Trajectory

I had a thought about trajectory: The common theory is that trajectory is a smooth arc from muzzle to target. Is that really true? Could it be that a bullet wavers up and down along that arc due to atmospheric changes? As a bullet follows trajectory it rises and drops. Along the rise to apogee and descent to target there are minute changes in air density which could cause the bullet to jump up or drop a little. Also, could the makeup of the ground over which the bullet travels have an effect? Different substances on the ground reflect heat in different ways thereby affecting air density which could affect bullet travel. Wind currents may play a part as well. We know the effects of crosswinds. Could wind move a bullet up or down as well as sideways? Bullet size, shape and speed may have an effect.

So, my question is: Is trajectory a smooth arc or a wavering path? How about some thoughts from any Physics types or ballistics experts out there.

I shoot o occasion f class so i am intetested in the subject. And your thoughts are right.
Add to this coriolis effect and spin drift. And the courve of trajectory definition complicates.
However, previos defonition fine arc is not wrong.
It only depends of which model you will need to use and what you want to achieve.

I will give you example.
In old greece, the beleif was the earth is flat. That model perfectly satisfied the needs of then chartography and with piece of mind, they could sail and keep trade accross mefiteranean.

By the time of christophor columbus, understanding was the earth is round. For the chartography in times of great geographical discoveries this model is ok.

Now model is that the earth is basel elipsoid, useful to the point. But for fine positioning like in offshore drilling other models are accepted. Corrections will have to be applied for northing and easting.

So, for ballistics, if you will shoot pistol at 25 or 50 yards, or rifle up to maybe 200 yards, definition of fine arc as a model of trajectory is acceptable. If you want to shoot at 1000 yards, other definitions or model of trajectory should be considered.

Luckily, we live in modern times, and target shooters can download ballistic appliances for such calculations.

I'm not a physics type nor would I consider myself (by strict definition) being a ballistics or any other type of expert, as there is no such thing as an expert, just more knowledgeable people.

To help answer your question:

Bullet/ projectile: size, shape, weight, mass, design, velocity all factor into trajectory.

Wind: direction, speed, current, temperature, also factor into trajectory.

Weather: humidity, temperature, air mass, elevation (altitude), also factor into trajectory.

Earth: gravitational force, terrain, curvature, also factor into trajectory.

Barrel: design, length, smooth or rifle bore, straightness, harmonics, material, density, to stock fit, also factor into trajectory.

Recoil, shooter's stance, shooter's position, fit to shooter, also factor into trajectory.

Firearm action: single shot, side x side, bolt, semi or fully automatic, bolt, rolling block, trap door: fit to barrel also factor into trajectory.

Now the question to you is: How scientific do you need or want to get to understand trajectory?

Mathematics is friend and foe and a bit of a pain the ass.

Try being an artillerymen where the projectile travels over tens of kilometers. Back in the day we worked manual met data; used slide rules developed for each type shell and charge fired; maintained and constantly updated internal ballistic data caused by the changing dimensions of the barrel of each artillery piece in the battery; adjusted range elevation based upon the relative altitude of gun and target; fired registrations to weed out anything else; and did it all again every few hours as conditions changed.

Fortunately, I am neither a long range target shooter nor sniper so a sub-MOA group two-inches high at 100 meters gets me where I need to be with almost every caliber I have to any range I am willing to take a shot at a game animal.

I can see from your responses that this could really become a complicated area of study. There is a lot more than meets the eye if it were delved into in some depth. Might be a good topic for a Master's or Doctorate thesis.

On another topic: I just talked to MRC about my 404 Jeffery and he weighed one. It came in at 8 lb. 2 oz. so scoped and loaded they should scale about 9.5 lbs. He said they were having all kinds of problems with the website and are in the process of updating it.

I think you can read into ballistics as much or as little as you want.

It's true that there is some variance in the bullet path due to atmospherics, but for most shooters they aren't a consideration.

Even for the competitive shooter, understanding the science only really helps for the relative constants; velocity, bullet bc, temperature, humidity, elevation, spin drift, coriolis.

Everything else is where shooting becomes an art. Wind is the main one of course, both at the shooter, at the target, and enroute. The best F class guys at the club are very adept at reading this from mirage and flags on the range and through years of practice build a pretty good idea of the net impact of all the different wind conditions over the 1000yds or whatever. Then it's just a case of adjusting getting your rounds off asap before conditions change again. It's always amusing to hear the heated debate at the firing point when the terrain means you have flags flying left at 100, right at 300 and then directly towards the shooter at 800 for example.

Other stuff like ground effects, the initial bullet instability before it 'goes to sleep', changes in path during the transonic zone and changing air temp do of course have some impact, but are not repeatable or readily measurable in real time, so have to be ignored, guestimated through gut feel (hm, maybe 1 click more left for luck etc) or at least relied upon to generally cancel each other out.

This then gets us to the smooth trajectory that most ballistic calculators use. It's not the true path of the bullet of course, but it's a pretty good 'line of best fit' for the overall data set, and about the best representation from the data that it's actually possible to gather in the conditions.

I'd say it works very well for pretty much as far as a man portable rifle will group, and certainly the other factors matter less than the slight body movement that comes as part and parcel of the human factor.

Few love the mathematics of ballistics more than I do.

IMO, any minute variations along the flight path of a given bullet are meaningless.

Many [ignorant people] believe ballistic arcs to be perfectly parabolic. I think they would be, absent an atmosphere, but deceleration can only be accurately measured with calculus, not algebra. Every second of bullet travel means a change in velocity, and thus a change in y-velocity relative to x-velocity. At the end of the day, your bullet falls where you predicted it to fall (assuming you breathe correctly and stroke your trigger and your pupil is perfectly aligned at the center of your scope and..., or it does not.

@ Newboomer

I can suggest a book "Their arrows will darken the Sun", by Author Denny Mark.

It is a book which explains ballistics well, without too much of formulas - that are in fact deterrent to an average reader.
It is available on internet.

I am a physics type (along with some other sciences), and IMO @mark-hunter nailed it. It's kinda like are we talking "minute-of-critter" requirements, true MOA requirements, or something on levels that require finer degrees of resolution with the most meticulously loaded cartridges and finest precision quality barrels?

A model is a model, there is some degree of "slop" in any model, as well as any manufactured item. Having insight to differentiate which model is appropriate to use in a given situation while keeping in mind the tolerable degree of "slop" inherent in a model, as well as degree of "slop" in any manufactured item and degree of measurement error, earned me a handsome salary during my career.

I've kept a primary focus on the "80/20" rule in most things in life even before I ever heard of the "80/20" rule.

@sgt_zim is also on the right track. Everything in this case is a really a set of differential equations in the time domain from the moment the projectile clears the muzzle, to the point of impact, to when the projectile and its fragments come to rest.

The ballistic arc model is an example of a low order model first or second degree), derived from the most basic fundamental principles. It can be refined through intense study of deviations from the simpler model behavior in the actual results, and modifying the simpler model with other predictions of behavior that actual results are applied to. Sometimes the modelers get lost way too deep in their model world though. For example, if someone tries to introduce rifle ballistics correction factors for the effects of gravity across the range of elevations humans and prey animals can both exist in, then further tweaked for relative position of the moon to correct for the effects of its gravity at different phases of the lunar cycle.

Newboomer,
As you can tell by the responses thus far, my first response to your original question, I left out/forgot to include several other factors.

In short, you can spend several minutes or several hours, even several days or a virtual life time calculating a shot trajectory that is only good for that one particular bullet, at a specific time and place, in which all conditions are astronomical to 1 to ever be repeated.

In essence, the time for the shoot to be made will have long past before the calculations are solved.

Math: , discovering a calculation error .

1+1=2: 1+1=3: 1+1=4: 1+1=___:
Factor in anomalies.

Mr. Zorg said:

@sgt_zim is also on the right track. Everything in this case is a really a set of differential equations in the time domain from the moment the projectile clears the muzzle, to the point of impact, to when the projectile and its fragments come to rest.

The ballistic arc model is an example of a low order model first or second degree), derived from the most basic fundamental principles. It can be refined through intense study of deviations from the simpler model behavior in the actual results, and modifying the simpler model with other predictions of behavior that actual results are applied to. Sometimes the modelers get lost way too deep in their model world though. For example, if someone tries to introduce rifle ballistics correction factors for the effects of gravity across the range of elevations humans and prey animals can both exist in, then further tweaked for relative position of the moon to correct for the effects of its gravity at different phases of the lunar cycle.

Click to expand...

Yessir, there's Accurate, and then there's Accurate Enough.

Even with "serious" ballistics, of the sort which are necessary in warfare, if the bullet will fall on a target 2' tall x 18" wide at any given distance to which the bullet might be effective on flesh and bone, that's Accurate Enough.

We can play all kinds of games at National Match or F-Class, and those are fun games for certain, but is a degree of accuracy that doesn't produce any better results on an enemy combatant than somebody who can consistently achieve 1.5 MOA with their setup and in most environmental conditions (which will hit that 2'x18" target with monotonous regularity out to 1000 yds).

I think, for all practical purposes, if we and our equipment can consistantly put rounds into an acceptable group at reasonable hunting ranges, that will suffice.

Pheew, I think I got a cramp

Once I got to a certain level of math and thermodynamics education, it became clear to me that some ways these principles are used are just games.

For example, if you have a refrigerator set to 34°F, and you put some room temperature say 75°F beer in the fridge, the beer will never exactly reach the same temperature as the fridge. The math says you have a decay function stretching out to infinity approaching, but never reaching, thermal equilibrium (same temperature) as the fridge. As humans, we just say "good enough for my purposes" at some point in time. We do the same with calculations and computer programs where once the model reaches a user specified limit of deviation, we call it done and move ahead in my field.

There's another example where you can create a funnel or horn shape stretching to infinity that by the math, has a finite volume. You can fill up the inside, but never paint the whole outside, even though the "walls" have zero thickness and the tip stretches to infinity.

There is a way to turn calculus / differential equation problems into algebra problems, called LaPlace transforms. I was one of the few that honestly liked this approach among my fellow students, I guess because it's a difficult concept for many. It's extremely applicable to the field of control devices and systems design and operation.

Somebody on this forum few days ago mentioned Carlos Hatcock - White Feather.

My memory is fading, but possibly it was from the book "Marine Sniper" (by author Charles Henderason) I have read years ago, when asked about ballistics, scope clicks, distance, bullet drop, windage corrections either for some of his best hits, or in general about technique of shooting, (I am not sure) etc...

Carlos Hatcock said something like... "yes, taking all this in calculation, and adding to that - a SWAG correction"

- what is SWAG?

- Some wild ass guess!

(I hope i did not quote wrong)

Even today with all this wonderful ballistic apps and gadgets, one who ever makes better SWAG correction, is the one who will at long range collect medal or score longer hit in real life, and it finally will have nothing to do with ever improving mathematical models, which are good for start, and excellent for training, and even fantastic for newbies, but never perfect and most importantly do not make for experience, training and talent of a human being.

Finally calcs are only good as our variables that we enter in calculation:

We can enter wind in app calculation, but how perfect is our reading (wind gauge instrument error)
We can enter bullet velocity, but what is the instrument error of chrony(?), and in fact we are entering only average values of velocity measured by chrony (having instrumental error) .. etc
Wind is not constant and can vary across the same field.
Finally there is no instrument reading of mirage effect - its always estimated (swag again)
etc

Possibly the word SWAG is first time mentioned there in that book, and then in that context, at least it is first time when I noticed that word in that book.

Later it cough up and remained in long range slang till now. On that, though, I am speculating...

SWAG: Scientific Wild Ass Guesstimation.

Was resurrected around 1977(?), by some Vietnam Vets I had the privilege to serve with while in the army.

Several other acronyms along with a nearly forgotten numeric code sheet used to reduce rather descriptive language while in the presents of female soldiers; during this awkward transitional time.

Formula for mission success:

5 P's of Planning + Knowledge+Experience + Alternate Plan Options+SWAG/Fudge Factor+ What if's....of Murphy's Law = Operational Success

I'd heard the 6 P's in my industry were

Prior
Planning
Prevents
Piss-Poor
Performance

and my definition of SWAG matches @mark-hunter; I first heard it in context that had nothing to do with shooting or ballistics.

SWAG can be tempered (narrowed) through education and / or experience (and works best IME).

Things that are unexplainable, or very tedious to explain, are attributed to FM (that was a military acronym as explained to me).

Mr. Zorg said:

I'd heard the 6 P's in my industry were

Prior
Planning
Prevents
Piss-Poor
Performance

and my definition of SWAG matches @mark-hunter; I first heard it in context that had nothing to do with shooting or ballistics.

SWAG can be tempered (narrowed) through education and / or experience (and works best IME).

Things that are unexplainable, or very tedious to explain, are attributed to FM (that was a military acronym as explained to me).

Click to expand...

The sixth P was replaced by a more aggressive term amongst us. Term(s) were/are operationally correct!! However, not public opinion, politically correct.

Not that we really give/gave a damn about what REMFs and/or AYOs, not in direct harms way, think/thought.