A Short Course in External Ballisticsby Fr. Frog
Well, ok, maybe this isn’t so short.
There is a lot of misleading information and myth flying around (“bull-istics”) on the subject of the external ballistics. The tables below will hopefully shed some light on how that bullet really travels once you’ve pulled the trigger. All tables are rounded to the nearest 10 feet per second and drops are rounded to two places, unless I am trying to show small increments. Greater precision is meaningless in the “real” world. Even for the best of marksman a ½ minute of angle difference is effectively meaningless at realistic ranges. The majority of information is presented on rifle cartridges but the principles hold true for shotgun and pistol as well.
Remember Fr. Frog’s Rules of External Ballistics:
- There ain’t no magic bullets! (Although some are better than others for a particular purpose.)
- Divide the range at which someone claims to have shot their deer by 4 to get the real range.
- Always get as close as possible.
- Don’t believe manufacturer’s claims.
- Velocity erodes, mass doesn’t
- In the battle between velocity and accuracy, accuracy always wins.
- Inconsequential increments are meaningless.
- Most gun writers are pathological liars.
The Bullet’s Path
Many people believe that bullets fly in a straight line. This is untrue. They actually travel in a parabolic trajectory or one that becomes more and more curved as range increases and velocity drops off. The bullet actually starts to drop when it leaves the firearm’s muzzle. However, the centerline of the bore is angled slightly upward in relation to the line of the sights (which are above the bore) so that the projectile crosses the line of sight on its way up (usually around 25 yards or so) and again on its way down at what is called the zero range.
Definition of Terms
Back Curve - This is that portion of the bullets trajectory that drops below the critical zone beyond the point blank range. Past this point the trajectory begins to drop off very rapidly with range and the point of impact becomes very difficult to estimate.
Ballistic Coefficient (BC) - This is a number that relates to the effect of air drag on the bullet’s flight and which can be used to later predict a bullet’s trajectory under different circumstances through what are called “drag models.” Technically a drag models applies only to a particular bullet, so using them to predict another bullet’s performance is an approximation - but the results can be very close if the proper drag model is used. The most commonly used drag model is the G1 model (sometimes referred to, not really correctly, as C1) which is based on a flat-based blunt pointed bullet. The “standard” bullet used for this model has a ballistic coefficient of 1.0. A bullet that retains its velocity only half as well as the model has a ballistic coefficient of .5. The G1 model provides results close enough to the actual performance of most commercial bullets at moderate ranges (under about 500 yards) that it is commonly used for all commercial ballistics computation.
Note that there are two standard sets of meteorological conditions in common use. The older one, is known as Standard Metro or Army Standard and the more modern is called the International Civil Aviation Organization (ICAO) standard. The characteristics of these two standards are listed below.
|Altitude||Sea level (0′)||Sea level (0′)|
|Temperature||59° F||59° F|
|Barometric Pressure||29.5275″ Hg||29.9213″ Hg|
While they are similar, the different parameters do have a slight affect on calculations and in effect change the standard atmospheric density by about 1.8 percent. Under ICAO conditions the speed of sound is 1116.5 f/s and under Standard Metro conditions it is 1120.27 f/s.
Since a quoted ballistic coefficient depends on atmospheric density, the same bullet has two different BCs depending on the conditions used. If a quoted BC based upon the Standard Metro conditions is used in a ballistics program based upon the ICAO standard, the BC needs to be modified by multiplying it by .982. Conversely, ICAO based BCs need to be multiplied by 1.018. While this is a very small change and has little effect at short range (under 600 yards), it does have an effect at long ranges.
A word to the wise. Many manufacturer give rather generous BCs for their bullets because: a) they want to look good--high BCs sell bullets; b) they were derived by visual shape comparison rather than actual firing data; or c) they were derived from short range firings rather than long range firings (which are more difficult to do). You should confirm your calculations by actual firing if you require exact data. Several manufactures have recently readjusted some of their BCs to more closely conform to actual firing data.
Bore Centerline - This is the visual line of the center of the bore. Since sights are mounted above the bore’s centerline and since the bullet begins to drop when it leaves the muzzle, the bore must be angled upwards in relation to the line of sight so that the bullet will strike where the sights point.
Bullet Trajectory - This is the bullet’s path as it travels down range. It is parabolic in shape and because the line of the bore is below the line of sight at the muzzle and angled upward, the bullet’s path crosses the line of sight at two locations.
Critical Zone - This is the area of the bullet’s path where it neither rises nor falls greater than the dimension specified. Most shooters set this as ± 3″ to 4″ from the line of sight, although other dimensions are sometimes used. The measurement is usually based on one-half of the vital zone of the usual target. Typical vital zone’s diameters are often given as: 3″ to 4″ for small game, and 6″ to 8″ for big game and anti-personnel use.
Initial Point - The range at which the bullet’s trajectory first crosses the line of sight. This is normally occurs at a range of about 25 yards.
Line of Sight - This is the visual line of the aligned sight path. Since sights are mounted above the bore’s centerline and since the bullet begins to drop when it leaves the muzzle the bore must be angled upwards in relation to the line of sight so that the bullet will strike where the sights point.
Maximum Ordinate - This is the maximum height of the projectile’s path above the line of sight for a given point of impact and occurs somewhat past the halfway point to the zero range. It is determined by your zeroing range.
Maximum Point Blank Range - This is the farthest distance at which the bullet’s path stays within the critical zone. In other words, the maximum range at which you don’t have to adjust your point of aim to hit the target’s vital zone. while hunting, unless there is some overriding reason to the contrary, shots should not generally be attempted much past this distance. An approximate rule of thumb says that the maximum point blank range is approximately your zero range plus 40 yards.
Mid-range Trajectory - This is the height of the bullet’s path above the line of sight at halfway to the zero range. It does not necessarily occur at the same range as the maximum ordinate height, which can be greater.
Minute of Angle (MOA) - A minute of angle is 1/60 of a degree, which for all practical purposes equates to 1 inch per 100 yards of range (actually it’s 1.044″). Thus, 1 MOA at 100 yards is 1 inch, and at 300 yards it is 3 inches. The term is commonly used to express the accuracy potential of a firearm.
Zero Range - This is the farthest distance at which the line of sight and the bullet’s path intersect.
The bore’s angle in relation to the line of sight is exaggerated in this drawing for clarity.
A Brief Discourse on Ballistic Coefficients
This is probably the best article I have read on ballistic coefficients. It was written by Jim Ristow of Recreational Software, Inc. and is reprinted here with his permission. It was designed to encourage a discussion about ballistic coefficients and to explain why good BCs are crucial to getting accurate results from ballistic software. The illustrations and tables were not part of the original article.
A Little History
In 1881 Krupp of Germany first accurately quantified the air drag influence on bullet travel by test firing large flat-based blunt-nosed bullets. Within a few years Mayevski had devised a mathematical model to forecast the trajectory of a bullet and then Ingalls published his famous tables using Mayevski’s formulas and the Krupp data. In those days most bullet shapes were similar and airplanes or missiles did not exist. Ingalls defined the Ballistic Coefficient (BC) of a bullet as it’s ability to overcome air resistance in flight indexed to Krupp’s standard reference projectile. The work of Ingalls & Mayevski has been refined many times but it is still the foundation of small arms exterior ballistics including a reliance on BCs.
The shape of the projectile used in the Krupp firings. It is 3 calibers long and has an ogival head with a 2 caliber radius.
Modern bullet designs. Much different than the Krupp bullet. Would you expect them to have the same drag characteristics?
By the middle of the 20th century rifle bullets had become more aerodynamic and there were better ways to measure air drag. After WWII the U.S. Army’s Ballistic Research Lab (BRL) conducted experiments at their facility in Aberdeen, MD to remeasure the drag caused by air resistance on different bullet shapes. They discovered air drag on bullets increases substantially more just above the speed of sound than previously understood and that different shapes had different velocity erosion due to air drag. In 1965 Winchester-Western published several bullet drag functions based on this early BRL research. The so-called “G” functions for various shapes included an improved Ingalls model, designated G1. Even though the BRL had demonstrated modern bullets would not parallel the flight of the G1 standard projectile, the G1 drag model was adopted by the shooting industry and is still used to generate most trajectory data and BCs. Amazingly, the G1 standard projectile is close to the shape of the old blunt-nosed flat-based Krupp artillery round of 1881!
The firearms industry has developed myriad ways to compensate for this problem. Most bullet manufacturers properly measure velocity erosion then publish BCs using an “average” of the calculated G1 based BCs for “normal” velocities. In other words, the only spot on the G1 curve where the model is correct is at the so-called “normal” or average velocity. These BCs are off slightly at other velocities unless the bullet has the same shape, and therefore the same drag as the standard G1 projectile - few modern bullet do.
Some ballistic programs adjust the BC for velocities above the speed of sound, others use several BCs at different velocities in an effort to correct the model. While these approaches mitigate some of the problem, BCs based on G1 still cannot be correct unless the bullet is of the same shape as the standard projectile. Also, the change to air drag as a function of velocity does not happen abruptly. Drag change is continuous with only small variation immediately above or below any point along the trajectory. Programs that translate the Ingalls tables directly to computer or use multiple BCs can produce velocity discontinuities when drag values change abruptly at pre-determined velocity zones. The resulting rapid changes to ballistic coefficient do not duplicate “real world” conditions. A BC based upon the correct drag model (which technically changes with every bullet) stays the same value. However, using a more modern drag model such as G7, the calculated ballistics comes closer to actuality than with the G1 and some manufacturers are beginning to supply G7 based BCs.
Shooting software is finally appearing based on methods used in aerospace with drag models for different bullet shapes. Results are superior to traditional “G1 fits everything” thinking, but now shooters must learn BCs are different for each model. Each bullet has a slightly different actual drag model and if the exact drag model for a particular bullet is used the BC does not change with changes in velocity. This could get cumbersome very fast with all the bullets on the market. However, most bullets actual drag models come pretty close to matching one of the existing standard drag models (as shown on the graph below), so we can get by with one of them and come much closer to real life performance than with the catch-all G1/Ingalls. Note that if the correct drag model is used (which technically is different for each bullet) the BC does NOT change with velocity, and if a drag model is used that more closely matches the actually drag model, the BC will show less of a change at different velocities than using a badly matched (G1) drag model.
This is a scary proposition for most bullet companies who know many shooters pick bullets based only on their BCs. For example, a boat tailed bullet with a G1 based BC of .690 may actually have a G7 based BC of only .344, since the G1 drag model much more accurately describes its performance. But, everyone “knows” that .690 is “better” than .344. However, using the wrong drag model will yield trajectory data that indicates incorrect drop. Fortunately the differences only become important at very long range (>500 yards) but there is a difference. As an example the GI M80 Ball bullet (149gr FMJ boat tail) has a verified G7 BC of .195. The commercial equivalents of this bullet are listed as having a G1 BC of between .393 and .395. You can see the differences in the plotted trajectories using both the G1 and G7 values and a program that handles both types.
|G1 = .393||G7 = .195|
Modern ballistics uses the Coefficient of Drag (CD) and velocity (actually the bullet’s Mach number) rather than the traditional Ingalls/Mayevski/Sciacci s, t, a & i functions. This avoids velocity discontinuities, and when combined with a proper drag model is far more accurate to distances beyond 1000 yards. A by-product of modern ballistics research is that the C.D. can be estimated fairly accurately from projectile dimensions as well as used to define custom drag models for unusual bullet shapes (see caveat below).
The drawing below shows how the various drag models vary.
Note the difference between the G1 and the G5, G6, and G7!
The Coefficient of Drag for a bullet is simply an aerodynamic factor that relates velocity erosion due to air drag, air density, cross-sectional area, velocity, and mass. A simpler way to view CDs are as the “generic indicator” of drag for any bullet of a particular shape. Sectional Density is then used to relate these generic drag coefficients to bullet size. The Sectional Density of a bullet is simply it’s weight in pounds divided by it’s diameter squared.
Sectional Density = (Weight in Grains) / (7,000 × Diameter²).
You can see from the formula that a 1 inch diameter, 1 pound bullet (7,000 gr.) would produce a sectional density of 1. Indeed the standard projectile for all drag models can be viewed as weighing 1 pound and having a 1 inch diameter.
Another term occasionally found in load manuals is a bullet’s “Form Factor”. The form factor is simply the CD of a bullet divided by the CD of a pre-defined drag model’s standard projectile.
Form Factor = (CD of any bullet) / (CD of the Defined 'G' Model Standard Bullet)
So What Is A Ballistic Coefficient?
Ballistic Coefficients are just the ratio of velocity retardation due to air drag (or CD) for a particular bullet to that of its larger 'G' Model standard bullet. To relate the size of the bullet to that of the standard projectile we simply divide the bullet’s sectional density by it’s form factor.
Ballistic Coefficient = (Bullet Sectional Density) / (Bullet Form Factor)
From these short formulae it is evident that a bullet with the same shape as the 'G' standard bullet, weighing 1 pound and 1 inch in diameter will have a BC of 1.000. If the bullet is the same shape, but is smaller, it will have an identical CD, but a form factor of 1.000 and a BC equal to it’s sectional density.
The following are the most common current drag models used in ballistics:
G1.1 - Standard model, flat-based pointed bullet, 3.28 calibers in length, with a 1.32 caliber length nose, with a 2 caliber (blunt) nose ogive.
Ingalls and G1
G2 - Special model for a long conical-point banded artillery projectile, 5.19 calibers long with a .5 caliber 6° boat tail. Not generally applicable to small arms.
G2 conical nose shape with radiused blend
G5.1 - For Moderate (low base) boat tails, 4.29 calibers long with a .49 caliber 7° 30′ boat tail and a 2.1 caliber nose with a 6.19 caliber tangent nose ogive.
G6.1 - For flat-based spire-point type bullets, 4.81 calibers long and a 2.53 caliber nose with a 6.99 caliber secant nose ogive.
G7.2 - For VLD type, or pointed boat tails, 4.23 calibers long with with a .6 caliber long 7° 30′ Tail Taper and a 2.18 caliber long nose with a 10 caliber tangent nose ogive. Most modern US military boat tailed bullets match this model.
G8.1 - Flat base with similar nose design to G7, 3.64 calibers long with a 2.18 caliber long nose and a 10 caliber secant nose ogive. The US M2 152gr .30 cal bullet matches this drag model. Close to the G6 model.
GS - For round ball, Based on 9/16″ spherical projectiles as measured by the BRL. Larger and smaller sphere characteristics are effectively identical.
RA4 - For 22 Long Rifle, identical to G1 below 1400 f/s.
GL - Traditional model used for blunt nosed exposed lead bullets, identical to G1 below 1400 f/s.
GI - Converted from the original Ingalls tables. Essentially G1.
GC - 3 caliber long flat nosed cylinder. Identical to G1 below 1200 f/s.
For Best Accuracy, Calculate Your Own Coefficients!
Accurate BCs are crucial to getting good data from your exterior ballistics software. A good ballistic program should be able to use two velocities and the distance between them to calculate an exact ballistic coefficient for any of the common drag models. While you should really simultaneously measure the velocities at the 2 points you can do very good work by measuring a minimum of 5 shots at the near and far ranges and average each group.
This method of calculating a BC is preferred for personal use and can be used to duplicate published velocity tables for a bullet when the coefficient is unknown or to more accurately model trajectories achieved from your own firearm. A lot has changed in shooting software. If your software is more than two years old, chances are it does not employ the latest modeling techniques or calculate BCs, and even some of the newest software is not perfect as you can see from the next section.
If the industry wants to stay with a single BC drag model with modern bullets they would probably be better off using the G7 model than the G1. While not necessarily a perfect match, the characteristics of modern bullets are much closer to the G7 drag model than the G1.
Bullet pictures courtesy Hornady Mfg. Drag curves courtesy Jim Ristow
We mentioned that CD can be estimated fairly well from certain bullet dimensions. However, because of the effects of bullet wobble (precession due to rotation), nose tip radius or flatness, nose curvature and boat tail, boundary layer interaction from cannelures and land engraving, etc… (all of which affect the wave drag, base drag and friction drag of the bullet differently), it is really impossible to predict with total accuracy the actual CD vs. Mach number. Also, while a ballistic coefficient can be computed from velocity measurements at two points, differences in bullet wobble diminishes the validity of chronograph testing for BC change over separate series of different muzzle velocities - it needs to be done by separate measurements at different ranges for each shot. Why? Read on.
An elongated bullet, as opposed to a round ball, is inherently unstable aerodynamically. When made gyroscopically stable by spinning, its center of gravity will follow the flight path. However, the nose of the bullet stays above the flight path ever so slightly because the bullet has a finite length and generates some lift. This causes the bullet to fly at a very small angle of attack with respect to the flight path. The angle of attack produces a small upward cross flow over the nose that results in a small lift force. The lift force normally would cause the nose to rise and the bullet to tumble as the nose rose even more. That is where the spin comes in and causes the rising nose to precess about the bullet axis. When the spin is close to being right for the bullet’s length, the precessing is minimized and the bullet “goes to sleep.” If it is too slow the bullet will not be as stable as it should. (That is why Jeff Cooper says it’s wrong to shoot groups at 100 yards for accuracy testing and suggests 300 yards. If your twist isn’t right for the bullet used your group size will be larger at long ranges than would be expected by extrapolation of 100 yard data due to bullet wobble.)
Of course, any other disturbing force such as a side wind gust could cause a difference in bullet nose precession but the effect would be quite small for a properly spin-stabilized bullet. Most of the lift force is on the nose of the bullet and is proportional to the square of the bullet velocity, as well as the nose shape and length. The new long-nosed bullets for long range match shooting can generate quite a bit more lift occurring farther ahead of the center of gravity, and can produce a nasty pitch-up moment. That is why they require a faster than normal twist to stabilize them. Pistol bullets, being relatively short and with little taper to the nose, require a slower spin for stability.
Let’s look at the rotational speed of a bullet. The formula for computing the rotational speed of a bullet is:
R = (12/T) × V
T = Twist
V = Velocity in f/s
R = Rotations per second
Now consider a bullet chronographed at about 2750 f/s muzzle velocity fired from a rifle with a 10″ twist. It is rotating at around 198,000 rpm. Let the flight velocity decay to 2000 f/s. Now what is the bullet rotational speed? Well, it doesn’t fall off much because the only things slowing it down are inertia and skin friction drag (which is pretty low). With the M80 ball bullet it has been measured about 90 percent of the original rpm (or in this case about 178,00 rpm), depending on the bullet. Then chronograph an identical bullet from the same rifle, this time with a muzzle velocity of 2000 f/s. Its rotational velocity will be 144,000 rpm. Its stability will be different from the bullet fired at 3000 f/s and allowed to slow down to 2000 f/s. It will not have the same drag at 2000 f/s although the bullets are identical. Therefore, two identical bullets fired from the same rifle at different velocities, will not have the same drag coefficient or ballistic coefficient just because of the way the measurements were taken. There are times when test data does not mean what you think it does. Again, radar range testing is the only way to fly for trustworthy bullet drag data (I am indebted to Lew Kenner for this lucid description of bullet stability).
Another factor is that it is not necessarily true that the drag coefficient of a particular bullet is proportional to that of another bullet of the same design across the Mach number range, but this is what a ballistic coefficient assumes.
Something else to worry about is the effect of the bullet tip shape/condition on the ballistic coefficient. Because modern bullet have soft points they are subject to damage and manufacturing tolerances that can alter the BC from bullet to bullet and across otherwise similar bullets, although this affect is small unless there is a great deal of deformation.
For truly accurate results, individual bullet characteristics need to be measured on radar ranges as is done by the military - much too expensive a procedure for the commercial bullet industry who doesn’t really care about great accuracy in BC calculations - and the drag model from those measurements applied only to the particular bullet tested (if you have a spare $100,000 + and would like to buy me such a setup, let me know).
The good news is that for normal rifle ranges the drag coefficients and ballistic coefficients can work satisfactorily for most purposes - so let’s proceed.
External Ballistics Continued
Continuing our discussion of external ballistics let’s look at what bullets do under different conditions and see what affects what. While the majority of the discussion below deals with rifles projectiles, the principles apply to shotgun and pistol projectiles as well. I suggest that you read the rifle section first. If you pay attention you will see that many effects are quite small and have no real affect at reasonable ranges. Too many people worry about inconsequential increments. I had one fellow brag to me that a certain load gave him less drop (according to published data) than the brand he had been using. According to the published data the difference amounted to less than 2″ at 500 yards. Huh? (And, he was hunting deer in the NY woods). If you are one of those rare people that really can hit consistently at extremely long ranges you may have some things to think about. For the majority of shooters, however, what happens inside of 300 yards is what really matters.
By the way, most folks tend to visually over-estimate the range and think that something is farther off than it is. If you have never done range estimation, try this. Find a large open field and laser or measure off a real 400 or 500 yards and then turn around and look at your buddy standing at the start point. If you don’t have a laser or long tape measure simply find a long straight road, set your odometer to 0 and then drive. Each 1/10 of a mile is 176 yards.
The tables below are based upon the assumption of a telescopic sight mounted 1.5 inches above the centerline of the bore. The velocities are for illustration purposes only and may not be attainable in your particular rifle. Unless stated otherwise the bullet used for these calculations is the GI M80 Ball bullet with a G7 ballistic coefficient of .195 as derived by the Ballistics Research Laboratory, Aberdeen Proving Grounds and using a program designed for the G7 drag model (mainly because I have verified data). While some of the tables go out to 1000 yards I have purposely cut many of the tables off at 600 yards since the data makes the point intended. Longer ranges are listed only where the effect under discussion only really shows up at extreme range. I have also rounded velocity and drop numbers to eliminate meaningless precision.
A note: as previously mentioned the drag models for G1 and G7 can’t really be compared. However, the G1 equivalent for the M80 type bullet is about .397 according to the published data for the Hornady and Sierra FMJ-BT 150gr bullet and this number will provide a fairly close match of trajectory for ranges under about 500 yards. For other bullets use your ballistics software to determine what works best for you.
Velocity vs Trajectory
Many people falsely believe that velocity makes a really big difference in the bullet’s trajectory. In the table below using the cartridge data described above, note how it takes a velocity increase in muzzle velocity of about 200 f/s to make a less than ½ minute of angle (1.5″) difference in point of impact at 300 yards when zeroed at 225 yards. Unless you have Superman’s eyes, you can’t see that difference at that distance, let alone hold to it. So much for worrying about that last foot-second for most field shooting! Spend your time practicing marksmanship.
However, for those with the skill (which are few and far between) to make use of improved trajectory and wind drift (which we’ll discuss later) at greatly extended ranges, the extra velocity can provide an edge. For the vast majority of shooters it is a moot point.
Effect of Velocity on Trajectory
(zero = 225 yards)
|Range||Velocity||Bullet Path||Velocity||Bullet Path||Velocity||Bullet Path|
Effect of Velocity on Point Blank Range
(maximum ordinate = 3″)
If you are sharp eyed you will notice in the table above a slight change in the maximum ordinate. This would indicate that if you crank up the velocity you can achieve a slightly longer point blank range. In the table below I have adjusted the zero range of the 2500 and 2900 f/s loads to match the 3.0″ maximum ordinate of the 2700 f/s load.
210 yd zero
225 yd zero
240 yd zero
With the 2500 f/s load the point blank range (-3" low) occurs at about 245 yards, with the 2700 f/s load at about 265 yards, and for the 2900 f/s load at about 285 yards As you can see, changing the velocity by 200 f/s gives us only about a 20 yard change in a point blank range. Is all that fuss getting an extra 200 f/s really worth it? Not unless we are talking about drop at very long ranges.
Zero Range vs Trajectory
Too many shooters zero their rifles at too short a distance and thus lose the advantages of a more useful trajectory. For the most efficient use of trajectory you want to keep the actual point of impact vs. point of aim within the vital zone of your target for the greatest distance possible. Your critical zone size will vary depending on your intended target but ± 3 - 4 inches is a good compromise for most uses.
The table below shows the effect of different zeroing ranges. You can see that a good zeroing range for the .308 / 150gr is somewhat greater than 200 yards (200 meters might be a workable choice but 225 - 250 yards works very well). Looking at the first set of tables (above) with the 225 yard zero you can confirm this. This data holds true for most other cartridges of similar bullet weight and velocity.
Effect of different zeroing ranges on trajectory
(muzzle velocity = 2700 f/s)
|Range||Zero = 100||Zero = 200||Zero = 300|
Ballistic Coefficient vs Trajectory
The ballistic coefficient of the bullet also effects the bullet’s trajectory, although at reasonable ranges not as much as some folks believe. The following table is based on commercial 165gr bullets in both flat based and boat tail configuration at a nominal muzzle velocity of 2700 feet per second zeroed at 225 yards with published G1 ballistic coefficients of .400 and .460 (+15% for the BT). We could get really picky and remember that the G1 coefficients are based upon the flat base model which is not a true match for the characteristics of the boat tail bullet which are more closely matched by the G7 model. However, it is probably close enough for field use (500 yards and less).
Note that until the bullet gets past 500 yards that a nominal 15% change in ballistic coefficient doesn’t do very much. (Yup! You read it right. At 1000 yards with a 225 yard zero you’ve got to hold between 31 and 35 feet high to hit your target.)
Effect of ballistic coefficient on trajectory
|Range||Velocity (G1=.40)||Bullet Path||Velocity (G1=.46)||Bullet Path|
Environment vs Trajectory
Ballistic tables are generally based upon what is called “standard conditions,” which is generally taken to mean sea level, 59° F, a barometric pressure of 29.53″ Hg, and 78% humidity (known as Army Standard or Standard Metro) and all of the above tables are based upon these conditions. In real life, on any given day the actual conditions may be quite different than the so-called standard conditions so there could be some interesting things that go on. Let’s see...
For example, at higher altitudes the air is thinner so there is less drag on the bullet. However, it is also true that at higher altitudes the air is generally colder, the speed of sound is thus lower, and therefore transonic drag occurs at a lower velocity. Also, on a warm day the barometric pressure tends to be higher which increases drag but the higher temperatures tend to decrease drag slightly. In effect most things tend to pretty much cancel each other out - but not quite.
The following tables assume the standard M80 ball ammo at 2700 f/s with a 225 yard zero as used in the first table above (all of which were done for standard conditions). They will give you some idea of what the various changes will do independently. In each table below only the particular item has been changed. The other conditions remain at standard.
Bullet Path in Inches (225 yard zero)
|Range||Sea Level||1000ft Altitude||5000ft Altitude||10,000ft Altitude|
|Range||Standard (59° F)||(32° F)||(80° F)|
It must also be pointed out that temperature changes also affect chamber pressure. While the affect varies with the type of powder, with IMR type powder there is about a 1000 - 1500 pound change for every 10° F change in temperature. This would give you about a 3% velocity change for every 20° F. Which, as we have seen, doesn’t have a big effect on trajectory until you are out past 500 yards. Ball powders are another matter. Their change may not be linear and at the extremes of temperature, and they may show dramatic if not catastrophic changes.
Since the effect of temperature changes on internal ballistics is difficult to predict, the only way to get accurate data for temperature variances is to actually test the load in question under the conditions expected.
Humidity & Barometric Pressure
I thought I’d be real complete and tabulate trajectory changes due only to humidity and barometric pressure changes. There was a difference between standard humidity (78%) and dry (20%), and between standard pressure and ± 1″ of Hg at normal rifle ranges. These changes amounted to less than .2″ at 500 yards and 3″ at 1000 yard for the humidity change, and 1.1″ at 500 yards and 19″ at 1000 yards for a 1″ change in barometric pressure. Thus it is really of interest only to statisticians and long range target shooters.
And Now The Real World
(Average conditions where you might be shooting)
The chart below will give you an idea of what you can expect in the real world. Conditions are based upon actual typical weather conditions during the month of January in NJ, and the month of August in Prescott, AZ, to give a good spread.
(10′ ASL & 31° F)
(5300′ ASL & 90° F)
Bullet Weight vs Trajectory
Well what about changing bullet weights? The table below shows the differences in trajectory between three different bullet weights in the same cartridge. The muzzle velocities are based upon published data and the different velocities are what can be expected for the different bullet weights in the .308 with a standard length barrel. The bullets are commercial soft point flat base bullets of the same design from the same manufacturer with published G1 ballistic coefficients of .358, .400, and .431. Zero in all cases was 225 yards. Notice that the 180gr bullet, even though it started out 200 f/s slower than the 150gr bullet, is traveling faster at 500 yards and beyond and it shows less drop at very long ranges.
Effect of changes in bullet weight on trajectory
Look at the data, if you zero for the same distance there is no real difference in the trajectory out to around 500 yards. So, if you know where your 150s go, if you switch to 180s and re-zero to the same distance you have an almost perfect trajectory match. However, we do know that when you switch bullet weights the zero usually goes out the window big time and if you don’t re-zero all bets are off. Why? Because of the effect of barrel timing and recoil, and barrel whip (how the barrel vibrates as the bullet passes through it). These two things have a big effect with most barrels.
However, as I intimated above, the difference in barrel timing and barrel whip with different bullet weights causes a much greater difference than one might expect and the differences show up in both the horizontal and vertical plane. I have seen several rifles that throw 165gr bullets several inches to the left and high and throw 180gr bullets several inches to the right and low at 100 yards compared to 150gr bullets. There are some particular barrels such as those on some SMLE Mk4 Enfields which by some magic tend to throw heavier bullets slightly higher than would be expected and lighter bullets lower and these gems seem to hold the vertical deflection quite constant when bullet weight changes.
The Answer is Blowing in the Wind
The final thing we’ll look at is wind induced drift. To keep things manageable we’ll look at the effect of both velocity and ballistic coefficient in this table. We’ll use the same commercial 165gr flat base and boat tail bullets with G1 = .400 and .460 as before.
Drift in inches for a 10 mph (90°) crosswind*
Flat Based / Boat Tail
*For 45° crosswind use ¾ value
A couple of things to notice. At reasonable ranges (300 yards) the effect of BC or velocity on wind drift is not all that great - a 200 f/s difference in velocity or a 15% change in BC gives about a 1" difference in drift. There is less than a 3″ difference between the worst case and best case at 300 yards with the average deflection being about 8" for the bullets under study. At 1000 yards a 200 f/s change in MV gives about 12" difference in drift and the worst case-best case difference is 23″ with an average deflection of 128″ for the flat based bullet and 106″ for the boat tailed bullet. If you’re a long range target shooter or a sniper, you need to worry about these things. Otherwise, unless you are shooting in a gale, don’t sweat the small stuff.
An addendum. I’m sure someone is going to ask, “How can you tell how fast the wind is blowing?” These rules of thumbs are taken from TC 23-4 on sniper training:
3 to 5 mph - Wind can just be felt on the face.
5 to 8 mph - Leaves in the trees are in constant motion.
12 to 15 mph - Small trees begin to sway.
Another method of estimating wind velocity is by estimating the angle between a flag and its pole and dividing that angle by 4. If no flag is available a small piece of cloth, paper, or some grass may be dropped from shoulder height and the angle between vertical on your shoulder and where it lands can be estimated.
Shotgun External Ballistics
Most folks think that the trajectory of the 12 gauge rifled slug is close to that of a mortar. Since they don’t think they could hit anything past 25 or 50 yards (which is probably true if they don’t have a set of sights on their shotgun) they zero for slugs at 25 yards. Unfortunately, this short zero severely limits the effectiveness of the slug firing shotgun. Surprisingly, a slug’s trajectory is quite flat out to about 125 yards (assuming the proper zeroing range). The biggest limitation of the shotgun slug is that penetration and trajectory drop off drastically beyond 125 yards due to velocity loss, so its maximum effective range is probably about 125 yards. (I still wouldn’t want to be hit by a slug at 200 yards though!)
12ga Foster Type Rifled Slug (G1 = .109)
(20″ barreled riotgun with ghostring sights)
|Zero = 75 yds
|Zero = 100 yds
While the 100 yard zero appears to be more useful than the 75 yard zero, the fact that most standard riotguns only will group into 8″ - 10″ at 100 yards makes attaining a good 100 yard zero difficult unless sighted in at a shorter range with compensation for the distance. For those individuals using a rifled barrel shotgun with slugs for hunting or those lucky folks with the occasional magical riotgun that really groups ’em an actual 100 yard zero may be preferred. A lot of folks find it easier just to zero them for 2″ high at 50 yards (at which distance group size is usually quite good) which gives about an 85 yard zero, which is probably the real optimum distance. Using a shotgun and slugs with a good set of sights one can completely control their environment within a 125 yard radius.
By the way, for those of you interested in such things (even though at typical buckshot distances it doesn’t matter) the ballistic coefficients for buckshot are approximately as follows. Note that the GS drag model should be used for spherical projectiles but since most programs don’t handle the GS model I have converted them to the approximate G1 figures.
|Size||Diameter||Weight||BC (GS)||BC (G1)|
The following is the data for 00 buckshot. A 75 yard zero is assumed.
00 Buckshot Trajectory
|Zero = 75 yds
Pistol External Ballistics
As with shotgun rifled slugs, many people believe that pistol bullets have such a curved trajectory that long range hits are next to impossible. (We’ll ignore the silhouette shooters for the time being since that is a specialized activity, usually with optical sights and with equipment that often stretches the definition of “pistol”). While the defensive handgun is designed for use at short ranges (50 yards - and usually much less), don’t feel under-gunned if your target is at greater ranges (assuming that you know how to shoot). The following table give the trajectories of typical 9mm 125gr, .357 Mag 158gr, and .45 ACP 230gr duty ammunition with a 50 yard zero and ¾ inch sight height.
Typical Handgun Ammunition Trajectories
|Range||9 mm 125 gr||.357 158 gr||.45 230 gr|
Defensive pistol shooting at long range is not recommended except in dire straits. However, when the goblin is way out there near Fort Mudge and thinks he’s out of harms way, if you hold on the head and carefully squeeze one off, you’ll easily get a chest hit (and ruin his whole day). I once wowed a couple of MPs by getting 5 solid chest hits with 5 shots on an IPSC “option” silhouette at 100 yards using a Detonics pocket .45 from the braced sitting position. If I can do it, so can you!
So what can we learn from all of the above? Know your rifle and ammo! It is more important to use/develop a load that shoots accurately and consistently in your rifle than to worry about getting the last foot-second or quarter inch of trajectory or group size and spending countless hours or dollars in a quest for the Holy Grail. Use your time and resources to load good ammunition and then practice your shooting. Learn where your chosen load(s) shoot at various ranges. You’ll be much better off than some guy with a Remingchester .392 Super-blotto Magnum with the Mk7 Laser-dazer zippo sight who can’t shoot and who has no idea where his bullets are going.
I hope all this information has been of some help and that you’ve learned not to sweat the small stuff. The other thing I hope you have gotten out of all this is that for truly accurate results you need to test your stuff where you use it.
Disclaimer - As far as I know all the information presented above is correct and I have attempted to insure that it is. However, I am not responsible for any errors, omissions, or damages resulting from the use or misuse of this information, nor for you doing something stupid with it.
from Fr. Frog’s Pad