due to spin is dependent on the ratio of spin velocity to linear velocity. In the present case, that means we will see a larger curve in the ball.
A prominent seam rather than a misshapen paintball is another issue. I suspect if a seam is perfectly uniform around a ball, there will be little effect on the spin of the ball. The reason is that no matter what angle the seam presents to the barrel it symmetrically contacts the barrel at two points (the raised seam). Gas will blow past the ball where the seam does not contact the barrel, but it will be uniform. (Actually there might be a pressure differential when a part of the seam is close to the barrel, but I believe barrel friction overwhelms this effect.) However, this type of problem should lead to vertical changes in the trajectory, because again the gas loss will randomly affect the ball velocity. There is a real possibility that portions of a seam may be more deformable than other portions. Take a look at some paintballs. Often, the seam is not completely symmetrical. Again, this could lead to an asymmetric fitting of the ball in the barrel, causing spin. Keep in mind that the seam will also affect the ball dynamics once the ball leaves the barrel (see below).
Just how much spin will be imparted to the ball by gas escaping nonuniformly around the ball is not clear. I did discuss some very crude estimates in the Results Section, but I don't have any reliable data on paintball spins. We do know from the references at the end of this article that roughness or shape affects both the drag and lift forces. Go back and examine the graphs for the drag and lift coefficients. Notice how golf balls behave with bottom spin. The rough surface makes a big difference in both the drag and lift coefficients. In terms of the drag coefficient, it makes the ball travel farther. In terms of the lift coefficient, the lift force gets larger. Thats great for helping the ball to go further, since we have seen that this causes the ball to curve upwards. For the flatest trajectory I calculated a spin of 31,000 rpm was needed. That again is based on smooth sphere data. If paintballs cannot be considered as smooth spheres, then much lower spin rates can cause the same behavior.
What about the other 359 degrees the spin axis can assume? Side spin and bottom spin are not you friends. They also will increase as the ball shape deviates from a spherical form or the surface of the paintball becomes rougher. The result will be greater ball curvature at lower speeds. In the case of bottom spin, the ball will drop faster. In the case of side spin, it will curve left or right earlier, ending up much farther off center than the calculations would predict. Worse, the curve may be complex, following even an "S" shaped curve, as I demonstrated in the Results Section.
Spin is not the only mechanism that causes a ball to have a curved trajectory. I have alluded to this problem in the Introduction Section. Have you ever heard of a knuckleball in baseball? Deflection from a normal trajectory can occur even when the ball does not spin or spins very slowly. If a baseball is thrown with the seam just at the right angle relative to the forward motion, a pressure difference results to disruption of the air flow around the ball. The baseball will experience the same type of effect as the Magnus effect. Anything that breaks the perfect spherical symmetry can cause unbalanced aerodynamic forces. This could be a misshapen paintball, a seam, a scratch or a blemish. The result will be an unbalanced pressure differential resulting in some sort of deflection.
However, I doubt that this force due to misshappen paintballs is really a significant contributer to odd trajectories. My reasoning is that the asymmetry of the ball would cause it to rotate even in the barrel and once this happens, it is likely that the effect due to this force will rapidly average out to zero.
Another element I neglected in the calculations was a decrease in spin velocity as a function of distance. The spin should experience a drag effect, but I have no direct evidence of how much. Heck, I don't even know the typical spin regime for a paintball, let alone the spin drag coefficient. There are a few estimates of this for other sports balls, and the effect does not appear that large. However, the lack of experimental information on paintballs did not warrant including a completely speculative additional calculation.
Effects Due to Air Pockets in
A last thought about real world paintballs is the asymmetry of the filling (mostly polyethylene glycols) inside a paintball. If the ball has an air pocket that is not exactly at the center of the ball, then the center of mass is not coincident with the ball center. In almost all cases, the air pocket will be near the inside surface, for the same reason bubbles rise to the surface in water.
For a spinning ball, the offset center of mass could conceivably affect the paintball's path. To put this in perspective, lets take a look at a couple of real world examples where the center of mass is shifted from the center of a system. First, imagine two different sized pucks connected together by a string sitting on a frictionless (e.g.,air hockey) table. We grab the bigger puck, start it spinning so that the little puck rotates around the bigger one, and then we give the system a shove away from us. What path does the larger puck follow? The system of pucks will move away from us, but the path will describe complex position and velocity oscillations around the center of mass. The net motion will be in the direction we pushed, but the actual path of the larger puck will oscillate from side to side and its forward motion will slow down and speed up as the smaller puck rotates around the larger one.
Another example is the hunt for planets in other star systems. The old way of doing this was to watch for oscillations in the position of a sun over a long period of time indicating that some large nearby body was pulling the sun out of position as the "planetary" body rotated around it. (The current way is to watch for Doppler shifts in a stars spectral lines.) Now a paintball may not be a sun, but the dynamics are similar. Just as with the sun/planet system the mass is not symmetrically distributed in space. Except in our case we don't have more mass to worry about; we have missing mass. That should not really matter. We have a center of mass that is not coincident with the center of rotation.
However, there is another, so far neglected, factor here which will make any further discussion of this bubble effect moot. Given time, the air pocket will rise to the top of the fluid and end up against the inside skin of the ball. The fluid in the ball is viscous, so that in a game, even as the balls are jostled and moved, the fluid will tend to stay on the inner wall. Once the ball starts spinning the air pocket experiences a centrifugal force and will begin to move to the center of the ball. This occurs because air has a much lower density than the fluid by several orders of magnitude. Its the same reason heavy particles settle in water in a centrifuge. As the air bubble moves toward the spin axis (nominally the ball center), the center of mass also moves to the center.
Although I will not show the calculation details (it is based on Stokes Law applied to particles in sedimentation centrifuges), if I assume the following conditions: air pocket = 4 mm dia., bubble starting at inner wall, fill viscosity = 310 centpoises, and fill density = 1.16 g/cc , and the spin is 1000 rpm, it takes about 0.6 s for the air bubble to come close to the center of the ball(0.1 cm). This relatively long time to reach center means that there could indeed be an effect on trajectory due to an air bubble. Look back at the trajectory data. You will see that for practical ranges this time is of the same order as that to hit a target. Thats not good news, it suggests there could be an effect. However, lets pursue this a little further.
Just how will this 4 mm bubble affect the center of mass? Imagine that we divide a paintball in two; one half contains the air bubble plus fill, and the other only fill. Now lets say we can place all the mass of each half at the edge of the ball. We can calculate that the center of mass will shift from the center of the ball 0.1 millimeters toward the mass with no air pocket. Thats not a very large shift and moreover, I have biased the center of mass calculation in favor of the worst possible case. The mass of the ball is really concentrated much closer to the center (remember its a sphere). In addtion, while spinning through the air the bubble will definitely move toward the center. (There are ways to get a more accurate answer, but I don't think its really worth the effort.) The result is that the center of mass really won't be much different from the center of the ball. /P>
Based on the above analysis, I believe the presence of air in the ball will not cause noticibly erratic trajectories. There will be no wild gyrations due to air bubbles in the fill.
However there is twist to the air pocket problem. Variable amounts of fill means variable mass for the ball. Assuming a marker always delivers the same amount of gas to produce the same amount of force on a ball, a lighter ball will leave the muzzle with a higher velocity. This will cause the ball to go further. This effect is proportional to the force. Based on our previous scenario with a 4 mm bubble, the change in mass is ~ 1.2%. This would mean a +3 ft/s change in velocity. To be honest, that big a change in velocity is unlikely. From our paintball statistics page, the mass variation is only 0.2 % for good quality paintballs.
Last Updated: Febuary 5, 2000