Originally Posted by 22WRF
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I see all of these published figures for various loads that give the velocity of a bullet at given distances after it has been shot from a rifle, and so I am curious as to the original function used to take the derivative to arrive at that instantaneous change (or velocity if you will)[1].

And then of course, they will sometimes say the total distance that the bullet will travel, and so I would imagine that they will take the integral (antiderivative) of the original function in order to arrive at that total distance[2].

But again, how do they come up with the original function for each bullets flight path?[3]

[1] Typically measured not calculated with some extrapolation for values between the observations. Typically the first measurement will be at say 10 feet from the muzzle (that is time of flight will be measured over a distance centered at 10 feet from the muzzle) with extrapolation to a muzzle speed MV. It's easy to measure V at points 8 and 9 and 10 and 11 and 12 feet from the muzzle and hope that over short distances for some value of short the loss of speed in traveling each foot is fairly close to a constant value. Similarly with the loss or gain of speed from shorter or longer rifle barrels taken in general.
[2] See Hatcher's Notebook for a discussion of firing trials to find maximum range at assorted muzzle elevations mostly for military applications.
[3]For a simple explanation from the historical perspective see again Hatcher's Notebook.

The two books by Bryan Litz and the Berger reloading manual have a great deal of information and implicit background for practical purposes. Notice especially the discussion of G1 and G7 ballistic coefficients for relating a projectile of interest to a projectile for which more complete firing trials exist. I don't know what the actual functions might be but I know they aren't smooth uniform everywhere differentiable - and I sure don't know what the ballistic coefficient might be for an unstable bullet that goes through the target sideways but I've seen that happen so I know it does.