all of the following is focused upon the problems associated with shooting over distance, e.g., the principle difficulty being, you just aren't tall enough to shoot down on things. this comment will be made clear. (more or less.)--
certainly not the brightest of isaac newton's intellectual progeny, nonetheless i once aspired to be an electrical engineer: i would attend oregon state university, and have my very own slide rule in a neat leather case on my belt, and a whole row of colored pens in a plastic shirt pocket liner, ... , in my shirt pocket of course.
i drew the line at wearing plaid long sleeved shirts buttoned to the throat. sheesh.
somewhere along the line i found whitman college and political science, and the rest if not history, has at least gone by.
i do remember, quite well, however, the old formula for working out the matter of how long it takes for something to drop from a given distance to the nebulous concept of "ground." it is d= 1/2 (a)(t)(t), where d is "distance," and 1/2 is "1/2,", and "a" is the physical constant of the "attraction of gravity" upon our little orb, (32 fps squared) and t is "time," as in seconds, unless great distances are involved. and, yes, in this instance, "distance" may be understood as "down to the ground, distance." *** http://en.wikipedia.org/wiki/Equations_for_a_falling_body .
it means precisely this, in terms of rifle ballistics.-- if you were a very dexterous fellow, and not at all bothered by the prospect of shooting your fool fingers off, and you held a bullet at the muzzle of your rifle, the barrel of which is held exactly parallel to a very perfectly flat surface, and shot your rifle without physical recoil causing the barrel to elevate relative to the ground, and released a bullet from between your fingers held exactly level to the plane of the barrel at its muzzle, why the bullet dropped from between your fingers would strike that level expanse, assuming no intervening obstacles, at precisely the instant of the bullet shot from the rifle, no matter its velocity.
this assume no wind, nor other disturbance of your physical surroundings. period.
the only difference is that the bullet dropped from your fingers would fall in a path perpendicular to its release point and the ground, and in an absolutely straight path, with only a vertical component (vector) to its "flight," and no horizontal component at all. (assuming no vectors muddling the matter by some aerodynamic influence on the bullet's attitude.) while, the bullet fired from the gun's muzzle absolutely parallel to the ground would have an identical vertical vector (component) to its "flight," it would also have a horizontal component to its "flight," and the velocity of the bullet as it moderated over the course of its "flight" would determine precisely just how great a distance would comprise the horizontal aspect of its "flight."
"flight" is italicized because bullets do not "fly," nor do they "soar," nor do they "climb" or "glide." they simply travel, and the path of that travel is determined by two things, horizontal speed and time spent obeying the dictates of gravity, which are "down." (even as a bullet goes "up" it is falling: it may travel above the line of sight, but, it is always falling as against the line or axis of the bore. always, the bullet obeys gravity.)
(get that fact down, get you head "around it," and you will figure out all that follows pretty easily.)
simple physics. (even electrical engineer wannabees know of these matters. some of you do not, and never have, and probably won't even if you trudge through all this.)
now, you cannot avoid the simple truths imparted above. you just cannot. not on this earth.
what you can do is manipulate the situation a bit.
let us take a battleship with a 16" rifle. if that rifle, say with a 650 pound charge of propellant behind the projectile shoots that rifle held perfectly parallel to the surface of the ocean, then the projectile strikes the surface of the ocean none too far from the ship and just at the same instance that our little rifle bullet does when dropped by some worthy sitting on the muzzle at the exact instance the artillery round leaves the muzzle of the barrel.
no matter how foolhardy, our friend will do this exactly once over the abbreviated course of his life.
but, we all know that battleship rifles can shoot over the horizon (16 miles on the ocean), some over twenty miles. but, not, my friends, with the rifle barrel held flat and parallel to the earth's surface.
one raises the muzzle. simple as that. by this simple tactic, the artillery man and the shooter of rifles does exactly one thing, and one thing only. he buys a little (t)(equals "time") against physical reality, by which the horizontal component of trajectory, that being the projectiles velocity is given a bit longer to work its magic.
i am given to understand, via my very fallible memory, that the most efficient purchase of this additional "flight" time is to elevate the muzzle, (which means to raise the line of the rifle's bore against the surface of the earth, or, consider it as raising the bore as against the horizon), about 33 to 34 degrees of arc against the surface of the earth.
that's it. to raise the muzzle any higher is to cheat the projectile of horizontal distance.
(now, the gunner may very well wish to do this, for very practical reasons. such as survival, in the terms of naval gunnery. when hms hood was sunk by the bismark, she was not steaming away from bismark to flee, but was steaming for all she was worth toward bismark to decrease the distance between them. what hood's commanders hoped to do, it is presumed, was to force the bismark's gunner to lower her guns so as to hit a closer target. hood's commanders knew that she was vulnerable to rounds descending upon her in a very vertical manner, such as found at the end of the bismark's gun's maximum ranges, rather than descending upon her in a flatter angle. the idea being, to get close enough to where bismark's shells might land against the armor plating on hood's sides, rather than descending down through her thin decks. hood could not get close enough, and was sunk turning to bring her aft guns to bear on bismark.)
for the following discussion, assume a rifleman of sufficient height so that his shoulder is 5 feet from ground, and that when he shoots his rifle it is more or less level with the ground. it is here that our equation above begins to reveal its more unpleasant aspect, in terms of the reality of musketry.--
the equation for drop from five feet. remember, d: 1/2(a)(t)(t).
so, the time for an object to free fall from five feet is solved thusly. (consider this the distance our hardy rifleman drops a bullet from his muzzle at the same time his bullet exits the muzzle.)
5= 1/2 (32)(t)(t), .... , so, 5= 16(t)(t), ... , so 5/16= (t)(t), or .3125 = (t)(t), so, ....
t= .5590 seconds. more or less. depending.
the problem of the short rifeleman. the little chart from the sierra handbook, edition 5, 3rd printing, page 1076 illustrates this problem quite nicely, that our rifleman is not tall enough to do whatever he wants to do.
this is taken as accurately as i can from the book, and i am trying not to perform error.--
calculation of maximum point blank range for a vital zone of: 10 inches
maximum point blank range is 343 (yards). set zero at 292 (yards)
trajectoy for sierra .257" dia. 117 gr. spt at 2900 feet per second (muzzle velocity)
at an elevation angle of: 0 degrees [this is not true, and cannot be, as explained: jjay.]
ballistic coeffiecent of: 0.388 0.383 0.362 (they vary of speed of bullet decreases, due to resistance)
........
range (yards) velocity (fps) energy drop (inches) bullet path (inches) time of flight (seconds)
0 2900.0 0.0 - 1.5" *** 0.000000000
50 2779.2 -0.53 +1.82 0.052837693
100 2661.6 -2.19 +4.02 0.107990399
200 2434.6 -9.3 +4.62 0.225830888
300 2216.9 -22.29 -0.67 0.354968217
400 2010.4 -42.38 -13.05 0.497081750
450 1911.7 -55.55 -22.36 0.573599874
*** it is assumed in such matters that the line of sight sits in the scoped optic 1.5" above the line of the bore in the barrel below. it is close enough, given the height of most scope mounts.
now, this little chart is very revealing of a certain truth often ignored in ballistics, when people talk of "flat shooting" cartridges. the hard facts of the matter are revealed in the two columns, "drop" and "bullet path," which seem oddly contradictory. if you will look at "drop" for the value of 450 yards, you will find the figure 55.55 inches. this is the "drop" away from the line of the bore, and is simply revealing that as to the line of the bore the bullet is subject to gravity the instant it leaves the bore, no matter the elevation of the bore. if you look at "bullet path" for the value of 450 yards, you will see that the bullet has dropped 22.36" below a certain value, and that value is the line of sight, which is parallel to the ground, there being no elevation of the line of sight relative to the ground.
this is the secret of small arms riflery. while the line of sight may be level, the bore is slightly elevated to the target in relation to the line of sight, which is straight to the target. (assuming flat ground, again.)
so, the bullet "drop" begins as soon as the bullet leaves the muzzle, relative to the line of the bore.
the "bullet path" intersects the line of sight twice, once very near the muzzle (within 50 yards on the chart above, you can see the transition going "up" when the values change from -1.5 to +1.82 under "bullet path") and once at the zero range of 292 yards, at the entry under the 300 yards of -0.67" at the value of 300 yards). and, once the bullet comes below the line of sight, the effect of gravity is noticed very quickly.
remember the time it took for the bullet to hit the ground when dropped from the hand at 5 feet? .5590 seconds, remember.
look at the chart at 450 yards. if the rifle barrel/line of bore were held perfectly parallel to the ground, then our bullet would have been hitting the ground around this point. right at 00.573599874 seconds for time of flight.
modern high speed bullets. the chart above does not reflect the most absolute values for velocity and high performance bullets, e.g., those that resist drag influences most efficiently, but they do show on a fairly representative basis how the modern bullet performs. for instance, i am working with a bullet/cartridge combination that shoots a 130 grain sierra 7mm boat tail hollow point bullet at about 3100 feet per second, the bullet having a ballistic coefficient of about 0.396. so, though the cartridge/bullet load combinations are not exactly equivalent, these are approximately the values i am looking at.
close enough for government work.
what this means, as a practical matter. even if you have a bullet sighted in for 300 yards, in an effort to stretch your trajectory, once your target moves into the 400 to 450 yard range & farther out, you are going to have to raise the elevation of the bore in order to hit it. and, as you might imagine, as the target moves into the 7 or 800 yard range, you are going to have to raise it quite a bit.
you could get taller, and shoot off a tower to compensate for the inevitable bullet drop. do you carry any 25 to 35 feet high step ladders around with you?
you can do it with "kentucky windage," e.g., by holding the cross hairs above the target. or, you can buy/must buy more expensive optics that will allow you to dial in the greater distances to make a hit on the target.
(keep something in mind. the "vertical drop" component/vector of the above charted trajectory amounts to 4.5 feet of drop, to achieve 1350 feet of horizontal component/vector. all that you have to do to achieve that, is to aim the barrel 4.62" above the line of sight, more or less, at 200 yards to "buy the added time" to get the extra distance. think of that, a little 4.62" hump over the trajectory to get 450 yards of pretty close to point blank shooting.)
the latter is the more preferred way to do it. it is just more reliable to have the cross hairs move, than it is to try and guesstimate the added values for elevation.
my own little project, riflery. i shoot at a local range that has targets at extended ranges, one of the benefits of living in the "open spaces" of the west. the range has leased a little ground which is otherwise useful for raising government subsidized grasses and the like (to shore up wheat prices by taking marginal wheat ground out of production, but, ... , that is another story), and as a consequence we have access to 6 or 700 yards of shooting ground. so, the club has seen an explosion of guys trying to hit little targets, shoot little groups, at very extended ranges.
in order to hit little targets at long range, this little group of shooters has acquired specialized rifles, usually remington model 700 with heavy barrels and fine triggers, and optics the likes of which were never imagined when i was a youngster, let alone owned by "the great unwashed," as our "betters" view us. my whole hunting life, i have been content with a leupold vari-x ii 2x7 power variable scope. i once made a killing shot on a spike elk with it at a measured 385 yards, a feat regarding which i remain proud, justifiably so in my view. the long range shooters among the club members are using 6x18 power variables, with all sorts of external dials by which they may raise the elevation of their barrels to be able to see and to reach distant targets.
(raising the barrel means to lower the scopes internals nearer the line of the bore, bringing the angle of the bore to intersect the line of sight closer to the rifle. when the scope is raised to the target, it "brings the barrel with it," and also elevates the muzzle.)
so, you might ask, just what sort of angles are we talking here? are the long range shooters raising the line of the bore to the 33 to 35 degrees talked about previously in relation to large rifles on naval war ships?
well, not hardly.
perhaps if riflemen were trying to engage targets 4 and 5 miles distant this might be done, but, then we would need an assembly similar to a gun carriage to carry the matter off. no, the hand held rifle, even in the hands of very skillful snipers and the like has its utility at ranges limited to perhaps 1200 yards or so. this seems to be expanding daily, and a canadian sniper and spotter have engaged a target in afghanistan to an astonishing distance, nearly 2000 meters, ... , if i remember correctly, it may be farther.
but, for right now, the envelope for predictable shooting using the .308 winchester/7.62x51mm nato in the model 700 is about 1,000 yards, for the hand held rifle, used by the rifleman from a bench or from the prone position on the ground, using bags and a bipod for forward support. no one shoots on their "hind feet" at such distances, seriously expecting to hit anything on the first shot.
well, i do not own such rifles nor can i afford to purchase them or the optics we are talking about: some people are spending $1,000 plus for the rifle, and as much as $1,200 for the optics, with an allied assortment of computers, handheld pda's with ballistics programs, and portable weather stations to measure wind, barometric pressure. some of the ballistics programs want the latitude and gps coordinates, in order to factor in the "coriolis effect", something that only battleship gunners and artillerymen worried about heretofore.
what's a poor boy to do?
well, you make do, that's what.
i have my elk rifles from a previous economic existence, and i decided to press one into service as a long range shooter. it is a 7mm mag, previously loaded w/ 175 grain nosler partitions to about 2850-2900 fps. it is, and has been, good elk medicine, but, sort of daunting to shoot repeatedly on a hot day.
so, i have taken "old tilly," and made her a long range shooter by the simple expedient of developing a load using the 130 grain 7mm/.284 caliber boat tail hollow point sierra bullet, otherwise called a sierra "matchking." it is sierra stock number 1903 in the catalogs. i have loaded it in front of 61 grains of hodgdon's h-4350, fired by federal 215 large rifle magnum primers, and i am getting about 3100 feet per second from it at the muzzle. it is proving reasonably accurate at distance, and it is largely conforming to predicted trajectory for a bullet with a ballistic coefficient of 0.396. look to the table above.
right now, with a 300 yard zero, is is shooting some 4 and 1/4" high at 100 yards, and 5 1/4" high at 200 yards, and i thoroughly expect it will be about 9 to 10 low at 400 yards, and perhaps 28 to 30" low at 500 yards, again maintaining the 300 zero.
past 500 yards, it will require sight correction to reliably engage targets.
what are the elevation values in a modern small arm? how far must you raise the muzzle of a rifle to shoot at distant targets? well, as it turns out, given the gist of the previous discussion, not that much, really.
i have used the following text in a previous article, to again illustrate a fundamental fact. and, that is, although a 1,000 yards is a long way to shoot a rifle, it is really nothing in terms of how far a rifle shot might travel if it were treated like an artillery piece. (i have made a cursory look via google to see if anyone has actually investigated what would happen, in terms of maximum bullet travel,if you elevated a hunting rifle's line of bore to say 33 degrees against the horizon, and touched a round off. my guess is you could probably get in excess of 4 or 5 miles out of such a round. keep in mind, that today's "very low drag bullets" are really nothing more than miniature artillery rounds, very similar in shape and form.)
at any rate, this text taken from, http://wintersoldier2008.typepad.com/summer_patriot_winter_sol/2011/12/for-those-who-think-that-the-280-british-and-the-276-pedersen-were-pipsqueak-cartridges-in-compariso.html , follows.--
the following is taken from chapter xvii of "hatcher's notebook", "random notes on various subjects," subpart 16, ballistic data, page 401 of the national rifle association edition, odysseus editions, inc., 1966--
pedersen rifle, caliber .276 [.284] (0.276-a-2)
125 grain bullet -- p.c. 48, (a.p.g.
instrumental velocity at 78 feet = [2690 f/s][muzzle velocity] (august 25, 1927
b.c. for j (v) = 0.2469 when v is greater than velocity of sound
= 0.4260 when v is less than velocity of sound
range, yards angle of departure time of flight/s remaining velocity maximum ordinate
min. of angle in time feet/s ft./lbs in feet
100 2.3 .115 2514 1758 .07
200 5.0 .239 2345 1529 .23
300 8.0 .371 2180 1322 .57
400 11.2 .514 2020 1135 1.09
500 14.6 .669 1864 966 1.82
600 18.7 .837 1711 814 2.84
700 23.1 1.021 1562 679 4.21
800 28.1 1.222 1419 560 6.00
900 33.9 1.444 1284 450 8.27
1000 40.5 1.691 1160 374 11.54
1100 48.1 1.960 1083 327 15.44
1200 56.8 2.242 1046 304 20.63
[end chart for .276 pedersen. note, "angle of departure" in minutes refers to the elevation of the bore above horizontal, to give the bullet sufficient "arc" to travel the required distance of the bullet's horizontal path in yards. think of it as the "elevation of a cannon barrel." but note, a degree of angle is comprised of 60 minutes: so to shoot a bullet even at moderate velocity 1,000 yards, the rifle barrel is not elevated even one degree, ... , about 2/3's of a degree in this case. that's not very much, is it.]
at these pages, similar charts are given for several models of .30-06 ammunition used by the u.s. armed forces for the last 100 plus years or so, one chart for the original 150 grain loading and another for a later adopted round with a 172 grain bullet, used quite a bit for machine gun fire because of its better ranging qualities. the values given are not so very different for all the rounds, and illustrate the basic premise that the military rifle is a creature designed for use at about 1,000 to 1,200 yards, maximum.
and, as i would hope all of the above discussion shows, the greatest utility for small arms, on the battlefields and in sporting & target use, is somewhere in the neighborhood of 4 to 500 yards.
this is why shooting at long range is so challenging, because it is so difficult. i have seen a friend shoot a 2.25" group at 600 yards, and do the shooting with relative ease. (now that he has "his eyeballs back," thanks to lasic surgery. will the miracles ever cease?)
john jay @ 01.24.2011
*** "ground" is such a perfectly wonderful word in the physical sciences. it is literally the "ground under our feet," so to speak, in many systems of thought. the foundation, as it were. a delicious word, full of many permutations. that is an irony by the way.
If I remember my high school physics accurately (after a gap of 50 years), the complete equation (ignoring the effects of drag)is
drop = u*t + a*t*t/2
where u is the initial vertical velocity if fps and a is 32.2 fps/s.
U = mv/sine(theta) where mv is the muzzle velocity in fps and theta is the angle of the gun in degrees. It has a negative sign when the gun is pointing upwards.
Drop = -t*mv/sine(theta) +
16.1*t*t when pointing up and
drop = t*mv/sine(theta) +
16.1*t*t when shooting downhill.
I'm sure you already know that but thought you might want to add it to make your explanation complete.
Posted by: Aloysious A Gruntpuddock | January 25, 2012 at 06:38 AM
aloysious:
laughing. your high school physics is a little older (not by much) than my high school physics, but your high school physics was obviously better than mine.
i just rely on sierra and others for the drop tables these days.
i am very glad to receive your letter. kinda fun.
as to the trajectory matter, i am afraid the market place has completely ruined any inclination i might have had to take slide rule in hand and figure things out.
it is all so dependent upon drag functions these days, in that bullets are no longer relatively simple things, but elaborate designs with ogives being very long, and the centers of arc describing them are so far away from the bullet as to be unbelievable. hornady is fond of ogives that are so long as to be almost flat.
several persons have kind of reinvented some things, guys like arthur pejsa who taught at annapolis, and others, and they are marketing programs to go into palm pda devices that simply calculate everything based on a few inputs.
the service sniper have all this, and they get the distance, and the device plots the trajectory and tells them the "dope" on the spot, e.g., it tells them the "come up" and how many clicks to dial into the scope.
this will be the next installment, as i learn a bit more about the whole thing.
wikipedia has a very good article on "external ballistics," complete with links to commercial sources.
and, again. it is all dependent upon accurate range assessment to target.
at the extended ranges, if range estimation is off by 10 or 15 yards, it is simply a miss, given the angles of descent of the bullet at long range.
it is all very interesting.
john jay
Posted by: john jay | January 25, 2012 at 12:04 PM