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.

While in the barrel, the paintball is accelerated by the pressure exerted upon it from the CO2 released by the bolt. The definition of pressure is force/unit area. Against this motivating force are several other forces. One is the friction of the ball against the barrel. Note that this force is not uniform on the ball and depends on the smoothness and material 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.

As the ball leaves the barrel the situation changes dramatically. The gas pressure behind the ball drops to atmospheric pressure almost instantaneously. 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 infinitely 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. 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 an object will not change its speed or direction. It is a given, from experience that paintballs can and do all sorts of strange things in the air. They drop; they curve, they can even turn corners. 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, provided we exclude the anomaly of a tailwind. In other words, the paintball is not going to magically pick up speed once it leaves the marker.

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 more 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 inifinitesmal rate of change of velocity with respect to 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.

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.

What forces could act on a ball? The most apparent is the force of gravity. Another significant force is the drag force. This is the air resistance that a ball experiences while in flight. Essentially 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 and examined forces that will affect the motion of the ball. However, there are several more forces that can affect the trajectory of a paintball, including temperature and humidity. Though the most significant are based on the aerodynamics of the paintball (the interaction of the air with the paintball). Two of these forces come into play if the ball is spinning. One is called the Magnus effect, and the other is the centripetal force. The last force has to do with the aerodynamics of misshapen paintballs. Although we will discuss this force later, it is the one force that we will not model, because it has an infinite number of variables and special cases it is too complicated to deal with easily. 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.

Fg- The Force Due to Gravity
For purposes of this discussion, 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 constant for 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 = vo:

where v0 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 v0x. From the diagram and a bit of trigonometry:

Substituting this into the equation for x:

In a similar manner:


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 projectile. Because it neglects to account for environmental forces, unfortunately this would only be the true results for a projectile fired in a vaccum.

FD - The Drag Force
The next force we consider is the air resistance the ball encounters. Unlike the gravitation force this one is 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 significant difference in the wind resistance. Thus, the drag force must have something to do with the amount of surface area presented to the wind.

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

One other factor is important here, but an alternate test vehicle is required to illistrate this point. 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 the viscosity.

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, 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*d2/4. 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. CD 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 CD, the more drag the object will have. For many objects such as balls, the drag coefficient is approximately 0.5. The problem is that CD 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.

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. 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 Reynold numbers with rougher surfaces, but notice that there are extreme changes in each curve. At certain speeds, there is a sharp decrease in the effect of the 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.

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 Reynold 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 104. 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.

FL - 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. The force developed by the Magnus Effect is very important in many sports. 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.

The air is flowing from the right. The flow lines around the top of the ball are closer together than at the bottom of the ball. This means that the air flows faster over the top of the ball than the bottom of the ball. When this type of thing happens, there is a net change in the air pressure between the top and bottom of the ball. As already mentioned, pressure is force per unit area. Hence, the ball experiences an additional force. In the present case, the net force would be upward. If the spin were reversed as in the second case, the net force is downward. The actual reason this happens is due to some very complicated things happening between the surface of the ball and a thin air layer adjacent to the surface (the boundary layer). On a smooth ball, this very thin layer of gas around the ball is affected by the spinning ball surface and is set in motion. This layer then affects the way the flow lines separate from the surface near the rear of the ball, and this in turn affects the entire pressure distribution on the ball. The Magnus 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 Magnus effect is harder to characterize. Through experiments and apparently analogy, it is often characterized in a similar fashion to the drag force:

If the ball has bottom spin, that is it spins counterclockwise, 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. The work by Davies 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, CL, is quite a bit lower than the drag force. Therefore, Magnus Effect contribution to the total force acting on the sphere or paintball is quite a bit less. At low V/U values, note that FL will be negative. This is called the antiMagnus 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.

FC - The Centrifugal Force
This force, developed by all rotating objects is very important for the stability of missiles, bullets, or artillery shells. However, a paintball is a centrosymmetric object. I am still working on what this force can mean to a paintball. It certainly has been ignored in almost all sports ball calculations. My best guess at this time is that this force would only come into play if the spin axis direction of the paintball changes during flight. The change in axis would be resisted by the spinning motion. This force would also slightly elongate the shape of the paintball perpendicular to the spin direction. How much this would distort it, I have not yet worked out. It certainly isn't going to turn the paintball into a thin disk. The distortion will affect the drag and lift coefficients, but more on this later. For the time being; we will ignore this force.

Other Forces?
There is at least one other aerodynamic force that has been observed with other balls that have prominent seams on their surface such as cricket balls or tennis balls. In these 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. 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 have to be as 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.

On To The Trajectory Equation Section

Trajectory Equations Section
Results Section
Relevance Section
Discussion Section
Explorations Section
Calculations Section


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Last Updated: 8/24/00