THE PHYSICS OF PAINTBALL

The Equations of Motion for a Paintball and the Trajectory Calculation (How do we figure all this stuff out?)
Having discussed the forces that might effect the flight of paintball, we are now in a position to write down the equations of motion. We actually proceed in a manner similar to the one shown for the two dimensional calculation of the force of gravity on an object outlined in the previous section. Because we will be taking into account the effect of spin, which can cause a ball to curve to the side, the calculation becomes a 3-dimensional problem. The coordinate system and

forces involved are shown in the images at the left and are complicated. Some staring at the images is usually necessary to understand the way this is set up. The velocity vector coordinate system is with respect to the ground with the x axis always horizontal. The spin velocity system is calculated in a coordinate system in which the x axis is always along the velocity vector and not the ground. The first image (flash image) shows the two coordinate systems. The one at the origin represents the coordinate system for the vector velocity. The second system further down the trajectory range represents the system for the spin vector. The x coordinate in this case is along the velocity vector.

Fg is the force of gravity, F_D is the drag force, F_L is the lift force, and F_C is all the rest. Think of F_C as a reminder that there are a host of additional interactions due to angular momentum interactions, vortex shedding, or seams that could be added in. As previously indicated we will neglect these forces because they typically will be small and difficult to derive equations for.

The drag force, FD:

For a projectile or particle, the equation for the drag force is usually given as:

ρ 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, π 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.

In vector form

is the unit vector. This can be rewritten with the first equation to:

The lift force - FL

The lift force FL through experiments and apparently analogy, is often characterized in a similar fashion to the drag force. The scalar version of the Magnus list force is:

If the ball has bottom spin, that is it spins counterclockwise to the velocity direction, 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.

The vector F L is more complicated to write down than the previous discussion on the drag force, even though the generalized scalar form looks like the drag force. Since the lift force is the interaction of a spin and linear system and we need to consider both. This type of relationship between a spin and linear system is described by a vector cross product. Discussion of a vector cross product is beyond this document. Refer to an advanced physics book or advanced mathematics book on vector multiplication to understand it. The vector form of the lift force takes on the form:

or

Final Form of the Force Equation and Solution

With both the drag force and lift force now characterized, the vector description of the entire system can be written :

where the spin ω unit vector is given by:

and the unit velocity vector is given by:

The vector cross product is given by:

To solve the problem we substitute in the definitions of the various forces into the vector description of the entire system.

Furthermore, in our coordinate system, the following relationships are also true which help reduce the cross product expansion:

The resulting dynamics equations in x, y, z are:

where:

and

We also need to keep in mind the following definitions to find the x and z components of the linear velocity:

We will simplify these equations a bit by applying some restrictions. If we restrict the spin axis to the xz plane (plane perpendicular to the direction of motion), then sin φ = 1 and cos φ = 0. Just how the angle Ψ, which is the angle between the spin axis and the direction of motion, affects the trajectory of the ball is not clear. Obviously, it will bring in Magnus force components, F_L, in all three directions. However, it is not clear if other aerodynamic effects would also develop. One thing does seem to be clear: Whatever the effect, the balls curvature will not be greater than that experienced when ψ is 90°, in other words, when the spin axis is only restricted to the yz plane. With the restriction that the velocity vector is always perpendicular to the spin axis ω , the spin axis, is then sin ψ = 1.

The final equations are then:

These are the equations of motion that must be solved. Notice that each equation has more than one velocity component. This means that these three equations are coupled differential equations. Calculating any one of the equations modifies another. In some cases, coupled differential equations can be solved exactly as we did in the simple case considering only the force of gravity. No exact solution currently exists for solving this particular set of equations. To find the trajectory, we have to use numerical integration methods. With modern computers, there are quite a few ways to do this. The basic operation is to calculate the acceleration change in an extremely small time interval, calculate the velocity over this interval, and then calculate the x, y, and z position changes. To do the latter, note that we also need to remember the relationship between distance and velocity as a function of time is dv = dx/dt. These new values are then used to predict the next interval and so on. In truth, the procedure is somewhat more complicated in order to get a good estimate of the changes in a small interval, but the essentials are as described.

Before we go on to the results of the calculations lets take an overall look at what these equations tell us. They should tally with our everyday experiences. To begin, remember from the first drawing above that x is the down range distance along the long axis of the barrel, y is the lateral or side to side distance from the barrel axis, and z is the vertical distance from the barrel axis. The left sides of the equations tell us that what we are calculating is how fast the velocity changes as time goes on. That is, each equation tells us how the ball accelerates or decelerates in the x, y, and z directions. Second, notice that the first two equations for x, and y; do not depend on acceleration due to gravity. This is because gravity only operates in the vertical or z direction. Third, all three equations depend on the mass (in the constant B), but the mass only effects the lift and drag parts of the equation. This makes sense. In a vacuum, where there is no air drag, a lead ball and a feather dropped at the same time, fall at the same rate. They both touch the bottom at the same time.

In addition, the mass is inversely proportional to both the drag and lift. The higher the mass, the lower will be the rate of change of the velocity in all three equations. This means that a very heavy ball will not slow down as fast as a very light ball if they start out at the same speed. Think about what happens when you throw a Nerf ball compared to, say, a baseball of the same size? If you throw each with roughly the same speed, the Nerf ball doesn't go nearly as far. That is just what the equations predict will happen. For the lift (Magnus) force, the inverse relation with mass means that less lift will be created as the weight of the ball increases. Thus, if we fire a paintball without any fill in it and it has bottom spin (the top of the ball moves toward the barrel) we would expect the ball would drop slower compared to a filled paintball because its mass is much less. Of course, the problem is that drag and lift have different signs in the vertical z direction, which means that these forces tend to cancel one another to some extent. A lighter spinning ball will drop slower, but there is more drag on it which tends to make it drop faster.