I can find the equation for it for G1: https://www.sellier-bellot.cz/en/products/ballistic-coefficient-calculation/

But I can't seem to find the actual equation for G7.

The longer backstory:
I'm using LabRadar for backing out BC's.  In particular, I'm interested in determining if excessive bullet deformation from too-high-tension bullet seating has an affect, and how much of an affect.  I have some LabRadar trace data for clean and for deformed bullets.    So far, it looks there is an effect - quite a bit in fact, based on the few calc's I've done.

On the same topic, there's a lot of noise to the LabRadar data, and most advise suggests >100 yard data is more ideal for backing out BC's, to help temper out the noise effects.  It presents an interesting delima on a few levels.  For one: LabRadar is what I have, so that's what I have - I don't have >100 yard trace data.  I can get traces out to 75-80 yards or a little further but that's it.  Part of my conundrum is:
1) do I do my analysis based on 0 yards to 75 data?
2) do I do my analysis based on first ACTUAL reading (like 8 yards typical), to 75 data?

But even better, and past all that:

I can find on-line calculators that will let me back out my BC's, but it's about 20 seconds/analysis by the time screwing around.  And yet, with a LabRadar in a few shots you actually can build up a pretty big body of data points, conceivably enough to do thousands of analysis.

(Noise + huge-body-of-data)*Statistical_Analysis = good results (sometimes).

So I'm looking for the actual equation, so I can dump that into Excel, and get a running BC for each of the 50 readings or so along the trace of each shot.  Maybe do rejection algorithm based on LabRadars reported single/noise ratio column.  And maybe weight the further span data more.  Not sure, but was going to play around with this.  And to do that, I need the actual equation so I can plug that into Excel.

In theory I can calc BC based on muzzle, based on first data point, or even based sequential between each data point; which would be darned near a finite-elements analysis - though the noise effect there has probably got to be huge - though I am curious on that one.

So anyway, before I can any of that, I need the G7 equation - and for the life of me can't find the damned thing!  Somebody please post it if you have it.  Thanks!

So I did a little googling...  thought this should be relatively easy to come up with...  yeah, NO!

Found an online calculator for BCs (you pick the drag function) - here

Also found a wikipedia page that made my eyes glaze over...  but maybe you can dig out something useful here

- R -

Thanks for the reply!  I'm still coming up with nothing as well.

As far as I can tell, G7 either didn't exist, or was ignored as obscurity until Berger scientist Bryan Litz promoted it (maybe he even developed it?).   Indeed, I have a 2000's era Sierra book where the authors wax on about how G1 is the best model still, like they never heard of
G7.  - that wasn't that log ago.

Anyway, apparently G7 is discussed at length in Litz' book Applied Ballistics (2009 ?), which apparently induced an industry wide attention to the G7 coefficients.  I'm assuming that book will have the equation.    He seems like an interesting guy, I'd drink with him!  I can buy the book for like.. $50.  Heck, I might.  But really, right now I'm just wanting the equation.  There are public scans of the book available, sort of - but most are Optical Character Recognition scans; which get the text right; but OCR will face-plant when it comes to equations.  Er, especially if the hosting of the book via OCR scan is something I'm not even sure is authorized by the copyright holder, or is a project of ill-repute.

You might find this article (Accurate Measurements of Free Flight Drag Coefficients with Amateur Doppler Radar) interesting...  I got more eye glazing over but maybe you'll get more out of it...

So many articles and youtube videos.  All say the same thing.  None show what the actual mathematical formulaic difference by equation - just by graphics.   And not a one shows the actual equation used for the velocity calculation via G7 model.

It's like...

Here's the value for the speed of Gravity
g = 9.81 m/s*2.   And the area of a circle is pi * r^2.  Take the first derivative, and the circumference of a circle = 2 pi *r

And here's the Space Shuttle.  

See!? use our on-line calculator!

200 articles and videos.  Annnd...

Every...

...Single...

....... One.

Yeah, I've run into the same thing...  It is probably something as simple as changing the constant in the numerator of the S&B G1 equation to a constant specific for the G7 drag function!!!  

Well, I was able to get Litz' book via wife's Kindle Unlimited for free, for now.  I scanned through it, and still don't see a clean equation for V=f(BC,V0,x,constant)...

Looks like an interesting read.

As to the question that maybe it's the same as the G1 math, but with a different constant.  Man I hope not, the whole point is that the G1 ballistics model doesn't have the right shape of the structure of the curve of the response.  Change a constant won't change the shape of the curve; the equation should be different in structure (I would hope).

Here's an idea...  E-mail Brian and tell him what you are looking for - he'll probably shoot it right out to you!  

Wait, that's an option?  Do you have his email address?

Pretty sure Rex explains exactly how to come up with the G-7 BC  numbers in this video...

I skimmed through that last night. I did not see a mathmatical description for backing out BC from doppler velocity data.  Do I need to look more closely?

Nope, if you didn't see it, then its not in there, I thought he had talked about it but I haven't watched that stuff in a few years ....Will keep looking in likely places...

Nevermind everything I said below the line lol. I re-read and see you are looking for the effect of deformation.

If I were doing this experiment with my 10th grade math education. I would just look at the flight time to 75yds and   muzzle velocity of the deformed bullet(say 10 individually)
Then the same with non-deformed bullets.

Then I would open my ballistic solver input every possible input accurately as possible. Then adjust the BC to get the flight times to match.......

eh I guess you could do the same with velocity but I'm not sure which is going to be more precise from the Labradar, velocity at 75 yards or flight time to 75yards or even measuring to 75 yards.

It sounds like fun, good luck.

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------'--------
Are you doing this for fun or because you need a G7 that isn't listed? I think using 60-100 yards of velocity loss is nearly useless in calculating BC.

If you just want the G7 BC....

You can get a pretty good idea of G7 just by the shape of the bullet provided it has a form factor close to 1.0 by using its sectional density(bullet weight in lbs devided by its caliber squared).

If you know the form factor, many bullets are getting into the mid to low .9s now, then just devide the sectional density by the form factor.
Attached File


Attached File


If you get out your measuring tools you can figure out the form factor.
Attached File

Hey thanks for the extensive reply!  And all good, I put up a lot of text, so no worries if you didn't catch exactly what I'm trying to do.  Basically, I'm looking for the simplified formula that the ballistic calculators use to calculate velocity loss as f(BC, V0, etc).  Every ballistics app and web-calculator out there has whatever that equation is, in it.  Then reverse the formulae to use LabRadar data to back out the G7 BC based on velocity (or flight time) data.  Nobody, (not even Litz based on a skim of his book; maybe I missed it), actually publishes that.  They all publish things like the equation posted based on a form factor, which require way more intimate detail than I'm going to have - and won't work on a deformed ogave anyway.  Which is interesting in and of itself - as the math is so simplistic this way; it doesn't account for the question of is there some irregular shape that might somehow be even better?  (I doubt it, but who knows - they said the Shelby Daytona wasn't shaped right either - and then got thoroughly schooled by it).  Most research I find is exactly what you posted - form factor etc and calculating G7 based on inputs nobody actually has.

The actual process should be very easy - drop in equation in Excel, then drag-copy and make it crunch away a few thousand times with every possible way to look at the data.  I just can't find the formulae anywhere - odd.

It's a curiosity project really, to determine:
1) Are some of the deformed bullets upon seating interchangeable with the rest of my ammo, or is that such a significant difference that it won't have the same flight characteristics?  
2) Are the factory BC's accurate? (e.g. S&B's BC for their 140 gr 6.5 CM is known to be way WAY off)
3) (and it's just one of those investigations on how-stuff-works, that might make me smarter on how-stuff-works.)

If I recall Litz does his BC testing, simply put, with microphones, one to trigger a clock and one that "listens" for the hit on target(or perhaps the sonic crack) to stop the clock. Then calculates BC based on flight time.

It is over very long distance to help reduce variables in measuring errors I suppose.
I saw him doing it on one of his videos.

I vaguely racall him mentioning something about the difficulty of getting BC from velocity loss. I could be wrong.

But I see what you're saying, if the B calcs can solve for velocity, tof, etc one should be able to swap the formula around.

I think that might get deep into kalculous, I can't even spell kalculous. It's all witch craft to me at that point.

So I actually DID hear back from Brian Linz, which I thought was cool.  But when I let him know what I was doing, and wanting the mathematical equation for G7, I got ghosted.  Ah well, no sore feelings there.

From what I can tell, G1 and G7 ballistics behavior and equations in the near-range, are essentially identicle (i.e. in the first few hundred yards for a rifle).  And since my data only goes out to 75 yards or so, I guess I can just do the analysis with the simplified S&B version.  Which, after converting from SI to Imperial units

G1 = 0.00875068843 * x /(sqrt(v0)-sqrt(vx))
where x is yards
v0 and vx are ft/s

Dumping into Excel for just one bullet trace, to get the game of statistics started, and one can see so many different things.

FACTORY published Hornady G1 ballistic for 143 gr SST: 0.625

BC average of a prestine bullet, and BC average of a seating-deformed bullet (ring around the ogave from compression of seating via the die).   All told, 20 shots, with 5 different analysis styles of each shot; so about 200 calculations done on each shot.  All based on the above equation.

Average G1 BC = 0.611
Average G1 BC of ogave crushed = 0.617

Conclusion: there was no meaningful change the BC of the bullet by compression of the ogave via the seating ring (based on the velocity data within 100 yards, at least)

But let's also dive a little deeper.  There were 5 methods for measuring the BC.  Here's an example of the amount of data each shot collected:

Time (s)Vel (fps)Dist (yd)SNR
02709.770-
0.0220212680.0919.7529.09
0.0230212677.0420.6528.43
0.0240212676.8521.5426.66
0.0250212674.4322.4325.15
0.0260212673.2823.3227.25
0.0270212672.5624.2124.52
0.0280212670.7525.126.78
0.0290212668.8225.9927.11
0.0300212669.2626.8823.72
0.0310212666.727.7723.49
0.0320212665.3128.6624.82
0.0330212663.7929.5523.16
0.0340212663.8530.4422.19
0.0350212663.5831.3320.69
0.0360212662.3732.2123.05
0.0370212659.0133.122.63
0.0380212658.7233.9922.56
0.0390212655.6534.8722.06
0.0400212654.4535.7620.29
0.0410212652.9836.6420.24
0.0420212651.4637.5318.85
0.0430212650.3938.4116.79
0.0440212652.1239.2917.24
0.0450212648.1440.1816.76
0.0460212648.4641.0619.55
0.0470212644.7241.9416.27
0.0480212642.142.8217.13
0.0490212643.3443.718.25
0.0500212644.2344.5816.19
0.0510212641.8945.4718.24
0.0520212637.2146.3515.2
0.0530212638.6547.2312.6
0.0540212636.7248.116.17
0.0550212635.248.9812.78
0.0560212632.3249.8615.63
0.0570212634.8850.7414.02
0.0580212631.7551.6213.62
0.0590212628.8352.4914.54
0.0600212631.0853.3715.38
0.0610212627.8354.2513.45
0.0620212629.7555.1213.69
0.0630212621.615610.85
0.0640212622.6856.8711.12
0.0650212626.1457.7511.12
0.0660212620.658.6210.4
0.0670212621.3859.4911.28
0.0680212618.8860.3714.53
0.0690212616.7161.2415.3
0.0700212618.4462.1112.54
0.0710212614.5962.9913.7
0.0720212617.1963.8611.41
0.0730212613.9864.737.83
0.0740212611.2765.611.95
0.0750212607.9966.4714.93
0.0760212606.6667.3411.45
0.0770212609.1868.2110.45
0.0780212604.9569.0810.51
0.0790212603.4869.9514.59
0.0800212604.1270.818.94
0.0810212608.0771.6811.94
0.0820212602.8872.5511.81
0.0830212605.0173.4211.26
0.0840212599.5474.2913.77
0.0850212596.6775.157.9
0.0860212598.4976.028.33
0.0870212593.1276.8811.56
0.0880212590.8677.759.92
0.0910212590.280.348.79

Graph that:

Note how the first actually tracked data doesn't show up until the bullet is 20 yards out, and the rig internally extrapolates to give you a reading for muzzle velocity (that lone data-point on the axis, high and left on the chart)

So let's run Ballistic coefficient serveral ways:
Method 1: Distance taken as full 80.34 yards all the way back to muzzle.
Method 2: Distance taken as the 80.34 yards, but ONLY back to the first actual reading (i.e. the distance and velocity change that occured starting at 19.75 yards out, and ending at 80.34 yards out)
Method 3: Distant taken as the LAST data point with a signal to noise Ratio (SNR) ratio of >10 (i.e. the 77.75 yard data point, taken back to 19.75 yards) - meaning ONLY actual readings are used, and only good quality ones.  At the trade-off of a shortened data-span (which can hurt the analysis)
Method 4: Measure the BC for EVERY distance referenced back to muzzle (i.e. calculate a new BC for every data point), and then average all 70 of those BC's.
Method 5: Similar to above, but reference it back to first measured velocity data point (i.e. 19.75 yards), and then average all 69 of those BC's.

In a perfect world, all 5 methods should give the exact same BC.  In the analysis, less so.  The method that gave the highest standard deviation out of 5 shot group, was method 1 and method 2; which are the two methods most people would use.  Method 3 was better than those 2, which isn't surprising, since it's only run good SingaltoNoise ratio'd data.  The method that gave the lowest standard deviation out of a 5 shot group, was Method 4.  This was relatively consistent with all 4 strings of fire.

This is an old thread, but the Labrador velocity number aren't really useful for tracking BC.  You need data from much further down range.  At the close distance the MV difference and slight BC difference in each bullet and any potential tracking errors are going to do nothing but make your head spin.

BC differences don't show up much until much longer ranges than the labradar can track and far what you dial and TOF.

To follow-up, I still can't find the actually equation for G7 BC.  There is the JBM site that does it for you

https://www.jbmballistics.com/cgi-bin/jbmbcv-5.1.cgi

How well does it work, and a LabRadar work, at backing out a BC?

I have some Berger 73 gr HPBT bullets.  It has a Berger Published
G7 BC = 0.178
G1 BC = 0.348
(personally, I've grown to question the 3 sig-figs of precision on such, but sure)

Ran that bullet in my LabRadar out of multiple different rifles.  I randomly picked one group, of 5 shots.  There is some noise, to be sure - but how would it look if you do the basic: pick one in the middle analysis?  Let's start:
I'm running the COSOLIDATION... xls form another member posted, which has built in G1 calculation (but not G7), then JBM for G7 calculation, based on MV vs farthest shot with green S/N.   THEN a shorter span G7 calculation using a high S/N ratio NEAR muzzle, compared to highest S/N ratio near farther out (which will have 50-75% span distance of Method B, but higher quality data).
So, to recap Calculation Methods are
A) .XLS LabRadar CONSOLIDATION spreadsheet built-in G1 (I'll try to link to the thread where that was posted later)
B) G7 via JBM website (so manually have to enter), based on MV vs farthest shot with a GREEN S/N ratio (i.e. greater than 10)
C) G7 via JBM website, based on closest NON MV (note, MV is not actually real, but calculated w/in LR based on extrapolation) vs farthest shot with a HIGHest S/N ratio in that clumping, (that's not likely the farthest Green BC)  (this route gives the highest quality measured input data, but at the cost of a lower span of distance)



Shot #   G1BC   G7BC (via JBM)  G7BC using lowest S/N ratio
BOOK     0.348     0.178                  0.178
--------------------------------------------------------------------
1           0.384
2           0.337
3           0.368         0.180                    0.179
4           0.366        
5           0.377
Avg        0.366

So in this snapshot, That actually went pretty good.  The G1 is a bit off, but the G7's came right on top of each other.  Obviously I need to repeat the exersize over numerous diffenent groups and barrel lengths evaluated; so my confidence, in the confidence of this analysis - I can't claim to be very high right now, but it's a snapshot.  I have the data already, it's just the JBM G7 manual entry is laborious (which is why I really really want to find the darned G7 equation somewhere.  I've tried reading it up in Litz materials, and even reached out to him, but I have yet to secure the simple formulae for G7!)  - and I'm late for a Range shooting fun-day scheduled today with a friend, so - time to go shoot'n instead!  

Found this....
https://bc.geladen.ch/labrabaco/labrabaco.html

There is no equation for it.  Drag functions are measured values of the CD (they can be multiplied by different constants depending on use and development).  It's a table of values for velocity or mach number for the standard bullet for the drag function.

The G7 values are https://jbmballistics.com/ballistics/downloads/text/mcg7.txt

Not true.  The G7 along with the G1, G2, G5, G6, G7 and G8 have been around for a long time.  Sierra did one thing and that's it.  I would not take their omission or lack of knowledge as the final word.  The G1, G5, G6 and GL drag function values were listed in a book by Winchester-Western copyrighted in 1965.

Your not going to find an equation for the BC.  There are some approximations, but they are just that.  They are typically found by measurement of velocity(s) and or time.

The JBM calculator (it's my website so I know how it works) does a binary search on the BC integrating the equations of motion until it finds a value that is close enough.  Not particularly hard once you can calculate a trajectory, but starting from nothing, it will take you some time to figure it out.  It's as good as the data you input.  The problem is getting accurate measurements.

If you want to calculate BCs from multiple range data points, it gets a bit harder.  To do it right, you have to do a non-linear least squares which involves derivatives of the integrals of the equations of motion.  I've done it, but it's not trivial.

I have to admit, I'm a little confused as to what the OP is asking for.  An equation for the drag function?  An equation for the BC?  A method for calculating G7 BCs?  I don't think simple methods for the first two exist (there are no references to this kind of thing in all my books except for approximate methods with limited applicability).  As for the third, I outlined my method above.

You're right to question the three significant figures.  I think (and this was backed up by Bob McCoy the last time I talked to him) most manufacturers know their BCs to within 5% if they are lucky.  I won't speak for Litz' measurements, I don't remember if he discusses his accuracy or not in his books.  Some of you may remember Sierra changing all their BCs years ago because they found a systemic error(s) in their process - unfortunately, they went down, not up.

Thanks for the calculator, I use it often...

The only thing I've seen for BC values is on my kilo 2400 RF, they provide a trueing page where you can adjust both velocity and BC for any given charge to make sure its matching your rifles cartridges actual velocity and BC...from muzzle out and as far as you shoot..

The trick is knowing how to change the values to minimize the error!

You pick a target, range it, dial the dope it has and shoot it, then record the hit versus what it called for dope, you can do it with both velocity and BC...

My point was that you typically have errors in both the BC and muzzle velocity (measurement).  Additionally, you have errors in muzzle velocity due to temperature.  There are also errors in air density, etc.   Until you quantify all of these, just changing BC or muzzle velocity is probably not the right thing to do.  It may work for some ranges, but does it work when shooting uphill and downhill?  What about temperature extremes?

You're error may also be due to low velocity error in the BC.   Adjusting the BC globally to account for this may give you more error than you want/need at other ranges.

With the AB program you list temps you shoot at and the results of it..As for up and down, so far yes its been dead nuts...better than nothing for sure...The only thing I have seen it not accommodate is velocity fall off for barrel wear...not a huge deal, I normal shoot mine past the labradar for everything but hunting so not a huge deal....

Neat!  Thanks for hosting that!

The reason for the desire to have the equation (if there is one), is that with a LabRadar, one can build a rather substantial body of admittedly noisy data.   Each shot has at least 20 readings.  There is a large ability of analysis within each shot, and one builds up 5 shots per group, and multiple groups, creating a very large body of data.  With that, if there were a simple equation, one could then run G7 BC calcuations on every velocity at  every other velocity distance on each shot, and build up hundreds of calculated G7's, on each shot.   Or a great number of other approaches, as the computer can do the the math.  And then do that for hundreds of shots.  And then cross compare what kind of number the different types of analysis deliver.   Might be pointless, might average out to pretty close to the real number.  

But I'm confused, a basic ballistic calculator uses the user provided G7 BC, MV, distance of shot, and environmental conditions, to then calculate the flight time and velocity at distance.  as well as drop.  Is that not an algebraic equation, to get that?  So What's that equation?  Can't one just do some algebra to flip it around, and so input the MV and distant velocity at a known distance, to back-out the G7BC from the same equation?

No. There is no simple algebraic formula that will give you good answers in a general sense.   In mathematical terms you have couple differential equations that have to be integrated numerically.  

There are approximations but none work as well or are as general as doing the integrations.  

The point mass equations I use are https://jbmballistics.com/ballistics/topics/cdkd.shtml. These are vector equations for acceleration that are integrated to get velocity and again to get position.

Well darn.. ok, thanks.  I've done PDE's and Orthogonal colocation of finite elements.  My gumption for doing that, and throwing in some LaPlace Transforms, isn't quite there for this right now!