Ironsight said:
First of all, I am very sorry for the individuals that have no capacity to discuss a technical matter in a civilized manner and the only way that they know is to use that kind of implicit and inappropriate language to hijack a well-founded forum. I have reported that incidence as a rude post and hope that the moderators will do something about it.
Back to the topic, I never said that the trejectory of a projectile is a straight line (linear) but almost everyone knows that it is a parabola (non-linear but not exponential). Therefore, in order to hit the target the axis of gun barrel has to make a slight positive angle with the axis through the front and rear sights (the line of sight). Since, the axis of sights has to be optimized for different shooting ranges. For the 50 yard range this angle of compensation is more than what is required for the 25 yard range. The bullet trajectory once fired crosses the line of sight two times, once close to the muzzle and again close to the target. What I calculated was to extrapolate from 25 yards to 50 yards using a common frame of reference which is M.O.A again with a certain approximation that is very very close to linear for the transition from 25 to 50 yards (close distances not for 100 to 1000 yards). I was just trying to answer the initial question using a very simple mathematical linearization.
And Slavex's point is that assuming a linear profile is incorrect.
We're talking about an object which is axisymmetric neither in form, nor in mass distribution, which is spinning while passing through a non-uniform medium. The assymetry in form will cause local shock formation to vary in the theta wise direction, not to mention simple drag, particularly variation in the turbulent transition and separation points in the axial dimension.
The small but non-zero distance between the bullet's mass centroid and axis of rotation will cause precession, which will cause the bullet's axial direction to "wobble".
Fundamentally, slight variations in the bullet's outer shape will determine what proportion of the gasses pass a certain location on the bullet's surface, which will impart a certain degree of initial wobble.
Incidentally, did you know that a parabola is a second order exponential expansion? So a parabola is an exponential. Q.E.D.
Nevermind that Slavex was not referring to the theoretical trajectory of the bullet (which is indeed a parablola... in the curved neo-rectilinear frame of this planet's gravity), but rather it's deviation from that path, as a function of x,y,z,a,b,c.. etc.
We can formulate this as follows:
Px = Px0 + V1 (t-t0)Cos(theta) - (1/2)(drag_x/mass)(t-t0)^2 + EpsilonX
Py = Py0 + V1Sin(theta)(t-t0) - (1/2)(drag_y/mass+32.2)(t-t0)^2 + EpsilonY
Pz = EpsilonZ
Where:
V1 = muzzle velocity
Px0 = initial distance downrange
Py0 = initial height
theta = muzzle angle in the 2D (x,y) frame
By definition, Pz0 = 0.
The Epsilon, or error functions of each parametric equation is a non-linear equation involving a large number of variables, some impossible to quantify except using some sort of stochastic analysis, and frankly, I'd rather lick the dark end of Slavex's knob than perform *that*.
Simply by inspection of the number of controlling variables involved in determining the error functions, it is relatively plain that they are not linear.
