Sunday, February 19, 2017

Gravity's Rainbow

Thomas Pynchon's 1973 novel depicts, among other things, the arc of trajectory of the German V-2 rocket.

It's a very good book, even though Time Magazine called it one of the best 100 books published between 1923 and 2005. I mean, shit, Time Magazine? On the other hand, and speaking of shit, the Pulitzer Prize Advisory Board refused to award their 1974 prize to the novel -- despite the up-vote of the Prize Jury -- because they were triggered by a single short passage involving coprophilia. So it's got that going for it.

Although it's a good book (I believe I'll reread it soon) it has a wide scope, a high level of complexity, and a zillion characters. You really have to work at the chore. It's worth expending the time and effort, but it's heavy sledding in places. Because Pynchon.

And speaking of gravity, arcs of trajectory, and chores that are worth the effort, I went shooting the other day.

It was a very nice day in this part of the world. Sunny and warmish, with the temperature flirting with 60 degrees and a light wind out of the south.

The chore I undertook was to sight in one of my AR's with a new reflex sight.

While the rifle is a genuine Colt and is most certainly built on the AR-15 system, it's actually an LE6920, the non-select fire law enforcement version of the famous M-4.



It was manufactured during the "assault weapons ban" which began in 1994 and expired in 2004, and that is why it is so marked. Once the sun set on the legislation, it became perfectly legal for civilians to own such infernal machines, and I bought this one within 24 hours of the ban's expiration. It was a right celebration, it was!

As it came from the factory the rifle was equipped with a classic AR-15/M-16 style carry handle/rear sight unit. With a bit of a twist. The carry handle/rear sight was detachable, affixed to a Picatinny rail mounted on the top of the receiver. It does retain the classic triangular gas block/front sight unit, permanently affixed.

For many years I left the carry handle in place and relied on the iron sights. It's a good sight setup and I could shoot that rifle, I could! Over the last couple of years, though, I've been experimenting with various optics.

Out in this part of the world the horizons are far away, and a scope brings faraway stuff into closer view, increasing accuracy. In theory. The rifleman must still do his job. I mounted a 3X9 power scope with a 5.56 BDC (bullet drop compensator) reticle and shot it for a year or so, but while it did a proper job I didn't really like it. Too heavy and bulky and it made the rifle unbalanced. It also came with the restricted field of view that rifle scopes bring to the party. I hate giving up that field of view. It's like looking through a soda straw, especially at the higher powers.

Also, the 5.56 NATO round isn't really a long-range cartridge, and a 16 inch barreled M-4 isn't really a long range rifle. It's a carbine. It's good -- really, really good -- out to 300 yards, and can be quite effective out to 600 yards.

For me, anything out to 300 yards has always been iron sight range, and I've generally been pretty good with the irons out to 600 yards. Except...

Well, I'm getting old and presbyopia has made it hard to focus correctly on the damme iron sights!

So what I really wanted was a good red dot or reflex sight. These sights are either a short tube or an angled glass in a robust frame. Both kinds project an illuminated dot or other aiming point onto a collimating glass the field of view of the shooter, somewhat akin to the trusty HUD.

Old eyes love red dots! There's almost no parallax with these sights so the aim point remains aligned with the bore axis, regardless of the position of the shooter's head or eye. So long as you can see the dot, it's lined up with the bore and shows where the bullet will hit. Well, to be precise, where the bullet will hit at the range the rifle/optic is sighted in for. That's why it's important to be able to eyeball ranges with reasonable accuracy and know the trajectory your rifle shoots with different loads. More on that in a bit.

Another plus of the red dot sight is that you can mount it farther forward on the rifle, which allows for a larger, less restricted peripheral field of view. You can keep both eyes open while pointing the rifle, just as with iron sights, and the dot naturally appears in your field of view. It's almost as if the sight disappears, leaving behind only a cheerfully illuminated dot superimposed on the target. There's no frantic searching for the target through a soda straw.

Unfortunately the good red dot sights are pretty pricey.

But I found one that's affordable and had good reviews from people I trust.

I bought the Holosun Micro with a chevron aiming point and BDC dots for 400, 500, and 600 yards.

It's got a clever motion sensor which automatically switches on the aiming point when you shoulder the piece, and automatically shuts it off when it's not in use. This extends the battery life to something like 50,000 hours. Battery life used to be the big bugaboo of red dot sights. Nothing like losing your dot at a critical moment.

I also bought a Vortex VMX-3T magnifier with a flip mount.

This fits on the rail behind the Holosun and stays flipped to the side, out of the way until you need it to magnify your sight picture for longer range shooting. Push a button and it unlocks, flips up, and locks in place behind the sight, magnifying the view.

The Holosun is very good for quick shooting, allowing you to pick up the target while maintaining an excellent peripheral field of view. It's light, and even in combination with the magnifier the rifle isn't overbalanced. It's also pretty easy to sight in. The adjustment screws have a very nice audible and tactile click, and each click moves the dot 0.5 MOA, or one-half inch at 100 yards. The screw caps have a handy driver blade molded into the top and can be used to turn the adjustment screws.

One of the delights of sighting in the 5.56 NATO round, which is similar to but not exactly the same as .223 Remington, is that it's a very flat shooting round.

What does "flat shooting" mean, and why is it a delight to sight in?

All rounds have a curved trajectory when they are fired. This is because from the moment they leave the muzzle and are no longer supported by the barrel (when shooting more ore less horizontally), they begin to fall. That's right, gravity affects bullets in exactly the same fashion as it affected the hammer and feather dropped on the moon by Astronaut Dave Scott during the Apollo 15 mission on August 2, 1971.

Here on Earth objects fall at roughly 9.81 m/s^2 (meters per second squared) or about 32.2 f/s^2 (feet per second squared). It's a little more at the poles and a little less at the equator but here at 41 degrees north latitude 9.81 m/s^2 is close enough for government work.

When I say objects fall, I mean that objects not at rest -- unsupported by the ground, a structure, or some other object. Gravity's acceleration is essentially constant at the surface, and that acceleration is the "force of gravity" that gives objects weight and holds them in place when at rest. A ball laying on the ground is at rest. The same ball sitting on top of a ladder is also at rest. Nudge the ball off of the ladder and it will fall toward the ground. When I say fall I mean accelerate toward the center of Earth's mass, which is roughly located at the center of the planet.

This is an important concept, that objects accelerate rather than fall at a constant rate. Let's say our ladder is very tall -- several hundred meters. In the first second the ball will fall 9.81 meters, but it continues to accelerate at the same rate, so that during the second second it falls twice as far/fast. At the end of two seconds, then, it has fallen nearly 30 meters -- 29.43 to be exact (9.81 in the first second and 19.62 in the second). In the third second the ball's velocity and distance traveled doubles again, adding an additional 58.86 meters of travel for a total of 88.29 meters. At the end of four seconds it will have fallen 176.58 meters and will have achieved a velocity of 176.58 m/s.

Well, the ball would behave that way if it were falling at Earth's gravitational acceleration in a vacuum. Earth's surface is not in vacuum, though, so the ball will be encountering air resistance the whole way. At some point it will reach terminal velocity, an equilibrium point where gravity's pull is balanced against the force of air resistance. In aerodynamic lingo air resistance causes one type of "drag."

Now wait! What?

Okay, physical law time. Let's look at the motion of objects on our planet. Balls or bullets, they both obey the laws of the real physical world.

So, Newton's laws of motion.

1. An object at rest will remain at rest, and an object in motion will remain in motion, unless acted upon by an unbalanced force. This law is also called the law of inertia.

2. When force(s) act upon a mass, the result is acceleration. The mathematical expression of this concept is F=MA, or force equals mass times acceleration.

3. For every action there is an equal and opposite reaction.


Newton's law of universal gravitation: All objects attract each other with a force directly proportional to the sum of the masses and inversely proportional to the square of the distance separating the objects.

Before a round is fired the bullet is at rest. When the round is fired expanding gasses from burning gunpowder apply an unbalanced force to the back of the bullet, causing it to accelerate. As the force pushes the bullet down the barrel, an equal and opposite force pushes the rifle against your shoulder. Once the bullet leaves the barrel its inertial tendency is to continue in a straight line at a constant velocity, however, it is subject to the action of the unbalanced forces of gravity and air resistance, including wind, so it begins to accelerate downward, shed velocity, and drift with the wind. The fact that the bullet is going really fast matters not a whit, it still obeys physical law.

Now that I've belabored the point, let's think about what happens to a bullet when fired from a gun. If we shoot it straight up (for reasons which should be obvious don't try this at home) it will have a great deal of initial velocity, and if it's a 5.56 NATO round out of my rifle that will be just over 920 m/s at the muzzle. From the moment it leaves the muzzle, however, it will begin to lose velocity as both gravity and air resistance apply force to it. A bullet isn't a rocket, it has only the kinetic energy it gained from being blasted out of the gun barrel by expanding gasses, and no more. For the 62 grain bullet fired from my rifle, that's about 1,690 joules or 1,247 foot pounds. From the moment of firing, gravity and air resistance slow the bullet's upward travel. At some point the bullet will stop climbing, and this is the point where all of its initial kinetic energy has been converted to potential energy (see the first law of thermodynamics).
This is a chart of an altitude/time plot of a 30.06 round fired vertically. The 5.56 would be similar, but I'm too lazy to crunch the numbers, so I'll cheat and use this, which I've cleverly stolen from the interwebs. S

At the top of the arc, for the briefest of moments, the bullet is at rest. But it can't stay there, because it has no more kinetic energy with which to overcome gravity. Gravity takes over and the bullet begins to accelerate toward the center of the earth, neatly turning potential energy back into kinetic energy. It accelerates until drag and acceleration balance at the speed called terminal velocity. When it hits the ground (or perhaps your head), most of the kinetic energy is transformed into heat energy, but that's a different post.

Okay, how does that apply to sighting in a rifle? The important bit to understand is that as soon as the bullet leaves the barrel it begins to fall. Well, that's not precisely correct, unless the bullet is fired precisely horizontally, or on a plane exactly perpendicular to the ground. If it is fired precisely horizontally it will fall at 9.81 m/s^2. If you fire the round horizontally from a bench height of one meter the bullet will strike the ground in about half a second (0.45s) with a downward velocity of 4.42m/s (14.5f/s) and about 0.04 joules of energy. The bullet's trajectory will be a continuously steepening downward arc.

Because of this, it's useful to set up our rifle sights so the bore axis is pointed very slightly upward with respect to the sights. That way, when the round is fired the bullet's trajectory begins with an upward arc. Its ballistic trajectory actually takes it above the line of sight initially. Gravity begins pulling on the bullet immediately, but just as with the example of firing straight up, it will climb until its upward inertia is overcome by the pull of gravity.

And why do we want the bullet to begin its journey on an upward arc? Primarily because we would like to be able to aim the sights dead on the target at the most likely or convenient range. If we fired with the bore axis aligned perfectly with the sight axis, the bullet would always hit below our aim point because, with no upward inertia, it would begin to fall immediately. The slight upward trajectory means that we're lobbing the bullet down range.

This is where bullet velocity comes in, and where we can define the difference between flat shooting and not-so-flat-shooting rounds. The higher the velocity, the farther the bullet travels before it hits the ground. If you fire a .45 ACP and a 5.56 NATO side by side and perfectly aligned over a flat surface, both bullets will hit the ground at exactly the same time, but the faster bullet will have gone a lot farther.

Therefore the arc of trajectory of a fast bullet appears to be flatter than that of a slow bullet. You can hit a target at 200 yards with both the .45 and the 5.56, but the arc of the .45 will have to be substantially higher.

Higher velocity means greater accuracy over distance, too, because the bullet is affected by unbalanced forces for a shorter time and and therefore moves a smaller distance from the point of aim.

Here's the trajectory plot for my rifle.

And here's the tabular data.

As you can see, between 50 and 300 yards the round will hit within seven vertical inches of the point of aim, so long as I do my part. That's handy. Here's my first three-round string at 200 yards. It's a three-inch spread, or about 1.5 minutes of angle (MOA) accuracy. How I managed to hit the center of the bullseye with one round I have no idea. Really just an accident.

Here I'm tinkering with adjustments.

And here's the final result, a 20 round group (one flyer) that I'm happy with.

And I suspect that's more than enough for today.

Happy reading and happy shooting!


  1. I find that somehow, for the past several days, I have failed to visit you here. Shame on me. I went back in you posts and read them and shall view the videos of those post as I have time ( which I don't today ). I have yet to fail to enjoy any of your posts and shall endeavor to remember to stop by daily.

    Paul L. Quandt

    1. Thanks Paul. You can hardly be expected to tune in daily if I can't be bothered to post daily. It's always here though, so no particular "bat time" is required. Thanks for stopping by and commenting.

  2. Gravity. Science. Shooting.

    Good stuff.

  3. I am glad your Holosun worked out. I have an EoTech on my Ruger AR556. Which needs to be sighted in.