*INTRODUCTION (The
Stuff Starts Flying Now!)*

The kind of information needed to explore the flight of a paintball
crosses several scientific disciplines. One is the classical dynamics of
Isaac Newton, another is the field of fluid dynamics, more specifically
aerodynamics. Ballistics is the name of the specific scientific field
that encompasses the science areas that we need to address. It is the
study of projectile motion. There are two separate areas of ballistics -
exterior ballistics and interior ballistics. The latter deals with what
happens to a projectile before it leaves the barrel or guiding
mechanism. The former deals with what happens after the projectile
leaves the barrel. Although what occurs before the paintball leaves the
barrel is crucial to the flight or trajectory of the ball, it is beyond
the scope of this article. A brief description as I see it from my
armchair will hopefully set the stage.

While in the barrel, the paintball is being accelerated by the
difference in pressure between the CO_{2} behind the ball and
the air in front of the ball. The definition of pressure is force/unit
area. Against this motivating force are several other forces. One is the
drag of the ball against the barrel. Note that this force may not be
uniform on the ball and depends on the condition (smoothness and shape)
of the barrel, and on the smoothness and shape of the paintball. A
second hindering force is the air drag in front of the ball. A third
force is rotational torque that the gas or barrel may impart to the ball
causing some sort of rotation.

Now as the ball leaves the barrel the situation changes dramatically.
The gas pressure behind the ball drops to atmospheric pressure (How
fast? I don't know. I guess at the speed of sound). There is no longer
any force pushing the ball in the direction in which the muzzle is
pointing. There certainly are forces acting on the ball, but there are
no major forces pushing the ball in the same direction that it is moving
after it leaves the muzzle. If the ball happened to be in a vacuum, such
as far out in space and away from any source of gravity, it would move
along for many years in a straight line at the velocity it had just as
it left the muzzle. This is often where many people believe some magical
force propels a paintball with an extra push. No! *The only time the
marker can influence the ball is when it is in contact with the
barrel*. Why are we so concerned about the forces pushing on the
ball. Without a force in the direction of motion nothing can happen. The
heart of the statements are embodied in Newton's Laws of Motion:

*Newton's First Law of Motion :
Every body persists in its state of rest or of uniform motion in a
straight line unless compelled to change that state by forces impressed
upon it. *

The key words here are persists and uniform motion. Unless acted upon by some external force (push) an object will not change its speed or direction. We already know from experience that paintballs do all sorts of strange things in the air. They drop; they curve. So they must be acted on by other than forces imparted to it by the marker when in the barrel. However, none of these forces will act to increase the speed of the ball in the direction of motion once it leaves the marker. In other words, the paintball is not going to magically pick up speed once it leaves the marker. I know I keep repeating this, but for some reason this is usually where many people seem to go wrong in their thinking.

Since we know from experience that the ball does not move forever in a straight line, there must be other forces acting on the ball. So how do these other forces work? Well lets first take a look at the Second Law of Motion:

*Newton's Second Law of Motion:
The change of motion is proportional to the motive power impressed
and is made in the direction of the right [that is, straight] line in
which that force is impressed. *

Expressed a little differently, the acceleration caused by one or many forces acting on a body is proportional in magnitude to the resultant of the forces, and parallel to it in direction and is inversely proportional to the mass of the body.

The difference between Newton's words and the second description is that Newton saw the concept in terms of the momentum, mass x velocity (mv). The First Law said that we need a force to make a change in the motion or direction of an object. The Second Law tells us how to calculate what will happen if we impose a force on an object.

The second law results in the formula that is the foundation of classical dynamics:

See that middle equality in the equation: For those that have not had
calculus yet, the meaning of the *d(v)/dt* term is the
infinitesimal rate of change of velocity with time. In other words,
acceleration is a measure of how fast the velocity changes with time. It
is know as the derivative of velocity with respect to time. This is part
of what calculus is all about. I know this is confusing for some readers
and I apologize. However, to obtain a reasonable answer to our problems
we have to use calculus. This rate of change thing is the cornerstone of
most advanced calculations in almost every scientific problem. Just bear
with me as best you can.

The second curiosity is the little arrow above some of the variables. This means the quantity is a vector quantity and therefore the variable is defined by both a direction and a magnitude. To use this equation, we have to know not only how fast a ball is going, but we have to know its direction as well.

First, I will list all the major forces that at this point I believe can have a significant impact on a paintball. I will then give a brief explanation and I will amplify the discussion of these forces later.

Force |
Description |

Gravity | Obvious |

Drag | Aerodynamic force generated opposite to the direction of the paintballs flight; due to air resistance |

Magnus (LIft) | Aerodynamic force generated when a paintball spins |

Rotational/Gyroscopic | Several other forces based on the rotational motion (spin) of the paintball coupled with gravitational or drag forces. |

Form effect | Aerodynamic force due to seams, splits or other large surface regularities |

Wake effect | Aerodynamic force generated by the wake or vortices formed behind a paintball |

Centrifugal/Centripetal | Force due to asymmetric fill of a paintball |

The most obvious is the force of gravity. We all know about this one. Another important force is the drag force. This is the air resistance that a ball experiences while in flight. You all know about this one too. It is the same force that pushes your hand back if you put your arm out the window of a moving car. These are usually the dominant, easily noticed forces that will affect the motion of the ball.

The other forces are rather subtle, but they can still significantly affect the trajectory, or flight path, of a paintball. Note that all these other forces have to do with the aerodynamics of the paintball, that is how the paintball interacts with the fluid (air around it). These other forces are also very complicated to describe and calculate. A well known force that certainly affects the paintballs trajectory is called the Magnus effect. It is also sometimes called the lift force. This force will only appear when a paintball spins. The wake effect force has to do with how air circulates around and behind a moving object. The gyroscopic force in this case has to do with how a spinning ball interacts with the drag force. Last is the other is the centrifugal or centripetal force. The last force has to do with the aerodynamics of misshapen paintballs. Lets write down then the forces that we can expect will have the greatest effect on the path of a paintball in air:

We are going to review each of these forces as they apply to our paintball trajectory problem. For those without a mathematical bone in your body, the important thing to take away with you from the discussion is what variables influence the trajectory of the ball, and by how much.

**F _{g}-
The Force Due to Gravity**

We all know about this force. For our purposes, it is constant and will not change during the ball flight. The equation describing the gravitational force on an object due to the acceleration caused by gravity is given by:

where g is the gravitational acceleration and *m* is the
sample mass. For those of us who have taken some kind of physics course,
we have all seen the calculation for an object with a certain initial
velocity set in motion at a particular angle. Please refer to any
advanced physics book to see a detailed explanation of what is presented
below. The calculation proceeds like this:

From the picture above, Newton's second Law gives us:

In this simple case, the variables can be separated and each side
integrated (a calculus operation) with the initial condition that for t
= 0,v = v_{o}:

where v_{0} is ball initial velocity. Since the ball is
moving in two dimensions there are two equations to describe its motion
in both the z and x directions:

; |

Velocity is defined as the rate of change of distance a body will undergo. This is written as:

If we substitute this into the two previous equations, we can again integrate them with the initial condition that at t = 0, x = 0 to obtain:

; |

Now in the horizontal x direction there is no acceleration, so:

The problem is what is* v _{0x}*. From the diagram and
a bit of trigonometry:

Substituting this into the equation for x:

In a similar manner:

and:

Eliminating t between the two equations gives us the final solution for the trajectory:

This is the equation of a parabola and from this you can find that if the ball is fired at a 45 degree angle from the ground, the horizontal distance will be a maximum. The magnitude of the velocity at any point on the curve can be found from :

The results of a calculation for a 45 degree trajectory are shown below:

Note that velocity is not constant even in this case, but changes due to the pull of gravity on the ball. The problem is that this parabolic equation does not express the real trajectory of a particle. It would on the moon or other airless planet, but doesn't work on earth because of air resistance.

Aerodynamic Forces

Before we get into the forces due to aerodynamics, there is a general point about aerodynamics that needs to be made. Newton's Three Laws apply universally for the way things work around us. However, when we step into the realm of fluids and interaction of solids with fluids, the world becomes much more difficult to describe and predict. Theoretically, it should not be. If we treat molecules as hard or slightly sticky spheres, and if we know the trajectory of every atom in the fluid, and something about the solid's surface roughness, we can predict exactly what will happen. In practice, we are still in the foothills of of storing and computing, in a reasonable amount of time, the positions, collisions, velocities, and trajectories of an immense number of individual atoms. Instead, we do the next best thing, we lump the fluid molecules together as ensembles (groups) and treat the ensemble as a continuum material (fluid) with its own set of properties. The idea of pressure is a simple example of this. We know that pressure is due to many individual molecules striking walls or surfaces, each with its own individual velocity and trajectory. In describing a molecular ensemble as a continuum we develop a whole new set of Effects, Laws, and Principles that act as summaries of the way a fluid acts on itself and other materials under certain conditions. Why is this a problem? Because there are often different ways of defining the same ensemble. How we write these new Rules and define new Constants often depends on the end result that the new description will be applied to. Just as individual artists would paint a mountain differently, scientists also take different perspectives to describe fluids, because often a particular methods has its own visual and mathematical advantages in describing and computing real world interactions.

Why are these different description methods important to aerodynamic forces? Because the forces are complicated not only by their very nature, but by our competing ways of looking at them. As I will mention later, even the concept of why an airplane can fly has drawn much controversy in the last ten years. It is not because airplanes should not fly, but that our different methods of trying to describe the process can be misinterpreted and badly explained. It turns out that those same controversies extend over to paintball aerodynamics.

**F _{D}
- The Drag Force**

The next force we consider is the air resistance the ball encounters. Unlike the gravitation force ,this one is complicated. (All the other forces that act on a ball are complicated.) Lets think about what affects the air resistance on an object. By going back to our experiment of putting your hand outside of the car, we can get a general idea of what is important. If the speed of the car increases, you will feel a lot more resistance on your hand. Therefore, velocity is important to drag. Note also that the direction of the resistance or drag is opposite and coincident with the direction of the cars speed.

Next, notice that whether you position your hand with the palm to the wind or thumb to the wind makes a big difference. Thus, the drag force must have something to do with the amount of surface presented to the wind. This sounds like maybe the area that the object presents to the wind is important.

A little more subtle is the shape of the object. A flat object presented to the wind seems to have more resistance than a round object of the same plane area.

One other factor is important here, but we have to leave our car to test this one. Get into your boat. Placing your hand in the water when the boat is going the same speed as the car, the drag increases dramatically, no matter how we position our hand. So some property of the fluid (air or water) is important. This could be density or even the viscosity. Well that seems about it.

Okay, so what is the relationship between the drag force and some of the variables we have noted. There have been many experiments on this, and it is a very complicated subject. For a projectile or particle, the formula is usually given as:

*p*(Greek rho) is the density of air, *A* is the shadow
area. The latter is the two dimensional area of the object, which for a
sphere is the same as the area of circle, pi*d^{2}/4.* *
The velocity of the projectile is "*v*". The square
relationship with velocity means that a little change in velocity can
have a large affect on the drag force. However, not all objects exhibit
a square dependency with velocity. From what I can surmise from my short
exposure to this stuff, the form is based on calculations by Stokes many
years ago. He was one of the first to develop a comprehensive theory of
particle motion in fluids (you may have heard of the Stokes-Navier
equations) that did a fairly good job of predicting reality (dragwise).
*C _{D}* is the drag coefficient. It is a measure of how
well an object can penetrate the wind. Thus, it takes into account many
factors, such as the shape and smoothness of the object. The higher the
value of

*C*, the more drag the object will have. For many objects such as balls, the drag coefficient is approximately 0.5. The problem is that

_{D}*C*is rarely a constant, and often itself varies with velocity in a complex way. The only recourse to obtaining decent drag coefficients is to measure them. Now you understand the need for wind tunnels.

_{D}

Reading the scientific and non scientific literature on the behavior of balls and spheres in air is fascinating. There have been quite a few studies on the aerodynamics of golf balls, tennis balls, baseballs and cricket balls. The reference section shows just a few of the more critical and understandable articles that I used in preparing this work. Search the internet for a lot more. The graph at the left, based on several papers, is taken from the review article by Mehta [Ref: :Mehta]. The interest for us is how the curves shift with different balls. First, we have to back up just a bit and explain the x axis. Re is the Reynolds number; it's value is derived from the equation:

where the only new variable we need to define are *d* the
diameter, and *n* (Greek nu) is the viscosity. Why is the
Reynolds number used here? It is not possible to test every object such
as spheres at every size and every speed. The Reynolds number provides a
way around this. It is essentially a scaling factor that allows data
taken for an object tested under one set of conditions to be applied to
other objects of similar construction and shape, but different size and
different speeds. For our purposes, just keep remembering that the
Reynolds number is proportional to velocity. The data in the graph
compares the drag coefficients calculated from wind tunnel data for
several different types of balls. Note the dramatic change in the data
as the roughness of the sphere changes. However, not only do the drag
coefficient curves shift to lower Reynolds numbers with rougher surfaces,
but notice that there are extreme changes in each curve. At certain
speeds, there is a sharp decrease in the dragginess of air on the ball.
Golf balls have quite low drag at very high velocity. This is
deliberate, and a consequence of the dimples that you see over the
surface of the ball. The very low drag means that the ball will go
further. The fact that the whole curve shifts to lower Reynolds
number means that the speed at which the drag passes into the low drag
region is also reduced. The reason for all this behavior has to do with
the change from a smooth air flow around a ball to turbulent flow as the
speed increases. This turbulent airflow reduces the drag on the ball.
Similar curves our found for other sports balls.

Lets take a closer look at one of the curves; the smooth sphere data by Achenbach is of particular interest to us. The adjacent graph shows just this data and defines some of the terms used to describe the Reynolds number regions. At the critical Reynolds number, the drag on the sphere suddenly drops by nearly a factor of five. Referring back to the drag equation, this means that a ball above the critical region will have reduced drag and fly further. Of course, once the velocity drops below the critical Reynolds number, as we will see it must, then the ball will slow down very rapidly, appearing to drop very fast (relative to the horizontal distance traveled).

So where does a paintball fit into this scheme. The Reynolds number
of a typical paintball with a velocity of 280 ft/sec is around 9.4 x
10^{4}. From the curve, we can see that this clearly falls into
the subcritical region. As far as drag force is concerned (and perfectly
spherical paintballs), there is no free lunch for well formed
paintballs.

**F _{L}
- The Force due to the Magnus Effect**

This force is often neglected in simple calculations of trajectories, but it is an important component for a spinning projectile. It is also known as the Robbins effect. A rapidly spinning ball develops another force at right angles to the spin axis. This is often called the "lift force", but it may not always "lift" the ball. Any aerodynamic force which act perpendicular to the linear direction of motion is considered a lift force. That means any force that moves the ball to the side, top or bottom from a predicted forward trajectory. The force developed by the Magnus Effect is very important in many sports.[Ref: Mehta] We all know about the spin pitchers place on a baseball to cause it to curve from a straight path and confuse the batter. Likewise, spins in tennis or golf balls are also important. Of course, in the latter sport it usually is not considered a good thing. The curve produced in all these cases is due to the Magnus effect. The cause of this effect is the interaction of the spinning ball's surface with the air around it. The schematic below is an example of the flow lines around a spinning ball. In order to describe this we are going to have to delve a little deeper into aerodynamic theory. I am going to go into quite a bit of detail on this, because in describing the Magnus effect, many fundamental concepts in aerodynamics are invoked. We will start by revisiting the drag on a ball

In the diagram at the left, the air is flowing from the right or the ball is moving to the right. There is a tiny layer of air around the ball that is not shown; it is known as the boundary layer. This layer is very important, because it is the interaction of the boundary layer with the surrounding air flow that causes a large fraction of the drag effect on the ball. In the drag force section, I glossed over the details of how drag comes about. Here we need to go a bit deeper.

The boundary layer of air is stagnant with respect to the ball surface, that is, it moves with the ball surface. If the ball surface is rotating, then the boundary layer also rotates. The boundary layer is caused by the interaction of air molecules and ball surface through molecular forces and irregularities on the ball. These molecular irregularities may be of atomic size or larger. These imperfections act like barriers and causes the air molecules trying to flow nearly parallel over the surface to be deflected away at all different angles. There is more going on, but for now that will be sufficient. Because of this interaction, the boundary layer is considered very viscous and resists movement. In addition, layers of air further from the ball interact with the boundary layer and experience a drag from it. If the air is moving fast enough past the ball, the air molecules cannot move back behind the ball in a laminar or streamline flow to fill the space behind the ball; the result is turbulence. This inability of the air molecules to quickly follow behind the ball causes a partial vacuum behind the ball. This, plus the boundary layer interacting with air layers further from the ball creates and the direct impact of air with the front of the ball produce what we call the "drag force".

For a non spinning ball, the flow lines break from the top or bottom of the ball at the same point Spinning balls are a little different. In the spinning ball image at the left, note how the flow lines around the top of the ball separate at a further distance around the back of the ball, then for the flow lines at the bottom. The ball spin causes the boundary layer to rotate with the ball surface. The spinning boundary layer drags the air layers near the top of the ball further around behind (left side) the ball. On the bottom of the ball, just the opposite occurs; the boundary layer slows down the air layers nearer the ball, and the air layers separate from the ball surface earlier.

Now comes the tough part. That the air moves around the spinning ball in an asymmetric manner is well understood and is observed in wind tunnel tests of both smooth and rough balls. However, the explanation of why the air follows these particular paths and why there is a net force change on a spinning ball is more challenging to explain. Three well known physical constructs or mathematical representations go into considering the air flow around a spinning ball and how a lift force is generated. The Bernoulli principle which states that as the velocity increases over a surface the pressure drops, the Coanda effect which states that a fluid has a tendency to attach itself to and follow a surface, and Newton’s Three Laws. I am not going into detail on the first two components. You can find many articles on the Coanda Effect and Bernoulli principle. In exploring these aerodynamic topics you will likely run into an interesting controversy over the use of these principles in trying to explain why a plane flies! The amount of abused explanations is high and often very biased. Even I could see the bias and I don’t even have an axe to grind!) I am talking about a controversy that is still being argued! I suggest two recent articles http://physics.ucsc.edu/~ccrummer/aero1.pdf and http://philsci-archive.pitt.edu/archive/00002573/01/aero.pdf, both of which after reading and re-reading raised me up enough to peer over the rim of the Hole-of-Overwhelming-Confusion. The article by Crummer I found particularly helpful as a chemist for understanding what the air molecules do.

The critical element that connects all these issues to our spinning ball problem is that they result in pressure changes. Pressure is defined as force per unit area…

But if the air on one side flows differently than the other (streamline difference), than that means there must be a force difference between the top and bottom of the ball. The streamline difference also moves the "partial vacuum" behind the ball from a symmetric to off-center position. Hence, the ball experiences an additional force that has a perpendicular component both to the spin axis (in or out of the page) and direction of motion (to left). The Magnus force follows what is known as the right hand rule. In the spin case shown above, which depicts bottom or back spin, the net force would be as shows, upward. If the spin were reversed ( topspin), the net force would be downward.

An alternative explanation for the Magnus effect is often described in terms of pressure differences on the top or bottom of the ball. When a ball spins, the side of the ball where the spinning boundary layer is moving with the air flow over the ball causes the air layers further away from the ball to move faster over the ball, causing a lowered pressure. On the opposite side, the boundary layer against the general air flow over the ball and slows the air down. This creates a higher pressure. The net result affects the entire pressure distribution on the ball.[Ref: Briggs], and is analogous to the Bernoulli effect. Since we have already seen that pressure is a force, the ball will experience a force. These descriptions are not necessarily mutually exclusive. It is quite possible both explanations are needed to explain the overall result we summarize as the "Magnus effect".

Regardless of the complexities of the ways we interpret why the ball moves, we all have experience at seeing this effect in action, whether watching a baseball or paint ball curve sideways. The Magnums effect is not the same thing that causes a ball to bounce off due to English when it contacts another object. We are discussing a purely aerodynamic force. Unlike the experience we were able to draw on for drag, the Magnums effect is harder to characterize. Through experiments and apparently analogy, it is often characterized mathematically in a similar fashion to the drag force:

If the ball has topspin, that is, it is spinning in reverse to the image above, then the force is a lifting force which is what the "L" stands for. However, the force can be in any direction depending on the tilt of the spin axis. That is why we have sliders and curve pitches in baseball.

As with the drag
force, the Magnus force lift coefficient must be determined experimentally.
The adjacent Figure shows data from several workers.[Meta] The work by Davies (curve labeled 1)
is often cited and used for a smooth spinning ball. The data here is plotted
differently than in the previous figure for the drag force. In this case, the
parameter *V/U* is used, which is the ratio of the surface speed of the
ball to the linear (forward) velocity. The surface speed is given by the formula*
V = d ^{} x rpm/60*. Note that both the drag force and the Magnus
force have the same equation form, but the lift coefficient, C

_{L,}is quite a bit lower than the drag force coefficient, C

_{D}. Therefore, the Magnus Effect contribution to the total force acting on the sphere or paint ball is quite a bit less. Another feature to keep in mind is that the at low V/U for a smooth ball, the effect is very small. As the spin rate of the ball increases relative to the balls velocity through the air, the effect becomes rapidly greater.

Also, at
low *V/U* values, note that *F _{L}* is negative.
This is called the anti or reverse Magnums effect, and has only been observed for smooth
spinning spheres. You should remember that this antiMagnus effect region is
where paintballs most likely get their curve from. There is still controversy as to how the reverse Magnus effect comes about. Although at one time there was controversy about whether it was real or not, the general impression is that it does exist. However, as far as I know, there has been no explanation of the effect. That could be a sign that are rationale for the Magnus effect are not on all that solid a ground. With the explosion of the internet since these pages were initially published, there are now many beautiful movies and images of the aerodynamics of objects in wind tunnels. When I started this, there were virtually none.

**F _{R}
- More Rotational**

**Forces and Gyroscopic Motion**

The rotational force, developed by all rotating objects, is very important for the stability of missiles, bullets, or artillery shells. When the rotational force interacts with other forces the result is a change in some aspect of the ball motion. This aspect of motion is rarely considered in sports ball calculations, although it is very important in the motion of objects such as Frisbees or discus throwing.

There are three separate cases we will consider here. In each case, the spin or angular momentum vector interacts with another force to cause a change in the spin axis. This new motion is called precession or gyroscopic motion.

When gravity interacts with the rotational force it produces a torque. The result will be a change in the angular momentum vector of the ball. Remember that a vector consists of an angle and a magnitude. If the spin velocity does not change, then the spin axis must change its position. The result is that spin axis precesses in a periodic fashion. That works for gyroscopes, but how about spheres? A
paintball is a centrosymmetric object. What that means is that the center of mass is at the exact center of the ball. Gravity interacts with the center of mass of an object, and for a centrosymmetric object that means there will be no precession. Most of you know that the earth wobbles or precesses on its spin axis. Well the earth is a sphere, so why should that happen? The earth is not a *perfect* sphere and has a very bulge, which makes it an oblate spheroid. The fattened equator breaks the perfect spherical symmetry and so the earth precesses. Unfortunately, a paintball is similar and rarely, if ever, a perfect sphere. The paintball statistics page on this site shows that. The result is that a paintball will experience precession due to the spin angular momentum of the ball interacting with gravity.

The spin or angular momentum-gravity precession is not the only change in angular momentum the ball can experience. Even if the ball was a perfect sphere and there was no precession due to gravity, there will be an interaction between the ball spin angular momentum and the differential drag forces on the ball. To even more complicate matters, we know that the drag force changes rapidly with the ball's velocity and therefore is constantly changing along the trajectory. The amount of precession will depend on the ball's spin velocity and the angle between the spin axis and the drag force (forward motion).

We have one more angular momentum based effect that can affect the trajectory of a paintball. In the previous cases, we considered that the spin axis is coincident with the angular momentum vector. What if the two are not coincident? This can come about if the mass of the paintball is not evenly distributed or moves as the ball rotates. This imposes yet another force imbalance on the ball and creates what is known as force free precession. However, I suspect any spin axis precession from this effect will be relatively small.

Based on this discussion of rotational effects we can expect that the direction of the balls spin axis will change. If the spin axis changes, the motion of the ball will be quite complex. Worse, if all three of the effects are operating, the motion of the ball may be very complex. We discussed how the Magnus Effect will modify the trajectory from a non spinning ball, but the Magnus Effect does not change the orientation of the spin axis. In other words, the force from the Magnus Effect is always acting in the same plane or dimension on the ball, and that is at right angles to the spin axis. If the spin axis angle changes the direction of the lift force vector also must change. What motion will the ball show? One possible exaggerated motion is a corkscrew or helical path for the ball imposed on the main trajectory path predicted by just gravity, drag, and lift alone.

In the calculation section, I do not consider any of these angular momentum interactions, even though I believe the observable effects will not be negligible. There are two reasons for this. The first is practical. Incorporating this level of detail in the trajectory calculations is a much more sophisticated mathematical analysis, and is far beyond what I initially intended. Please keep in mind that people get PhD and masters degrees and spend entire careers on this kind of stuff. The mathematics and concepts here are not simple. Although this is not new territory for someone who is well acquainted with dynamics, it is not my specialty. I cannot simple sit down and write out the equations and solutions; I do not have the expert "feel" for this type of derivation. This does frustrate me and maybe as time permits I will dive into this in more depth. The second reason for not modeling these interactions is that, although significant, they are not likely to be major players in most paintball shots. Certainly, any of the angular momentum variations are not going to completely reverse the spin axis.

**Forces Due to Seams**

This aerodynamic force has been observed with balls having prominent seams on
their surface such as cricket balls or tennis balls. In such cases, depending on the angle that
the seam makes with the air flow, the boundary layer flow around the ball can change more quickly
on one side of the ball than the other. The result is similar to what happens in the Magnus effect.
Take a look at this link and the links
within it for a nice description and images of this force. A differential pressure occurs and the ball
experiences a force in the direction of low pressure side. This force is a real wild card. A perfectly
round paintball with no seam will not experience this effect. However, reality is quite different. Most
paintballs do show seams and, in addition, are somewhat ovoid. I cannot model this force. There is data
for baseballs and cricket balls, but these are not really comparable to paintballs. I have not found any
reference that measures the force on slightly misshapen smooth spheres or balls with a single seam. In
the case of cricket and baseballs, the impact of this force on the trajectory is nearly the same as the
Magnus Force. This force is also complicated because it will also be a function of spin and
depends critically on orientation of the seam or imperfections on the ball. For now, all I can do is make
you aware of the problem and ignore this effect.

**Vortex induced Forces**

Another aerodynamic force that was brought up in a long running, excellent discussion on the Air Gun Designs forums at the end of 2002 and beginning of 2003 was the possibility for random aerodynamics effects produced by what are called von Karman street vortices. (This thread may no longer be around.) At low Reynolds numbers, vortices are shed from a blunt object in a series of periodic waves. These can be seen as the swirls produced behind your hand as you move it through water or smoke. This same phenomenon is what probably caused the famous Tacoma Narrows Bridge in Washington to collapse. A wind of just the right speed caused the suspension bridge to begin to oscillate violently and ultimately the bridge collapsed. However, at very high Reynolds numbers the von Karman vortices become unstable and are shed in a chaotic fashion. This fluid dynamics effect on bodies is called vortex induced vibrations (VIV). The problem is that there is not a lot of work on the motion of free spheres moving through a fluid, especially in the Reynolds region that is typical of a paintball. Moreover, most of the work is on tethered objects, especially cylinders. Unfortunately, much of that work is not applicable to spheres because of their 3-dimensional nature.

Just what would be the impact of these vortex forces on a paintball is still open. However, on the discussion group a number of the participants felt that this was the reason for paintball inaccuracies. One group of contributors was convinced that this was the major cause of inaccuracy. Certainly in some of the high speed photographic evidence shown, it was not possible to resolve the trajectories captured by the cameras just in terms of spin and drag forces alone. That does not mean that the Magnus Effect is not operating for paintballs. Some recent barrels which are designed to impart a controlled spin on the ball, prove conclusively that the Magnus Effect is a real effect in paintball dynamics. The question is whether it solely describes the target spread of successive shots for paintballs.

Explaining these vortex aerodynamics is an entire discipline known as bluff body dynamics. These are bodies where a substantial amount of fluid flow over the body is turbulent. For us, spheres certainly qualify. One problem for us is that there just has not been a lot of experimental work done on the vortex shedding dynamics of spheres at high Reynolds number. The problem is partly due to the mounting a sphere in a fluid (air in our case) without the apparatus used to mount the sphere causing secondary effects in the vortices.

The vortex shedding can be axisymmetric, where the vortices are is shed from both sides in a periodic relationship; this only occurs at very low Reynolds numbers. In a larger range of Reynolds numbers, the vortex street will be asymmetric and periodic, being shed from alternate sides. At high Reynolds numbers the vortices are shed randomly and periodically. There are many web sites which discuss this, but you will find most of the information is on cylinders. Although there is a similar relation ship between the different types of periodic motion and symmetry of the shed vortices, spheres shed very different shaped vortices. The shed vortices have a similar appearance to a bent hairpin and cannot easily be visualized in two dimensions. (In fact, they are called hairpin vortices.) This type of vortex is commonly found in many types of flows around or through objects. A beautifully rendered calculated vortex can be seen here. An example of how these hairpin vortices are shed can be seen here.

**Other
Forces?**

Well, you have a lot to digest in terms of the forces that might affect the paintball. Are we done? I think
I have outlined the major players. There are some aspects that
I know we have missed discussing. For instance, it is quite likely that the spin of the ball does not
remain constant. Baseball spins do slow down. However, the drag forces on the baseball are different then
on a smooth paintball. It is likely the spin decay would be less. That change in spin will create some
very complicated secondary interactions between some of the forces, for instance, the change of the spin
axis will give rise to secondary changes in angular momentum, which interact with the drag and lift forces
and develop even more complicated motions.

**Why does the ball even get to the target?**

So now you are probably wondering why the ball even gets to the target with all the possible forces that may
come into play. However, the major changes in the path of the ball are dictated by two main forces,
gravity and drag. The other forces developed from the aerodynamics and spin are smaller. Among these other forces my belief is that the Magnus force is the most dominant. Thus, in modeling the dynamics we will only consider these three forces.
The rest are smaller forces and will compete and even interact with one another to influence the path of the ball. What these forces do suggest is that even if we rigidly mounted the gun, and exactly controlled the gun pressure, not every ball would exactly hit the same spot on the target. Moreover, the further we shoot the ball or the more time the ball spends in the air, the more pronounced the effect these small forces will have on the ball.