Complex ballistic formula?

[backbencher]

B = ballistic coefficient
W = weight of bullet
M = muzzle velocity
H = height of sights above bore
Z = zero range
D = distance of bore sight

Given a bullet w/ a known BC, weight, & MV, the height of sights above the bore, & the desired zero range, what is the distance at which the bore sight and the sight line coincide?

Farmers Fight!

backbencher

 

[Col]

surely the bore sight and sight line are one and the same

 

[scythefwd]

COL -
If the sight plane was identical to the plane the bore axis was on, then the two would never converge. The sight plane is slightly inclined so that in the distance it does converge with the bore axis at a single line (in geometry, a plane is a 2 dimensional structure with no boundries, a line is 1 dimensional with no ends... there is exactly 1 line formed where to planes intersect).

Bullets don't actually rise when they are shot... they are shot at a slightly upwards angle so that the bullet crosses the line of sight through the sights. This happens at 2 instances, when the bullet rises up past the line of the sights and again when it decends back down through that line.

Back bencher, the bore line and the sight line are not relevant. Its the bullet trajectory and where it crosses the sight line that matters (the bullet is affected by gravity, so you shoot it up through the line of sight (the sight line, which is not effected by gravit, so it is perfectly straight). Like I said above, this will happen twice, once where it goes up though the line of sight, and once when it drops back through it on its way to earth. Theoretically, you could adjust the sights so that the apex of the bullet path is exactly where it meets the line of sight, but fuctionally it would not work out as well as having the bullet pass through the line of sight twice. It would require your sights to be set so differently than how they were designed to be set that you would run out of adjustment at a much closer shot. You might not be able to zero at 100y, or maybe even 200 yards if the sights are set at the apex of the bullet arc due to adjustment limitations on your sights.

 

[scythefwd]

back - I may have misunderstood your intent...

Could the math you want be done and give the desired result?? Yes. I'd rather spend 6-12 rounds and just zero once I've gotten bore sighted and on the paper.

 

[ranger335v]

"surely the bore sight and sight line are one and the same "

If the line of sight and line of bore were parallel the lines of sight and bore would never cross; the line of bore and line of sight cannot cross unless we 'tilt the rifle' and adjust it to do so. Bullet drop begins immediately when the bullet clears the muzzle and continues according to the laws of gravity and momentum until it hits something. Meaning we zero the piece to compensate for bullet drop so we can hit where we want it to.

So far as I know there is no simple formula we can use to calculate the first cross point for any given ballistic path. However, most any computer generated trajectory chart will allow for sight height so you can find where the intersection occurs. Given a conventional hunting round in the 2,700 - 3,000 fps range with a line of sight about 1.5" above the bore and zeroed to hit about 1.5" high at 100 yards, the first intersection will occur very close to 45 yards, being a little less than half way to the target.

 

[LiveLife]

If you don't want to do the complex math, there are online ballistic calculators (plug in your numbers and click "calculate").

This one even factors sight height, wind speed and direction - http://www.handloads.com/calc/

 

[Robert101]

Use a ballistic calcuator. Many are available on the internet. I use one called Strelok on my Droid at the range. Great tool.

They have values for wind, temperature, sight height, etc., etc. Much better than 3 hours with a pencil and paper.

 

[brickeyee]

surely the bore sight and sight line are one and the same

The "sight line" is the sights.

The "bore line" is usually below since sights are mounted on top (above) the barrel.

They are not even parallel in most cases.

The bore line is tilted slightly upwards so the bullet crosses the sight line on its curved trajectory twice.

Once rising closer to the muzzle, than again falling further away (the common 'zero distance').

 

[Rule3]

What about altitude, ambient temperature, humidity and the phase of the moon?

There's an App for that

 

[gamestalker]

Every cartridge with varied velocity, B.C., and sight axis is different in where the cross over occures. The formula is rather complex, but this information is readly available to us in books and internet sites.

My books are full of various perpetual exterior ballistic performance senario's and most are probably pretty consistent with actual cross over, at least the ones I've used. But other factors can come into play that will pose an impact on actual zero with a specific M.V. and B.C., atmosphereic conditions such as humidity, elevation, and temparature are all effecting elements. But based on those elements being constant most formual's will provide a close enough projection to be useful in a hunting enviroment.

I did encounter a situation that had a pretty big impact on actual zero when I zeroed my rifle at an elevation close to 8,000', and then hunting with it at 2,500' elevation. The variation wasn't enough to cause me to miss the animal, but it did produce a very low shot placement. But that is probably extreme circumstances considering the shot was pretty far out there. Now I'll factor in the elevation variance to some extent, if that variation is going to be significant.

 

[rcmodel]

The thing to know however is, whatever the math, or ballistic table, or ballistic software tells you is only a scientific WAG.

Bullet BC can change at different starting velocity, or different bullet stabilization in flight, as well as change over various ranges in flight.
Then you have different barrel harmonics, bedding, and a host of other differences in every different rifle.

Bottom line is, all the math in the world can get you fairly close, but it is no substitute for actually shooting the rifle and load on a paper target at distance.

rc

 

[Zak Smith]

There is no single closed form formula. Ballistic calculators run a set of differential equations and iterate over the bullet flight on the order of 1000 times per virtual flight second.

 

[brickeyee]

"There is no single closed form formula. Ballistic calculators run a set of differential equations and iterate over the bullet flight on the order of 1000 times per virtual flight second."

Most of them use the ballistic tables in an automated look up fashion.

 

[Zak Smith]

Most of them use the ballistic tables in an automated look up fashion.

Ah, no.

 

[Jim Watson]

I figure the early ones were a lookup system based on the Ingalls Tables.
I wonder when the home computer got good enough to do numerical solutions.

 

[Zak Smith]

On a 3 GHz processor from roughly 5 years ago, my ballistics program (which is not optimized for speed) can step 20,000 times (20 seconds of flight time) in 0.063 seconds of cpu time, or roughly 3.2 msec per second of real flight time. Even if the processor was 300 times slower, it would still calculate in "real time speed." If you're willing to wait 10 seconds to get results for data from 0-1000 yards, you could have a processor 3,000 times slower. Outside of loop administration, there are maybe 15-24 math operations (think assembly level math instructions) per iteration of the diff eq.

 

[Funshooter45]

here's the one I use. It does everything except calculate how many calories your lunch has in it.

http://www.jbmballistics.com/cgi-bin/jbmtraj-5.1.cgi

 

[brickeyee]

Ah, no.

Ahh YES.

You have decompiled them to look?

I actually have.

The tables are usually easy to find in a hex program listing.

The seven degree of freedom software (search on 'aeroballistics') is touchy stuff (I use it also).

It often takes days of tweaking to get it to come close to the tables, and they are measured truth.

 

[Zak Smith]

They use the BC drag function tables, but those are not extensive lookup tables such as the ones Ingalls published. Modern ballistics programs do not depend on extensive lookup tables of pre-calculated data. A simple program using the Siacci method and no a priori data other than the appropriate drag function is accurate enough to make first round hits at small arms distances.

If you meant they look up the mach region to get the Cd appropriate to each iteration, I do agree with you.

ETA: The best counterexample of your claim is McCoy's "mctraj" program (published in his book).

 

[gamestalker]

In real application, these calculations can help to get you close enough to make finding your true ballistic zero a little quicker. So I do use them. Right now I'm using a Leupold TBR BDC that interfaces with the optics MOA. So far the TBR has been right on the nose when I locate true zero or cross over.

Brickeyee said there are 2 cross over points, I respectfully disagree. If the optical line of sight is properely zeroed to the bore line of axis, the bullet path should converge with your optical line of sight, one time and then drop away, this is true dead zero. However, sighting in at closer distances than true zero, will deffinitely produce a cross over, maybe that's what he meant.

 

[Zak Smith]

It can cross once or twice (or none), depending on the circumstances. Here is an example of a typical LR .308 load with two different zeros dialed

from
http://demigodllc.com/articles/practical-long-range-rifle-shooting-optics/

 

[scythefwd]

Thanks Zak.. .that image says exactly what I was trying to say... but so much better.

 

[mbopp]

Eons ago (1980-ish) TAR published a ballistic program written in BASIC. Yes it used lookup tables and information from Hatchers notebook. I painfully typed it into my VIC-20 that had an ASR-33 teletype as a printer.
You could start a program listing, eat dinner, and it would still be printing after dessert. But it did work.

 

[backbencher]

Thanks all for posting. Yes, I would rather go to the range & zero, but my unit doesn't get range time. Yes, I'm in Iraq, and yes, it would be really nice to get range time, but no, it's not going to happen.

The zero range is relevant, as my adjustable sight cam is set for the 62 gn SS109 bullet out of the M4 carbine w/ the 14.5" bbl, based on a 300m zero. Yes, we shoot @ 25 yrd to approximate the 300m zero, but I'm trying to approximate the 300m zero w/ an admittedly inaccurate laser boresight. My eyes are not good enough to hold above or below @ even 25 m - I tried that - so I'm looking for an exact distance to place the rifle & check my sights. Given that the distance has to be calculated, it would be handy to have the formula IOT use it again someday. I take it there is no el neato formula available - it just has to be sussed out from ballistic calculators, which I have to learn how to use.

Farmers Fight!

backbencher

 

[Seedtick]

...Yes, I'm in Iraq....

backbencher,

Thank you Sir, for your sacrifice and service and that of your family!

Seedtick