#6660306  07/06/12
The derivative and Integral

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Registered: 05/06/06
Posts: 3671

I have done very little reading about reloading and ballistics beyond what is in the Nosler reloading manual I use. But I am sure there are some very knowledgeable folks on here who would know the answer to my questions.
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).
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.
But again, how do they come up with the original function for each bullets flight path?

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#6674870  07/12/12
Re: The derivative and Integral
[Re: 22WRF]

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Registered: 06/25/10
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Loc: polar orbit

It sounds like you know calculus, so I recommend Robert McCoy's book Modern Exterior Ballistics. The revised 2nd edition of Modern Exterior Ballistics just came out, and you can get a copy from Amazon for under $70.

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#6684412  07/15/12
Re: The derivative and Integral
[Re: MacLorry]

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Registered: 01/11/06
Posts: 6

Good recommendation.
.22WRF,
No such function as you are looking for was ever derrived, AFAIK, because the drag function is different for every bullet shape. Also because the huge swings in drag in the transonic velocity range make it difficult to devise one. As close as it comes is recognition that in the subsonic range the drag function is proportional to the square of velocity times a form factor for the bullet. That's also true above about Mach 5. But inbetween, as soon as a bullet is going fast enough that some portion of the air moving over it reaches the speed of sound there is a huge jump in drag way beyond the v² proportionality as energy starts to go into shock wave formation. That extra drag drops off at still higher velocities until it is diminished to near insignificance as it approaches Mach 5 (the fastest bullets seldom fly above Mach 4). So the standard way of handling the shock wave drag is to calculate the v² drag for all velocities, then multiply it by the what is called the drag coefficient, which is a correction factor function of velocity that is 1.0 at subsonic velocities, then climbs as the transonic range is entered, peaking at about the speed of sound then falling off in a vaguely 1/v proportional relationship until its significance is diminished.
Ballistics software uses several strategies to arrive at drag. Mostly the ballistic coefficient is used with tabular data for the drag coefficient taken from real measurements of fired projectiles, so this is curve fitting rather than calculation from first principles. The BC scales the bullet's behavior to that of a standard projectile weighing 1 lb and have 1" diameter (sectional density of 1.0) and a particular shape whose drag is being modeled.
The first of those was what is now called the G1 projectile, which was fired literally for years and thousands of times at different velocities over 19th century electromechanical chronographs until how much time it took to traverse every yard was known for a wide range of velocities. The Army Ballistic Research Lab ran the data on other shapes for the military using more modern equipment and completed that work and was shut down near the end of the 60's, IIRC. McCoy's book goes into this in more detail, though it's been awhile since I worked my way through it. Be aware that the book was published just after Robert McCoy passed away and before he could edit the proofs, apparently, as there are a number of errors in it. Don Miller and some of his other students have an errata list and corrections available on line for free.
Edited by Unclenick (07/15/12) Edit Reason: typo fix and added information

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#7037005  11/04/12
Re: The derivative and Integral
[Re: 22WRF]

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Registered: 01/15/07
Posts: 745
Loc: A stones throw from Henry's Fo...

.... 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.

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#7037515  11/04/12
Re: The derivative and Integral
[Re: MacLorry]

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Registered: 10/11/08
Posts: 871
Loc: Arizona, USA

Note that the link I provided is for the "Revised 2nd Edition" of Robert McCoy's book Modern Exterior Ballistics. I don't know for sure that the corrections were incorporated, but if it were just a second printing I wouldn't think Schiffer would use the label "Revised 2nd Edition” on the lower right of the cover. My copy is the Revised 2nd Edition as you linked to. It does incorporate the corrections list at this pdf: http://www.jbmballistics.com/ballistics/bibliography/articles/modeb9lc.pdfI have not gone through _everything_ on the list, but at least mainly the revision addresses the errata listed.

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#7041287  11/05/12
Re: The derivative and Integral
[Re: MZ5]

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Registered: 06/25/10
Posts: 3390
Loc: polar orbit

Note that the link I provided is for the "Revised 2nd Edition" of Robert McCoy's book Modern Exterior Ballistics. I don't know for sure that the corrections were incorporated, but if it were just a second printing I wouldn't think Schiffer would use the label "Revised 2nd Edition” on the lower right of the cover. My copy is the Revised 2nd Edition as you linked to. It does incorporate the corrections list at this pdf: http://www.jbmballistics.com/ballistics/bibliography/articles/modeb9lc.pdfI have not gone through _everything_ on the list, but at least mainly the revision addresses the errata listed. Thanks for that information. The other thing missing from the 1st edition is an index. Does the 2nd edition include an index?

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#7044831  11/06/12
Re: The derivative and Integral
[Re: MacLorry]

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Registered: 10/11/08
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