John Farnam writes [via

Conversations with “precision riflemen” often get into tedious, esoteric mathematical realms, and usually very quickly. Many factors influence the bullet during the brief fraction of a second (sometimes longer) while it is in free-flight between when it departs the muzzle and the point of ultimate impact (exterior ballistics), but the two main ingredients are:

• (1) air friction.
• (2) gravity.

Density of air varies, of course, with elevation above sea-level, humidity, temperature, and dust/smoke content. But gravity relentlessly pulls the bullet toward the center of the Earth. Wind blowing against the flying bullet, from any direction, will push it (to some degree) off its still-air trajectory.

There are other variables to consider, from the rotation of the Earth, to “ballistic drift,” to the “Coriolis effect” [ED: aka the aforementioned rotation of the earth] and the gravitational pull of Jupiter. The bullet’s shape, diameter and density [ED: aka ballistic coefficient] also affects the way it passes through the air, as well as intermediate impact with grass and brush.

All of these are of concern mainly to the artillery officer; they’re generally irrelevant for the practical rifleman.

With modern, autoloading military rifles, the bore-line/sight-line span is six centimeters (2.5 inches). Any bullet that impacts within a circle with a six-centimeter radius is sufficiently accurate for most serious purposes.

So, the instant our bullet departs the muzzle, it is already six centimeters low. With sights adjusted correctly, the bullet will then climb until it crosses the sight-line. With 5.56×45 (223) caliber, I recommend this intersection take place at 40m. This is the “initial intersection.”

The bullet then continues to rise until, under the influence of gravity, it starts to fall back toward the sight-line. This highest point in the bullet’s flight, the “maximum ordinate,” is reached at 140m. At this range, the bullet is six centimeters above the sight line, but no more.

As it continues downrange, the bullet then falls back toward the sight-line, crossing it once more (this time on the way down) at 240m. This is the “second intersection.” The bullet then continues to fall until it is, once again, six centimeters below the sight-line. This takes place at 260m.

So, between point-blank and 260m, the bullet is never more than six centimeters away from the sight-line, “all other things being equal.”

We call this “maximum point-blank range,” or MPBR. Beyond 260m, we are in the “back-curve zone.”

The foregoing holds true only when the rifle, as it is fired, is perfectly vertical. When a shooting firing a rifle from a “rotated” position, the six-centimeter standard collapses after 150m. Thus, I tell students that their M4 is basically a 150m gun. Within that range, it is deadly accurate, no matter the shooting position, no matter the wind, no matter most other factors.

The “back-curve zone” is that portion of the trajectory which lies between maximum point-blank range and the ultimate point of bullet impact. Any time a rifleman is working in ranges that involve the “back-curve” of the trajectory, that is, after the bullet has fallen more than six centimeters below the sight line (after the second intersection), the path of the bullet becomes harder and harder to predict.

The rotating bullet is designed to remain stable as it travels through air, but only so long as it remains supersonic.

The speed of sound in dry air, at sea level, at 70 degrees F, is 342 m/s (1,125 f/s). Modern rifle bullets are not designed for stability at subsonic velocities. When bullet velocity degrades to the point when it transitions into subsonic velocities, stability progressively suffers.

Today’s snipers use the back-curve zone; and they earn my admiration. But under “field conditions,” using generic military-style rifles, most of us will squander a lot of ammunition trying (mostly in vain) to get reliable hits in the BCZ.

The “academic/theoretical range” of your rifle may extend into the back-curve zone, with the issues described above. I suggest therefore, that the “practical/usable effective range” of your defensive rifle excludes the back-curve zone. The “practical/usable effective range” of your rifle is confined to MPBR, often a good deal less (as noted above).

When the rifleman tries to sight-in his rifle for a particular, extended range (beyond MPBR), he must adjust the sighting device so that the second intersection will occur at a range that necessitates a maximum ordinate in excess of six centimeters. When he subsequently attempts a shot at a shorter range (without a sight adjustment), it will probably be high.

In fact, the rifle will shoot “dead-on” only at the exact, extended distance for which it has been sighted. Without another, subsequent sight adjustment, “holding over” or “holding under” will be necessary for all other ranges.

The shooter will thus find himself with a “single-purpose rifle.”

When training for real fighting and equipping ourselves with an M4, XCR, SCAR, et al, fitted with a red-dot, in 5.56×45 caliber (we can probably include 300 Blackout also), so long as we stay within the MPBR, you’ll enjoy great success, and stop squandering ammunition on dubious shots.

About John Farnam & Defense Training International, Inc
As a defensive weapons and tactics instructor John Farnam will urge you, based on your own beliefs, to make up your mind in advance as to what you would do when faced with an imminent and unlawful lethal threat. You should, of course, also decide what preparations you should make in advance, if any. Defense Training International wants to make sure that their students fully understand the physical, legal, psychological, and societal consequences of their actions or inactions.

It is our duty to make you aware of certain unpleasant physical realities intrinsic to the Planet Earth. Mr Farnam is happy to be your counselor and advisor. Visit: www.defense-training.com

Robert Farago is the former publisher of The Truth About Guns (TTAG). He started the site to explore the ethics, morality, business, politics, culture, technology, practice, strategy, dangers and fun of guns.

1. The direction a pinwheel spins has nothing to do with the coriolis effect or its location relative to the equator. The direction a pinwheel spins is dependent upon the direction and pitch of its blades and the direction of the wind.

• I saw that graphic and immediately wondered what a pinewheel was. I googled it and found nothing. I was hoping to learn a new word. And some new physics based toy to demonstrate the Coriolis effect. Alas, I did not.

• Yeah, that was easily the dumbest misunderstanding of coriolis I’ve ever seen. Bonus points for being labeled ‘for dummies’. 😛

• There’s a small chance (approximately 100% +/- 0%) that my wife made a VERY strong gin & tonic and I’ve taken the liberty of editing that silly graphic out and adding a couple little editor’s notes in that section. TTAG will follow up with a blurb on our good friend, coriolis effect, in a post tomorrow. Spoiler alert: it’ll be shockingly similar to my comment made just below…

• Easiest way to understand the North/South effect of coriolis effect is to envision one of those playground merry-go-rounds (roundabouts). You’re on it and a friend is on it and you’re sitting across from each other on opposite sides and facing inwards. It’s spinning fast, and you throw a ball across to your friend. What happens?

If the roundabout is spinning counter-clockwise, you’ll miss to the right. So will your friend. It doesn’t matter who’s throwing the ball across, the ball will miss to the right of the target. This is the same as shooting North OR South if you’re in the Northern hemisphere*. If the roundabout is spinning clockwise, you’ll both miss to the left. This is the same as shooting North OR South if you’re in the Southern hemisphere*.

If you’re shooting East or West, regardless of hemisphere, coriolis effect makes you miss high or low as the earth is either coming towards your bullet or moving away from it. If you’re at the equator, the earth is spinning 1,037 miles per hour and at 45 degrees latitude (half way between the equator and the north pole, aka along Montana’s southern border) you’re moving about 733 mph. Were you to zero your rifle shooting North or South, if you take a shot due East you’ll hit high and due West you’ll hit low.

In most of these cases we’re talking about like 2 to 6 inches at 1,000 yards depending on caliber and, in the case of East/West, latitude.

At the North Pole, your only option is to shoot due South. And vice-versa. 😉
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Edit: hey RF, copy and paste the above into a new article, eh? 😛

• ‘In most of these cases we’re talking about like 2 to 6 inches at 1,000 yards depending…’

In other words, this is nothing anyone should ever worry about when hunting, even at longer ranges. (I’m assuming the effect at 500 yards is probably a quarter of the effect at 1000?) And inversely, it also shows how amazing it is when military snipers make a 2000 yard shot. Even with a spotter, a couple tries and the best equipment in the world, it’s just amazing that anyone could make a shot like that with a rifle. Or for that matter that a century ago (loooong before computers) every navy on earth had accurate tables for shooting the big guns out 14 miles.

• [Naval artillery]

” Or for that matter that a century ago (loooong before computers) every navy on earth had accurate tables for shooting the big guns out 14 miles.”

They most certainly did have computers back then, very intricate mechanical ones that compensated for the earth’s rotation, and a bunch of other variables –

“Gears of war: When mechanical analog computers ruled the waves”

https://arstechnica.com/information-technology/2014/03/gears-of-war-when-mechanical-analog-computers-ruled-the-waves/

“Even without digital computers, the Iowa could fire 2,700-pound “dumb” shells nearly 30 miles inland with deadly accuracy, within a circle of probable error of around 80 meters. Some of its shells had circles of destruction larger than that.”

• As a native Iowan, that just about brought a tear to my eye!

Still, just fricken amazing the accuracy of fire they were able to bring in WWI.

• I think the easiest way to understand the Coriolis effect is to try stirring a fluid (e.g., a bucket of water) as fast as you can clockwise, then try it counterclockwise. It’s more easily sustained counterclockwise, at least in North America. The further north you are, the more apparent the difference.

• But that has nothing to do with the effect the earth’s rotation has on bullet trajectory, which is the only reason to bring it up in the context of shooting…

• Blammo, the effect is *far* to weak to be noticeable in that bucket.

Somewhere I recall reading it takes a standing body of water at least 1 km in size before the effect can reliably start on its own.

It doesn’t work on as small a body of water like a toilet, for example…

• I have a little nit to pick here, Jeremy. Couple of days ago I watched a video about Coriolis effect on shooting high or low and they explained it same way you did:
“If you’re shooting East or West, regardless of hemisphere, coriolis effect makes you miss high or low as the earth is either coming towards your bullet or moving away from it.” That didn’t make sense to me since your projectile starts with the same eastward speed as the whole earth, so it cancels itself out.

The reason the hits on due east target will land higher than on due west target even if the distance to both is the same is centrifugal force. Gravity pulls us down with 1G of force, but rotation of Earth tries to swing us off of surface and works against gravity. Think swinging dead rat on a piece of string around and letting go. (You are lightest at equator and heaviest at poles.) The faster you or projectile travels east, the more rotational speed it adds to Earths own speed and the “lighter” it gets, hence high hits.
Traveling west, you decrease the rotational speed with which you started by standing still and with that the you decrease the centrifugal force and hit lower.

Isaac Asimov I’m not, so this might help if I’m not clear enough: https://en.m.wikipedia.org/wiki/E%C3%B6tv%C3%B6s_effect

2. I wish people would explain the whole “bullet climbing” thing better. Looking at that chart, it makes you think that the bullet will actually rise out of the muzzle for a while. It doesn’t, that’s impossible. What’s happening is that your rifle is actually pointed slightly upwards. That’s how you compensate for muzzle drop, the rifle (slightly) lobs the bullets like an artillery shell.

A rifle that was perfectly horizontal would fire a bullet that would experience no rise whatsoever; that bullet would only fall (due to the relentless application of gravity). The only reason a bullet “rises” at all is because the gun is pointed slightly upwards.

• “What’s happening is that your rifle is actually pointed slightly upwards.”

^ That. Right there.

You can see if in a very exaggerated form on ultra-long distance setups.

• So a ballistic arc then?

It’s super easy.

At some point your bullet rises to your line of sight (aka zero), after it hits apogee if falls back through your zero. Knowing those two points and how far it rises and how far it falls is helpful. The flatter the trajectory the easier it is. Knowing that your gun hits with only X rise/fall inches at Y distance means you can put your sights on the target and hit within X of your aim point.

So basically with your hypothetical gun put your crosshair/dot/irons on target and within 300 yards and you can make a reasonabe hit within 5″ of your aimpoint.

• “I wish people would explain the whole “bullet climbing” thing better.”

The way it was explained to me is, the bullet trajectory is like an underhand softball pitch. You ‘lob’ the bullet to meet the target.

Or was that explanation erroneous?

(Knowing the range of the first ‘crossover’ point is damn handy to know, I must admit…)

• Indeed. My question is though, is this because the barrel is slightly canted upward, or, are sights set up so that you slightly cant the gun upwards. Also, when did this start in firearms manufacturing?

• My understanding is that this is due to the distance between the sight (line of sight) and the bore axis. If a scope is mounted 2 inches above the bore, the two lines (bore and line of sight) must cross to converge at the point of ‘zero’. So, if the scope is level, the ‘bore line’ must tilt up some. Pardon my non-technical verbiage. Think of something more extreme, a rifle with a scope mounted 12 inches above the barrel-imagine how much tilt there would have to be for it to be zeroed at 5 yards. it is all about the distance between the sight and the bore.

It has always ben this way.

• Yeah, I generally stop taking people seriously when they refer to the bullet “climbing” out of the muzzle instead of referencing that the muzzle is pointed up relative to the line of sight.

• As a corollary to this, you never adjust your sights – they’re always pointing at the target. When you make sighting adjustments, you’re really moving your rifle.

Once people understand this, their ability to adjust any sighting system becomes much more intuitive.

3. Guess I’m weird but I always figured max point blank range was the end of my barrel device. After that it’s effective range on a point or area target depending on target.

• You and me both. Mr. Farnam is on my short list of people to never get training with, not least because there’s apparently a whole new vocabulary required.

• Well, today I learned that ‘point blank’ can mean so many things it’s almost useless. Instead of “MPBR” let’s just stick with “battlefield zero”.

4. Alright, I found this very confusing. Is the upshot of all of this that he is basically recommending zeroing at 40m??

5. Here’s a cool trick, given enuff land. You can use a 22. Have barrel parallel to the ground, hold an object( Rock) with a couple fingers. Pull trigger and drop rock at the same time, the bullet will hit the dirt the same time the rock does. No matter what the caliber is , fast or slow. MPBR depends on size of the target.

6. but does he make ‘turtle face’ when he does stuff? then he’d look like an utra cool hi-speed operator

7. A 40m zero? Why? I stick with what I used in the Army. 25m zero. Dead on at 25m and 300m. It’s always served me well.

8. “…when the rifle, as it is fired, is perfectly vertical.”

Vertical means pointed straight up……

which I’ve only done once, out in some dunes miles from anyone else, just to see if we could get the bullet to land near us.

• Not the best explanation of vertical (I thought of Julian Hatcher shooting his 1903 straight up and looking all over the beach for the returning bullet).

I believe this author means “don’t gang grip your firearm”. Magazine, chamber and sighting system in vertical alignment (unless you have a Sten).

• This is where the Coriolis force will have an effect. It is caused by the difference in the radial velocity and the linear velocity as you gain in altitude. The projectile inherits the linear velocity from its position on the surface of the earth but requires additional linear velocity to maintain the radial velocity as it gains in altitude needed to stay over the same portion of the earth. If I could draw or use a graphics program I would provide a picture. I can’t so my poorly worded description must suffice.

9. He discusses an AR with a sight center 6 cm above the bore center and then shows a chart using a hunting rifle with the sight center 1.5 cm above the bore. This is really poor writing and editing; it took a re-read to realize that the chart and text do not match. The gundata.com calculator and chart is available free. I am not sure what round he is using, because I get a longer second zero distance (using 55 grain Fiocchi FMJ as a proxy for cheap enough to afford for practice.)

10. “When bullet velocity degrades to the point when it transitions into subsonic velocities, stability progressively suffers.”
Not quite exact. This insinuates that the bullet’s accuracy degrades steadily as it approaches the point of going subsonic. It is actually the precise point that the bullet drops below Mach 1 that the accuracy suffers. The accuracy does get progressively worse from there on, but only because the bullet has suffered an unpredictable and uncontrollable yaw at the subsonic point. If launched subsonic, like the Sharps guns of the ‘Buffalo’ hunting days, the projectile suffers none of these effects.
Although not quite clear on this point, the rest of the item rates an A+.

11. The effect of coriolis on shooting is actually pretty simple. The thing to remember is that, because the earth rotates on its axis, both the shooter and the target are moving. Their speed depends on their distance from the earth’s axis of rotation. It is highest at the equator and decreases with distance from the equator. At 60 degrees north or south latitude, the speed drops by half. At the poles, the speed is zero. You aren’t going anywhere. You are just spinning in place.

If your target is north or south of you, its speed is different from yours. It’s exactly like shooting from a moving vehicle at a target in another moving vehicle. If the target vehicle is moving faster than you, you have to lead the target. If you are moving faster than the target vehicle, you need to aim behind to score a hit.

A rigorous analysis of coriolis involves advanced mathematical operations like the cross product of vectors. Spend an hour studying that stuff and you will be ready for a stiff drink.

• While the cross product of vectors is the most compact expression of the calculations needed, only simple trig is required. If you can do a back of envelope calculations using trig, you can calculate the Coriolis “force”. No advanced math needed.

Math Major, physics minor.

• Sadly, in today’s world, anything beyond long division is “advanced math”. I do basic trig almost every day using whatever paper is handy and the calculator on my phone to figure out ballistic problems.
I’m not smart. I’m certainly no math wiz. This is 10th grade math that anyone can do if they just apply the basics and think a teeny tiny bit.
But people generally think it’s witchcraft.

• After spending years calculating, I can promise you that with 5.56, if you aim a tenth of an inch to the left, everything will be fine.

12. That was the worst article I ever read on maximum point blank range
It’s use of 40 m as the near zero is a peculiar distance
Standard in the United States is the use of a 25 yard or 50 yard near zero
For years hunters have used the traditional 2 inches high at 100 yards
to get a maximum point blank distance of approximately 250 yards.
no matter which of the traditional hunting calibers you are using
The discussion one commentor used at Chuckhawks is an excellent discussion of maximum point-blank distance

13. It seems like the term “point blank” originally had very clear and practical meaning.
Early cannon had externally tapered barrels; larger on the breech end and smaller on the muzzle end. Sighting was done lengthwise along the top of the barrel, so the sightline and the bore axis crossed due to the angle.
Point blank ranges, max and min, referred to the distance where the trajectory of the shell crossed the sightline.
In modern firearms, the sightline and bore axis crossing is arbitrary and adjustable. Therefore, “point blank range” seems to have lost its clear meaning and I believe it is confusing and should be avoided.