RESULTS (Ack! Too much
theory; just give me the facts!)
We have already showed the results for the idealized paintball shot on an airless Earth. So we will directly launch into the more realistic results for the world we live on. There are a couple of things that need to be mentioned first. All the calculations are presumed to be in an environment at 750 mm pressure (slightly below 1 atm) and a temperature of 25 C. As we saw in the previous section, air density and air viscosity are important parameters in calculating drag and lift coefficients. Both of these variables depend on temperature and pressure. Most of the calculations start with an initial velocity of 280 ft/s. Different fields have different velocities, but around where I live this is about the average allowed speed. You can develop your own curves from the trajectory calculator. The paintball diameter is taken as 0.679 inches. This is based on the data in the paintball weight and size statistics pages in the Paintball Section.
This is a big section. Therefore, it is broken into subsections which are linked below::
Maximum Distance Trajectory
The first case we will examine is one that everyone wonders about, but is not practical in the sport of paintball.
The figure above shows the trajectory and velocity change for a paintball fired at an angle of 31 degrees with a speed of 280 ft/s. Why such a strange angle? Well, from trial and error I determined that this is the angle (to the nearest degree) that gives the maximum distance. Lets take a look at the trajectory first. According to the calculations, the maximum distance is 302.6 ft and the ball rises to around 86 ft. The ball follows an almost linear path initially, then drops rapidly. This trajectory is typical of non powered flying objects. Many of you have heard that the maximum angle is 45, so how did I come up with 31. Drag! This goes back to what I mentioned about when most people stop learning physics. Without considering drag the trajectory can be predicted using algebra and a few equations. Once we enter the real world and allow for the drag force, the entire problem becomes more complicated. Calculus is needed to provide accurate calculations. If you want a real simple but powerful image of this trajectory, just reach into your everyday experience. A stream of water from a garden hose will show exactly the same kind of trajectory.
Now in the Figure look at the change in velocity as a function of distance. The most striking event is the drop in speed. It is especially rapid just after the ball leaves the barrel. By the time the ball has gone 100 ft it's speed is one half of the initial speed. Moreover, by the time the ball could hit an object on the ground some 300 ft away from the marker, the velocity is down to 60 ft/s. That is roughly half the speed a pitcher in baseball would throw a ball. (However, a baseball carries a lot more momentum.) Also remember, the chance of the ball breaking at this velocity is going to be lower. This is why this trajectory is not practical for the sport of paintball. Personally, I knew paintballs dropped in velocity, but I was surprised the velocity fall off was this large. Notice that the ball speed rises near the end of the trajectory. If you look at the trajectory, you can understand why it speeds up again. The ball is now dropping and no longer "fighting" gravity.
The "Useful" Trajectories
We now need to take a minute to think about what we really want to investigate in this section. If you haven't already done so, think about the distances where you face most of your opponents on a typical field arrangement. I know each field is different, but from my experience on both indoor and outdoor fields, in the early part of a game the opponents are in the range 50 to 125 ft apart. We all try "long balling" once in a while, maybe out to 175 to 200 ft, but as we will see, in truth the chances of hitting something and breaking a ball at that range are not very high. (See the section on paintballs.) On the other hand, at less than 50 ft the inaccuracies that can affect the ball drop off to the point they don't matter much, unless they are extreme. Next time you go to a field, keep this in mind and see if I am right. First, measure off some distances around your home and place a friend or family member at the points to get a feel for the ranges.
So from our practical perspective we are only concerned with the distances between 50 and at the most 175 feet. What do these trajectories look like? Here they are:
The vertical axis in the top graph has been deliberately squeezed so that the height and distance axes have the same relative scale. This makes for a poor looking graph but does gives a better idea of the real path you would see. All these trajectories start at the same point, from 5 ft off the ground. This is roughly the height where the marker would be at the shoulder of a 6 ft person. The small "figures" on the above graph shows where a paintball would hit them. They are placed so that they would be hit at a height of 4 ft. This is just about the lower chest level on a 6 ft person.
The lower figure shows the velocity curves. They look like one continuous curve, but it is all the velocity curves superimposed on each other. There are slight differences in the velocities for each angle, but they are too small to be noticed.
Some of you may be surprised that these angles are very small. The largest is only slightly more than 5 degrees. If you can't believe that, just watch someone else playing. The marker indeed does not rise very high as the calculation suggests. Since we will be using these trajectories for the balance of our discussion, the Table below gives you some detail about them.
|Small Angle Changes in Velocity, Distance and Time Relationships|
|Parameters to reach 4 ft height||Maximum range parameters|
|Angle||Dist.||Vel. At Dist.||Time||Dist.||Final Vel.||Time|
|Initial paintball speed = 280 ft/s; starting height = 5 ft;|
The first distance column is set at nice neat values except for the 0 deg case; this is deliberate. By trial and error, the angles were set to achieve a height of 4 +/- .05 ft at these distances. The negative angle at the 50 ft distance comes about because of the close distance between gun and target, and the fact that the marker is fired from a 5 ft height at a 4 ft high target point. This table contains some interesting information that you may not have thought about. The trajectories, although curved, look approximately linear for much of the range. It reinforces the point about the large drop in ball speed at various distances. Also, note the times, especially for the fixed height set of data. It takes a long time to reach 100 ft. If the person is running, you better have a long lead. For example, for someone running across your field of view at 6 miles/hour and a hundred feet away, you would have to lead him by ~4.5 ft to hit him. That is roughly 3/4 of his body length.
In addition, the long travel times for a paintball to reach, say 100 ft (0.5 seconds), also can explain why sometimes players think that they are deliberately being shot after they called themselves out. In a tense game, even a half second can seem like a long time. Even if an opponent's mind and body registered your call within 0.1 seconds, balls could still be coming your way for 0.6 seconds after you yell out. Now I am not trying to justify overshooting here. There are definitely times when some social midget does it deliberately. However, realizing that there can be a long natural lag time should make you a little more tolerant of other players motives.Effect of Paintball Speed Variation on Trajectory
So now we have established some baselines trajectories that we can use to look at other effects. One of these is the question of what happens if the ball speed isn't 280 ft/s. I am sure all of us have noticed that a chronograph never shows the same velocity for your marker all the time. There are two reasons for this. One is the accuracy of the chronograph itself. However, compared to variations due to your marker condition, I believe this is a fairly small problem.
What can affect the markers ability to maintain a constant ball speed?. Just about everything you can imagine. The size and shape of the paintball will affect the muzzle velocity because gas may leak around the ball causing a drop in velocity. You don't really believe all paintballs are perfectly round do you? Any scuffing on the ball or in the barrel can cause increased drag. Especially with some markers, the presence of liquid in the holding chambers will have a big impact on the velocity.
Another big factor is repetitive shooting, and where the marker is stored between games. Gas expansion, especially when the gas is carbon dioxide, is an endothermic process, that is, it takes heat to expand a gas. This is a result of gases being what we call non ideal. (In other words, an ideal gas wouldn't do this.) When a gas expands it takes heat from its surroundings. The result of any rapid gas expansion is that the gas absorbs heat from its surroundings causing local cooling. This effect is known as the Joule-Thompson effect; It is the same principle used in your refrigerator or freezer. As you fire your gun, the chambers and lines that store and move the gas cool down. This cooling affects the pressure. How? Well, there is a famous Law known as the Gas Law which is written PV=nRT. P is the pressure of the gas, V is the volume of gas in the chamber, n is the amount of gas, R is a constant and T is the temperature. If the volume of the chamber doesn't vary and about the same amount of gas goes into the chamber each time (it may not), then we can use the Gas Law to monitor the changes caused by temperature by equating the two states of the gas before and after cooling. This gives us, P0/T0 = PL/TL, where the subscript 0 is the pressure at the start of firing and L is at some later time. We can rearrange this to PL=P0(T0/TL). It is the temperature ratio that really interests us. Now T is not in Fahrenheit or Centigrade temperature units, but is measured in Kelvins. At 25 C = 77 F that is 298.15 K. If the temperature of the chamber drops to say 10 C = 50 F, then the temperature ratio is (283/298) or 0 .95. This means that there will be about a 5% drop in temperature, and PL will also be 5% less. If this pressure drop is directly proportional to a corresponding drop in velocity, we can expect our starting velocity of 280 ft/s to drop to 266 ft/s. That is quite a change! Also, if the marker is in the sun very long, we may see the pressure rise with a corresponding increase in velocity. I am sure that anyone who has left their marker in the hot sun (not a good idea for several other reasons) has experienced this when they went to chrono their marker for a game.
Despite the lengthy discussion about the Gas Law, the real pressure drop and corresponding change in ball velocity will be much worse. Considered as only a gas phase problem, the Ideal Gas Law seems like a reasonable choice. However, we really are not discussing just the gas phase. The tank is filled with liquid CO2. Second, CO2 is not an ideal gas, especially at the tank pressures used. A better solution to the problem can be found by examining the observed pressure versus temperature data for the CO2 vapor/liquid phase diagram. In this case, the effect of temperature is even more dramatic. I will not show the detailed PT phase diagram. However, a 5 deg. F change in temperature from 77 deg. to 72 deg. F will change the pressure by 6%. Again, assuming mass transfer of the gas will remain the same, the velocity will drop from 280 ft/s to 262 ft/s. A drop in temperature from 77 to 50 degrees as used in the Ideal Gas calculation will produce a 30 % drop in pressure, or a drop in ball velocity from 280 ft/s to 196 ft/s. Even this may be optimistic. If the temperature drop also produces a reduced mass flow into the firing chamber then the pressure will be even lower.
So what happens to the trajectory when we have a variation in speed. By simply holding the gun angle constant and varying our initial ball velocity, we can find out. The following figure shows what happens:
Each vertical line in the left figure represents the total vertical range of variation that would be observed if the speed varied between 260 to 300 ft/s at the distance marked on the the horizontal axis. The longer the distance, the more affected is the accuracy of the ball. Again we are centering are target location at four feet high. Looking at the envelope of variation, we can see that it is nonlinear. The length of the vertical lines do not change linearly with distance. The right figure gives us another way to examine the same data. Here we are plotting for a fixed gun angle the height of the ball as a function of the initial ball speed. The greater the slope of a line, the greater the impact a small change in speed has on the final height reached. Now this isn't new. We all intuitively know this, but now we can see how such variations arise, and the type of control we need on our equipment. Taking the slope of the lines near the 280 ft/s line provides us with an idea of what kind of change we will experience per change in 1 ft/s. The table below tabulates the data:
|Effect that a 1 foot per second velocity changes has on target strike position at various distances.|
|Change in target point (inches per 1 ft/s change)||0.06||0.14||0.29||0.52||1.33|
So at a target 125 feet away, a 1 foot per second difference in velocity causes roughly a ½ inch variation in vertical. But that is from less than a 0.5% change in velocity at 280 ft/s. At 5% or roughly +/-14 ft/s the change is 10.5". Admittedly, this is not a very big change, but it will make a difference if your target is sideways to you.
The Effect of Spin on the Trajectory of Paintballs.
We have already discussed that paintballs will curve if they are spinning. This is due to the Magnus effect. Even our intuition tells us this must be happening. Anyone who has played paintball has seen the balls curve. Sometimes, to our consternation, it's a very dramatic curve. Have you ever seen a ball drop and then come back up. I haven't, but my son was shot out of a game recently by one. Very frustrating, but entirely possible! Even more so now that a barrel designed to force a spin is in production (see below).
Before I show the data it would be useful to have some idea of how much spin a paintball can have. This is not a simple question. It depends on what happens inside the barrel, and I said earlier I was going to avoid getting into that aspect of paintball physics. Well, I lied a little bit. We need to apply some reasonable constraints on the velocity that the ball can achieve. Therefore, we want to have some rough idea of the possible velocities the paintball might achieve.
First, some useful (useless?) facts: The ball is being accelerated from 0 to 280 ft/s along a distance of roughly 1 ft. We can calculate that the acceleration of the ball is roughly 39,200ft/s2. That is over 1200 g's! The time to traverse the barrel is roughly 0.007 s.
Now as far as spin is concerned lets consider the most extreme case: The ball is frictionless on all barrel surfaces except for a line that has the texture of sandpaper running down the length of the bottom ID of the barrel. That line will produce an awesome friction and a net rotation of the ball. Its just like rolling a ball down an inclined plane. Assume there is no sliding to reduce the spin rate, that is linear motion is directly converted to spin on the ball . A 0.679" diameter ball has a circumference of 2.13". Let's assume that the ball achieves its maximum speed of 280 ft/s almost instantaneously. The ball will then make one revolution in (2.13 in/12 ft/in)*(1/280 ft/s), or it will make 1574.8 rev/s; this is 94488 rpm. That is quite a spin rate! Of course, it is very doubtful that a ball would ever be able to achieve this kind of speed for several reasons. First, in a real barrel the ball would slide as well as roll down the barrel. Second, the ball is not instantly accelerated to the final velocity. These two points are so important that they put the maximum spin rate estimate well beyond reach of any kind of normal paintball marker.
Unfortunately, it is not so easy to predict what is realistic. From my own field experience, watching the antics of my wayward paintballs and playing with variable spin rates in the dynamics equations, I believe that most of the paintballs that go wild, probably don't get much over 10,000 rpm. For most of the results given here, I will therefore restrict the spin to values less than this spin rate. However, because of some attempts by others to deliberately apply spin to paintballs, I will address high spin rates at the end of the spin section.Changes in Trajectories with Spin Present.
So lets take a look at one of our typical trajectories with spin present on the ball:
In the figure above we are using an angle that without spin would have a trajectory that would hit a 4 ft height at 100 ft. We apply either back spin (spin angle of 0 degrees) or top spin (spin angle of 180 degrees). According to the direction of the ball in the figure above, backspin is a counterclockwise spin on the ball. As I mentioned before, in the initial speed range of most paintballs the balls will exhibit an antiMagnus effect. Our intuition tells us that back spin should cause a ball to rise because the air is "sped" up over the top of the ball compared to the bottom by the spin. However, for smooth balls at low spin rates, the opposite will occur. So in the present case, the top spin ball produces a lifting force and the ball goes farther. Not only that; look carefully at the trajectory of the top spin ball. Notice that it is flatter for a longer range than the other two cases. We shall return to this flatter trajectory and its ramifications later in the high spin rate case.
Gee, maybe those people who claim that their paintballs travels with a flatter trajectory could be correct. Unfortunately, its because their barrel is probably damaged or dirty, so that its producing top spin.Lateral or Side Spin Effect on Trajectories
The next problem is side spin. In the previous case, we examined paintballs with either pure top spin or pure back spin. The spin axis was oriented either at 0 or 180. Now we will consider pure side spin. The paintball's spin axis is oriented at -90 or +90 degrees. (Note: These spin angles are a little confusing. From our coordinate system setup, the angle we input into the equations is actually the direction of the "lift" force rather than the spin axis. The spin axis is at 90 to the lift force vector.)
The spin axis in this figure is set to 90 deg. The figure represents the trajectories looking from directly above. We are only interested in the side motion in this case. There are a several interesting points about this graph. Although the ball curves continuously, the curve is most prominent in the later part of the trajectory. Remember that the lift coefficient which gives rise to the yaw, first becomes more negative as V/U (spin velocity/linear velocity) increases. Our calculations assume that the spin velocity does not change at all, but we already have seen that the linear velocity drops very rapidly from the initial velocity of 280 ft/s. Hence, the V/U value begins to change rapidly with distance, and has a progressively greater effect on the curve. Second, even at the low spin rate of 4000, there is a quite large deviation from a straight line. The shift is over 1 ft at the 100 ft distance mark. This would shift the ball from a dead center chest hit to a right arm hit. At 8000 rpm, the ball would likely miss everyone, but an overweight player. The most curious trajectory is obviously the very high spin case of 27,000 rpm. Here the ball actually curves out and then comes back in and crosses to the opposite side of the center line. This behavior is very similar to a curve ball in baseball. What's going on here? Well, lets take a look at the way the lift coefficient changes with distance in this particular case:
Remember again that the lift coefficient is controlled by the ratio of the spin velocity to the linear (forward) velocity V/U. We see that CL is first negative so we have an antiMagnus effect which forces the ball to the left. However, very soon the ball's speed drops; this forces the spin ratio to increase. From the Davies/Maccoll data, we showed in the Theory Section, which was used in the calculations found in the Calculations Section, at high V/U we shift to a pure Magnus effect, and the ball experiences a rapid and large force to the right.The Complete Effect of Spin on Trajectory
We have looked at the effect of spin only in the vertical and horizontal directions. What about other spin angles? By varying the spin velocity and the initial spin angle the "hit" or target region that a paintball with random spin effects will exhibit can be determined. The following Figure shows the data:
There are three sets of data at ranges of 75, 100 and 125 ft distance. Again, we used the angles so that if there were no spin the ball would pass through the 4 ft height point. The starting velocity was 280 ft/s. Each range has five lines associated with it. These are iso-spin lines, or iso-strike lines. Each represents where a paintball with a given spin rate would pass through at that particular range. Each of the iso-spin lines describes a circle. A paintball with any spin rate lower than that at which the line was calculated would fall within that circle. Another way to look at these circles would be as defining the accuracy range of spinning paintballs. Look at the smallest set of circles. For a distance of 75 ft, even 8000 rpm does not produce enough variation to miss a frontal target. Of course, if the figure was sideways, there would be a greater chance of missing. At 125 ft, the situation is very different. The chance of hitting a target even with a low speed of 6000 rpm is going to be much lower. As a rough measure of the probability, just look at the area occupied by the figure and the area encompassed by the 6000 rpm iso-spin line. Roughly about 2/3 of the area is outside the figure. This suggests that if the ball were randomly spinning with any spin up to 6000 rpm at random angles from the gun, the probability of hitting a person 125 ft away is about 33%. Definitely not a good recommendation for long balling.
From another perspective, at 2000 rpm, the chance of striking a body part is nearly 100 % at all distances from 75 ft to 125 ft. Even at 4000 rpm, the chances are well above 50%.
Another point to notice is that the accuracy circles shift to a lower position at longer distances. This is due to the effect of gravity. Protect that groin!The Effect of High Spin Rates on the Trajectories of Paintballs
So far we have been discussing relatively slow spin rates that I believe apply to random spin effects we might see with just about any paintball gun. Here, we are going to examine the effect of much higher spin rates on paintball trajectories. This section was prompted by several recent events that have come to my attention:
I have learned that several manufacturers have produced bolts that produce an asymmetric flow of gas in the barrel, which may or may not impart a deliberate spin to the ball.
Several barrels have appeared by Tippmann Pneumatics Inc. or BT Designs that deliberately incorporate a mechanism to impart a backspin on the ball. The Flatline barrel is "S"-curved so that the muzzle is higher than the firing chamber. The curve dictates both the direction of the spin, and the magnitude. The oversize barrel is needed so that gas will blow past the ball in an asymmetric manner causing differential friction and thus promoting a spin. Now despite the seeming revolutionary character of this barrel, I want to emphasize that the concept is by no means new. Robbins,(the Magnus Effect is sometimes called the Robbins Effect) who did some of the early work on spinning balls, used bent musket barrels to show that the curved barrel imparted a spin to the musket ball, causing them to curve. Nevertheless, Tippmann engineers do deserve credit for realizing the potential for the sport of paintball, and for developing the barrel itself. Since this is a relatively new development, only time will tell its worth. The second case, the BT Apex barrel is a variation on this theme, but modifies the spin at the front end of the barrel using a half rubber piece in a modified muzzle to produce friction on one side of the ball. I find this design amusing and clever. It is amusing because when I first put this page up in 1999, I suggested putting tape on one side of the muzzle to provide the friction. Certainly the BT design is more flexible and sophisticated than that, but is basically a takeoff on that idea. I think it is well established now that these ideas are not pure fiction; they do work and can be shown based on the physics developed here. In the discussion section, I do offer several other ways to impart spin without a special barrel. (Even though I hang my head in shame that when this document was first published on the web in March of 1999, I missed the obvious case that Tippmann developed in the Flatline barrel.)
Whereas our previous results at relatively small spin rates with back spin kept the paintballs in the anti-Magnus regime, we are going to look beyond that range here. If you go back and look at the curve for the lift coefficient, you will see that at low ratios of the spin velocity to the forward velocity, the lifting force is going to be negative, but at high ratios the value becomes positive. That means that at high spin rates, there is a "lifting" force on a spinning paintball. I also showed in the side spin section that at higher velocities this can lead to some strange side spin motion. The same is true for the vertical component of the trajectory. Here are some examples:
The figure shows a series of trajectories developed from different spin rates. In all cases, the balls have back spin on them. This means that the ball is rotating counterclockwise when viewed from the right side. There are several things to note. At low spin rates, the paintball drops closer to the gun than the non spinning paintball. This is due to the anti-Magnus effect. As the velocity increases, the point at which the ball hits the ground begins to increase. Not only that, but the vertical component of the trajectory begins to take on rather strange shapes. No longer do the trajectories have an initial long, nearly flat portion and a rapid drop off. The trajectories become very complex with a distinct dip in the middle. The paintballs begin to take a real roller coaster ride. At even higher spin rates the ball actually rise up and then drop. Pay particular attention to the impressive distances the ball can now attain. Given the right spin, the trajectories are quite flat with little variation in vertical height, and can double the distance.
Another point about the curves is the big change in the shape of the trajectory between 25,000 and 30,000 rpm. This is due to the fact that in this range we "break through" the anti-Magnus effect region (negative lift force) into the the Magnus effect region (positive lift force) very quickly. If you go back again and look at the lift coefficient diagram in the Calculations Section, you will see that there is a rapid increase in the lift coefficient from negative to large positive values. It is the ball entering this positive region very quickly at high spin rates that causes the sudden change in trajectory shape.
The value at 31,000 rpm is particularly interesting and is shown for a specific reason. This spin value turns out to be where the trajectory changes the least in vertical height, that is, has the flattest trajectory and still has excellent extended range. If such a trajectory could be achieved, then it would not be necessary to angle the barrel up to hit a far away target. One of the frustrating elements of "long balling" is the number of shots it takes to find the right angle to raise the barrel and not drop paintballs too short or too long. Even then, at long distances, I often cannot see if I really am dropping the ball where I want it. Worse, the element of surprise is certainly gone when the opponent sees balls dropping around him. There is a downside to these wonderful long trajectories. By the time the ball hits the ground its speed has dropped to 45 ft/s. Even when it finally begins to drop to the ground and is just below 4 ft height, its velocity at ~250 ft is 52 ft/s. I have doubts whether a ball hitting a soft object at that speed will break very often.
In most of the previous discussion of spin, I went out of my way to indicate that spin is not your friend. We now see that it can indeed be your friend under the right circumstances. Of course, the spin rates needed to achieve these effects are very high. However, if you look back at the maximum attainable spin rate I calculated, you will notice that the present values are still far below that limit. That is encouraging.On To The Relevance Section