The Equations of Motion
for a Paintball and the Trajectory Calculation (How do we figure all this stuff out?)
Having discussed all 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 above. 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 drawing below.
In addition, to the three axis, there are three angles (θ theta,φ phi,υ eta) that describe the position of the force vectors; θ (theta) is the angle in the direction the ball is traveling measured relative to the xy plane; φ (phi) is the angle the spin axis makes with the xy plane and measured in the yz plane from the positive z axis, and υ (eta) is the angle the spin axis makes relative to the xy plane and measured in the xz plane. This latter angle is the case where the spin axis would be pointing forward or behind the yz plane measured through the center of the ball. Just how this angle affects the trajectory of the ball is not clear. Obviously, it will bring in Magnus force components, FL, in all three directions. However, it is not clear if it would be that simple since we are dealing with an aerodynamic effect. One thing does seem to be clear. Whatever the effect, the balls curvature will not be greater than that experienced when φ eta is 90 degrees, in other words, when the spin axis is only restricted to the yz plane. Therefore, we will neglect the φ eta angle .
We can now write down the force equations:
FL is more complicated to write down than the previous discussion would imply. Since it is the interaction of a spin and linear system and we need to consider both, the vector cross product is involved. 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 description of the entire system is :
where is the spin vector. The unit spin vector is given by:
is the angle that the paintball's spin axis makes with the positive z axis. We then substitute in the definitions of the various forces, and also substitute the following definitions:
The resulting dynamics equations are:
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 changes. 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 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, we see 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 the ball stays at higher speeds longer. So a very heavy ball will not slow down as fast as a very light ball. 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 say. 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.
Last Updated: April 14, 2000