Ah. But the angle at which you fire it does not matter in my study. The only thing that matters is the range of angles that will hit a target. The target could be below the ground, above the ground...it really doesn't matter. As far as my study is concerned, the Civil War could have taken place on the moon. The range of ANGLES = Upper angle - lower angle. I just removed the step of actually calculuting what that upper and lower angle are at specific distances.Hancock The Superb wrote:And therein lies your problem. There is nothing on the other side of the equation to cancel. The target is motionless, it is not growing into the ground while the bullet is in flight. From the perspective of the bullet, the target is actually accelerating upward. At 100 yd. if you aim between the target's knees and feet, you will miss.the angle added due to gravity (as the projectile takes an parabola instead of a line) is cancelled on both sides of the equation.
Adding the lower angle to both sides gives you Lower angle + range of ANGLES = Upper angle...if we break the lower and upper angles into their velocity/acceleration components, you get 1/2at^2 on both sides, and since the t is roughly the same...they can be cancelled. This leaves me with the fantastically simple law of cosines that works for a target anywhere. The target just has to be 5'10" tall, but as far as the math is concerned, it may as well be sinking into the soil at 9.81m/s^2. To shift the angles up, I just add a factor to the top and bottom that accounts for gravity, but like I suggested earlier, there is no need when I only care about the difference between the two.
So realistic projectile motion, no. But I was never trying to accomplish that. It is safe to assume that the mean soldier will aim his weapon at the correct angle to hit the target, with his peers aiming in a normal fashion a little too high or a little too low. I am trying to find out how many of his peers will aim their rifles at about the correct angle based on 3 standard deviations (99% confidence interval).
If this does not successfully convince you that my fashion works, I guess I'll have to do the work with projectile motions and show that both ways produce the same result...a conclusion my math above indicates.