“The overturning moment MW tends to rotate the bullet about an axis, which goes through the CG (center of gravity) and which is perpendicular to the plane of drag….
Ruprecht Nennstiel, a forensic ballistics expert from Wiesbaden, Germany, has authored a great resource about bullet behavior in flight. Nennstiel’s comprehensive article, How Do Bullets Fly, explains all the forces which affect bullet flight including gravity, wind, gyroscopic effects, aerodynamic drag, and lift. Nennstiel even explains the rather arcane Magnus Force and Coriolis Effect which come into play at long ranges. Nennstiel’s remarkable resource contains many useful illustrations plus new experimental observations of bullets fired from small arms, both at short and at long ranges.
Shadowgraph of .308 Winchester Bullet
A convenient index is provided so you can study each particular force in sequence. Writing with clear, precise prose, Nennstiel explains each key factor that affects external ballistics. For starters, we all know that bullets spin when launched from a rifled barrel. But Nennstiel explains in greater detail how this spinning creates gyroscopic stability:
“The overturning moment MW tends to rotate the bullet about an axis, which goes through the CG (center of gravity) and which is perpendicular to the plane of drag, the plane, formed by the velocity vector ‘v’ and the longitudinal axis of the bullet. In the absence of spin, the yaw angle ‘δ’ would grow and the bullet would tumble.
If the bullet has sufficient spin, saying if it rotates fast enough about its axis of form, the gyroscopic effect takes place: the bullet’s longitudinal axis moves into the direction of the overturning moment, perpendicular to the plane of drag. This axis shift however alters the plane of drag, which then rotates about the velocity vector. This movement is called precession or slow mode oscillation.”
Raise Your Ballistic IQ
Though comprehensible to the average reader with some grounding in basic physics, Nennstiel’s work is really the equivalent of a Ph.D thesis in external ballistics. You could easily spend hours reading (and re-reading) all the primary material as well as the detailed FAQ section. But we think it’s worth plowing into How Do Bullets Fly from start to finish. We suggest you bookmark the page for future reference. You can also download the complete article for future reference and offline reading.
Shoot 101 Quiz
How much of an expert are you when it comes to firearms and ballistics? Test your knowledge with this interactive test. Guns & Ammo magazine created a series of features called Shoot 101. These articles provide “how to” information about shooting, optics, and outdoor gear.
On the Guns & Ammo website, you’ll find the Shoot 101 Ballistics Quiz. The 15 questions are pretty basic, but it’s still fun to see if you get all the answers correct.
You don’t need a lot of technical knowledge. And it’s not all about flight ballistics. Roughly a third of the questions are about projectile types and bullet construction. Note, for some reason the layout doesn’t show all the possible answers at first. So, for each question, be sure to scroll down using the blue scroll bar on the right.
You can do your own experimental calculations using JBM Online Ballistics (free to use). Here is an extreme example, with two printouts (generated with Point Blank software), one showing bullet trajectory at sea level (0′ altitude) and one at 20,000 feet. For demonstration sake, we assigned a low 0.2 BC to the bullet, with a velocity of 3000 fps.
One of our readers asked “What effect does altitude have on the flight of a bullet?” The simplistic answer is that, at higher altitudes, the air is thinner (lower density), so there is less drag on the bullet. This means that the amount of bullet drop is less at any given flight distance from the muzzle. Since the force of gravity is essentially constant on the earth’s surface (for practical purposes), the bullet’s downward acceleration doesn’t change, but a bullet launched at a higher altitude is able to fly slightly farther (in the thinner air) for every increment of downward movement. Effectively, the bullet behaves as if it has a higher ballistic coefficient.
Forum member Milanuk explains that the key factor is not altitude, but rather air pressure. Milanuk writes:
“In basic terms, as your altitude increases, the density of the air the bullet must travel through decreases, thereby reducing the drag on the bullet. Generally, the higher the altitude, the less the bullet will drop. For example, I shoot at a couple ranges here in the Pacific Northwest. Both are at 1000′ ASL or less. I’ll need about 29-30 MOA to get from 100 yard to 1000 yards with a Berger 155gr VLD @ 2960fps. By contrast, in Raton, NM, located at 6600′ ASL, I’ll only need about 24-25 MOA to do the same. That’s a significant difference.
Note that it is the barometric pressure that really matters, not simply the nominal altitude. The barometric pressure will indicate the reduced pressure from a higher altitude, but it will also show you the pressure changes as a front moves in, etc. which can play havoc w/ your calculated come-ups. Most altimeters are simply barometers that read in feet instead of inches of mercury.”
As Milanuk states, it is NOT altitude per se, but the LOCAL barometric pressure (sometimes called “station pressure”) that is key. The two atmospheric conditions that most effect bullet flight are air temperature, and barometric pressure. Normally, humidity has a negligible effect.
It’s important to remember that the barometric pressure reported on the radio (or internet) may be stated as a sea level equivalency. So in Denver (at 6,000 feet amsl), if the local pressure is 24″, the radio will report the barometric pressure to be 30″. If you do high altitude shooting at long range, bring along a Kestrel, or remember to mentally correct the radio station’s pressure, by 1″ per 1,000 feet.”
If you buy one book about Long Range Shooting, this should be it. Based on sophisticated testing and research, this 356-page hardcover from Applied Ballistics offers important insights you won’t find anywhere else. Modern Advancements in Long Range Shooting – Volume II, the latest treatise from Bryan Litz, is chock full of information, much of it derived through sophisticated field testing. As Chief Ballistician for Berger Bullets (and a trained rocket scientist), author Bryan Litz is uniquely qualified. Bryan is also an ace sling shooter and a past F-TR National Champion. Moreover, Bryan’s company, Applied Ballistics, has been a leader in the Extreme Long Range (ELR) discipline.
AUDIO FILE: Bryan Litz Talks about Modern Advancements in Long Range Shooting, Volume 2. (Sound file loads when you click button).
Volume II of Modern Advancements in Long Range Shooting ($39.95) contains all-new content derived from research by Applied Ballistics. Author Bryan Litz along with contributing authors Nick Vitalbo and Cal Zant use the scientific method and careful testing to answer important questions faced by long range shooters. In particular, this volume explores the subject of bullet dispersion including group convergence. Advanced hand-loading subjects are covered such as: bullet pointing and trimming, powder measurement, flash hole deburring, neck tension, and fill ratio. Each topic is explored with extensive live fire testing, and the resulting information helps to guide hand loaders in a deliberate path to success. The current bullet library of measured G1 and G7 ballistic coefficients is included as an appendix. This library currently has data on 533 bullets in common use by long range shooters.
Bryan tells us that one purpose of this book is to dispel myths and correct commonly-held misconceptions: “Modern Advancements in Long Range Shooting aims to end the misinformation which is so prevalent in long range shooting. By applying the scientific method and taking a Myth Buster approach, the state of the art is advanced….”
Bullet Dispersion and Group Convergence
Part 1 of this Volume is focused on the details of rifle bullet dispersion. Chapter 1 builds a discussion of dispersion and precision that every shooter will benefit from in terms of understanding how it impacts their particular shooting application. How many shots should you shoot in a group? What kind of 5-shot 100 yard groups correlate to average or winning precision levels in 1000 yard F-Class shooting?
Chapter 2 presents a very detailed investigation of the mysterious concept of group convergence, which is the common idea that some guns can shoot smaller (MOA) groups at longer ranges. This concept is thoroughly tested with extensive live fire, and the results answer a very important question that has baffled shooters for many generations.
Part 2 of this Volume is focused on various aspects of advanced hand-loading. Modern Advancements (Vol. II) employs live fire testing to answer the important questions that precision hand loaders are asking. What are the best ways to achieve MVs with low ES and SD? Do flash hole deburring, neck tension, primer selection, and fill ratio and powder scales sensitivity make a difference and how much? All of these questions are explored in detail with a clear explanation of test results.
One of the important chapters of Part 2 examines bullet pointing and trimming. Applied Ballistics tested 39 different bullet types from .224 through .338 caliber. Ten samples of each bullet were tested for BC in each of the following configurations: original out of the box, pointed, trimmed, pointed and trimmed. The effect on the average BC as well as the uniformity in BC was measured and tabulated, revealing what works best.
Part 3 covers a variety of general research topics. Contributing author Nick Vitalbo, a laser technology expert, tested 22 different laser rangefinders. Nick’s material on rangefinder performance is a landmark piece of work. Nick shows how shooters can determine the performance of a rangefinder under various lighting conditions, target sizes, and reflectivities.
Chapter 9 is a thorough analysis of rimfire ammunition. Ballistic Performance of Rifle Bullets, 2nd Edition presented live fire data on 95 different types of .22 rimfire ammunition, each tested in five different barrels having various lengths and twist rates. Where that book just presented the data, Chapter 9 of this book offers detailed analysis of all the test results and shows what properties of rimfire ammunition are favorable, and how the BCs, muzzle velocities and consistency of the ammo are affected by the different barrels.
Chapter 10 is a discussion of aerodynamic drag as it relates to ballistic trajectory modeling. You will learn from the ground up: what an aerodynamic drag model is, how it’s measure and used to predict trajectories. Analysis is presented which shows how the best trajectory models compare to actual measured drop in the real world.
Finally, contributing author Cal Zant of the Precision Rifle Blog presents a study of modern carbon fiber-wrapped barrels in Chapter 11. The science and technology of these modern rifle barrels is discussed, and then everything from point of impact shift to group sizes are compared for several samples of each type of barrel including standard steel barrels.
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On LongRangeHunting.com, you’ll find a good article by Shawn Carlock about wind reading. Shawn is a veteran law enforcement marksman and a past USPSA national precision rifle champion. Shawn offers good advice on how to estimate wind speeds and directions using a multitude of available indicators — not just your wind gauge: “Use anything at your disposal to accurately estimate the wind’s velocity. I keep and use a Kestrel for reading conditions….The Kestrel is very accurate but will only tell you what the conditions are where you are standing. I practice by looking at grass, brush, trees, dust, wind flags, mirage, rain, fog and anything else that will give me info on velocity and then estimate the speed.”
Shawn also explains how terrain features can cause vertical wind effects. A hunter on a hilltop must account for bullet rise if there is a headwind blowing up the slope. Many shooters consider wind in only one plane — the horizontal. In fact wind has vertical components, both up and down. If you have piloted a small aircraft you know how important vertical wind vectors can be. Match shooters will also experience vertical rise when there is a strong tailwind blowing over an up-sloping berm ahead of the target emplacements. Overall, Shawn concludes: “The more time you spend studying the wind and its effect over varying terrain the more successful you will be as a long-range shooter and hunter.”
This Editor, as a life-long sailor, also has some suggestions about wind. Many folks may not realize that wind can cycle, both in direction and in speed (velocity). If you are patient, you should be able to sense the timing of the cycles, which will help you predict shifts in wind direction and velocity. While it is tempting to shoot in the lulls, sometimes the true wind vector (angle + speed) may be most constant when the wind is blowing stronger.
Another tip for hunters is to orient your shot, when possible, in alignment with the wind direction. Try to face into the wind, or have the wind at your back. This is especially effective when shooting in a varmint field. Use a string of tape on a pole to show wind angle. Then shoot directly into the wind or with the wind directly at your back. This will minimize horizontal deflection caused by the wind.
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Winchester just unveiled a completely updated website at Winchester.com. The new, mobile-friendly website offers comprehensive information on Winchester ammunition. In addition, the upgraded Winchester website now boasts a full-featured, interactive Ballistics Calculator which runs on web browsers as well as mobile Apps. This new Ballistics Calculator offers an innovative “Shooter’s Eye View”, shown above. You can change the magnification level on the “scope”, and adjust variables (such as temp and range) using the red sliders. Try it out — it’s fascinating to see how the calculated Point of Impact moves as you adjust the sliders.
NEW Winchester Ballistic Calculator Features:
— Calculator provides precise trajectory for hundreds of cartridge types and bullet weights
— Calculator includes library of Ballistic Coefficients.
— Calculator offers visual graphs showing trajectories — with calculated point of impact as well as trajectory curve chart.
— Calculator variables include sight-in range, target range, air temperature, crosswind speed, sight height, and elevation.
— Calculator offers side-by-side comparisons among five separate rounds.
— Calculator offers detailed statistics chart for fine-tuning your shooting.
— Calculator can print handy, small Drop Chart you can attach to your rifle.
The Winchester Ballistic Calculator is available as a free download for iPhone and iPad through the Apple iOs app store, and for Android phones and tablets through Google Play.
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Thomas Haugland, a Shooters’ Forum member from Norway, is a long-range target shooter and hunter. He has created an interesting video showing how to gauge wind velocities by watching trees, grass, and other natural vegetation. The video commentary is in English, but the units of wind speed (and distance) are metric. Haugland explains: “This is not a full tutorial, but rather a short heads-up to make you draw the lines between the dots yourself”. Here are some conversions that will help when watching the video:
.5 m/s = 1.1 mph | 1 m/s = 2.2 mph | 2 m/s = 4.5 mph
3 m/s = 6.7 mph | 4 m/s = 8.9 mph | 5 m/s =11.2 mph
For many riflemen, reading the wind is the toughest challenge in long-range shooting. Wind speeds and directions can change rapidly, mirage can be misleading, and terrain features can cause hard-to-predict effects. To become a competent wind reader, you need range-time and expert mentoring. In the latter department, Frank Galli, founder of Sniper’s Hide, offers a detailed digital resource: Wind Reading Basics for the Tactical Shooter.
Wind Reading Basics is much more than a 47-page eBook — it has charts, instructions for ballistic calculators, and even embedded videos. Galli explains: “We break down the formulas, walk you through using a ballistic computer, and give you all the information in one place. From videos, to useful charts, we make it simple to get started. It’s all about having a plan, and we give you that plan.”
Galli’s Wind Reading Basics, priced at $7.99, can be downloaded from iTunes for iPads, iPhones and iOS compatible devices. Here are sample sections from the eBook (which includes videos):
Need a simple, easy-to-use drop chart for your rifle? Something you can tape right to the buttstock? Then check out Hornady’s handy Online Ballistics Calculator. This user-friendly calculator will compute your drops accurately, and output a handy “Cheat Sheet” you can print and attach to your rifle. Simply input G1 or G7 BC values, muzzle velocity, bullet weight, zero range, and a few other variables. Click “Calculate” and you’ll see the full chart (shown below). Then if you click “View Cheatsheet”, you can generate the simpler, 4-line Drop Chart (shown above).
The online ballistics caculator is easy to use. You can select the basic version, or an advanced version with more data fields for environmental variables (altitude, temperature, air pressure, and humidity). You can also get wind drift numbers by inputing wind speed and wind angle.
Conveniently, on the trajectory output, come-ups are listed in both MOA and Mils — so this will work with either MOA clicks or Mil-based clicks. There are more sophisticated ballistics solvers available on the web (such as the outstanding Applied Ballistics Online Calculator), but the Hornady Calculator is very simple and easy to use. If you just want a basic drop chart, you may want to check this out.
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The Coriolis Effect comes into play with extreme long-range shots like this (2100 yards at Raton, NM). The rotation of the earth actually moves the target a small distance (in space) during the long duration of the bullet’s flight.
When you’re out at the range, the Earth seems very stable. But it is actually a big sphere zooming through space while spinning around its axis, one complete turn every 24 hours. The rotation of the earth can create problems for extreme long-range shooters. During extended bullet flight times, the rotation of the planet causes an apparent deflection of the bullet path over very long distances. This is the ballistics manifestation of the Coriolis Effect.
Bryan Litz of Applied Ballistics discusses explains the Coriolis Effect in his Ballistics Books and Seminars. Bryan notes that Coriolis is “a very subtle effect. People like to make more of it than it is because it seems mysterious.” In most common shooting situations inside 1K, Coriolis is not important. At 1000 yards, the Effect represents less than one click (for most cartridge types). Even well past 1000 yards, in windy conditions, the Coriolis Effect may well be “lost in the noise”. But in very calm conditions, when shooting at extreme ranges, Bryan says you can benefit from adjusting your ballistics solution for Coriolis.
Bryan explains: “The Coriolis Effect… has to do with the spin of the earth. You are basically shooting from one point to another on a rotating sphere, in an inertial reference frame. The consequence of that is that, if the flight time of the bullet gets significantly long, the bullet can have an apparent drift from its intended target. The amount [of apparent drift] is very small — it depends on your latitude and azimuth of fire on the planet.”
Coriolis is a very subtle effect. With typical bullet BCs and velocities, you must get to at least 1000 yards before Coriolis amounts to even one click. Accordingly, Bryan advises: “Coriolis Effect is NOT something to think about on moving targets, it is NOT something to think about in high, uncertain wind environments because there are variables that are dominating your uncertainty picture, and the Coriolis will distract you more than the correction is worth.”
“Where you could think about Coriolis, and have it be a major impact on your hit percentage, is if you are shooting at extended range, at relatively small targets, in low-wind conditions. Where you know your muzzle velocity and BC very well, [and there are] pristine conditions, that’s where you’re going to see Coriolis creep in. You’ll receive more refinement and accuracy in your ballistics solutions if you account for Coriolis on those types of shots. But in most practical long-range shooting situations, Coriolis is NOT important. What IS important is to understand is when you should think about it and when you shouldn’t, i.e. when applying it will matter and when it won’t.”
The Coriolis Effect — General Physics
The Coriolis Effect is the apparent deflection of moving objects when the motion is described relative to a rotating reference frame. The Coriolis force acts in a direction perpendicular to the rotation axis and to the velocity of the body in the rotating frame and is proportional to the object’s speed in the rotating frame.
A commonly encountered rotating reference frame is the Earth. The Coriolis effect is caused by the rotation of the Earth and the inertia of the mass experiencing the effect. Because the Earth completes only one rotation per day, the Coriolis force is quite small, and its effects generally become noticeable only for motions occurring over large distances and long periods of time. This force causes moving objects on the surface of the Earth to be deflected to the right (with respect to the direction of travel) in the Northern Hemisphere and to the left in the Southern Hemisphere. The horizontal deflection effect is greater near the poles and smallest at the equator, since the rate of change in the diameter of the circles of latitude when travelling north or south, increases the closer the object is to the poles. (Source: Wikipedia)
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With the Berger Southwest Nationals underway this week, we thought we’d steer our readers to a very useful resource, courtesy Berger Bullets. This online Stability Calculator helps shooters determine the optimal twist rate for their choice of projectiles.
Berger Twist-Rate Stability Calculator
On the Berger Bullets website you’ll find a handy Twist-Rate Stability Calculator that predicts your gyroscopic stability factor (SG) based on mulitiple variables: velocity, bullet length, bullet weight, barrel twist rate, ambient temperature, and altitude. This cool tool tells you if your chosen bullet will really stabilize in your barrel.
How to Use Berger’s Twist Rate Calculator
Using the Twist Rate Calculator is simple. Just enter the bullet DIAMETER (e.g. .264), bullet WEIGHT (in grains), and bullet overall LENGTH (in inches). On its website, Berger conveniently provides this info for all its bullet types. For other brands, we suggest you weigh three examples of your chosen bullet, and also measure the length on three samples. Then use the average weight and length of the three. To calculate bullet stability, simply enter your bullet data (along with observed Muzzle Velocity, outside Temperature, and Altitude) and click “Calculate SG”. Try different twist rate numbers (and recalculate) until you get an SG value of 1.4 (or higher).
Gyroscopic Stability (SG) and Twist Rate
Berger’s Twist Rate Calculator provides a predicted stability value called “SG” (for “Gyroscopic Stability”). This indicates the Gyroscopic Stability applied to the bullet by spin. This number is derived from the basic equation: SG = (rigidity of the spinning mass)/(overturning aerodynamic torque).
If you have an SG under 1.0, your bullet is predicted not to stabilize. If you have between 1.0 and 1.1 SG, your bullet may or may not stabilize. If you have an SG greater than 1.1, your bullet should stabilize under optimal conditions, but stabilization might not be adequate when temperature, altitude, or other variables are less-than-optimal. That’s why Berger normally recommends at least 1.5 SG to get out of the “Marginal Stability” zone.
In his book Applied Ballistics For Long-Range Shooting, Bryan Litz (Berger Ballistician) recommends at least a 1.4 SG rating when selecting a barrel twist for a particular bullet. This gives you a safety margin for shooting under various conditions, such as higher or lower altitudes or temperatures.
Story idea from EdLongrange. We welcome reader submissions.
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In our Shooters’ Forum, there was an discussion about a range that was threatened with closure because rifle over-shoots were hitting a farm building over two miles from the firing line. One reader was skeptical of this, asking “how’s that possible — were these guys aiming at the stars?” Actually, you may be surprised. It doesn’t take much up-angle on a rifle to have a bullet land miles down-range. That’s why it’s so important that hunters and target shooters always orient their barrels in a safe direction (and angle). Shooters may not realize how much a small tilt of the barrel (above horizontal) can alter a bullet’s trajectory.
How many degrees of muzzle elevation do you think it would take to hit a barn at 3000 yards? Ten Degrees? Twenty Degrees? Actually the answer is much less — for a typical hunting cartridge, five to seven degrees of up-angle on the rifle is enough to create a trajectory that will have your bullet impacting at 3000 yards — that’s 1.7 miles away!
Five degrees isn’t much at all. Look at the diagram above. The angle actually displayed for the up-tilted rifle is a true 5.07 degrees (above horizontal). Using JBM Ballistics, we calculated 5.07° as the angle that would produce a 3000-yard impact with a 185gr .30-caliber bullet launched at 2850 fps MV. That would be a moderate “book load” for a .300 Win Mag deer rifle.
Here’s how we derived the angle value. Using Litz-derived BCs for a 185gr Berger Hunting VLD launched at 2850 fps, the drop at 3000 yards is 304.1 MOA (Minutes of Angle), assuming a 100-yard zero. This was calculated using a G7 BC with the JBM Ballistics Program. There are 60 MOA for each 1 degree of Angle. Thus, 304.1 MOA equals 5.068 degrees. So, that means that if you tilt up your muzzle just slightly over five degrees, your 185gr bullet (2850 fps MV) will impact 3000 yards down-range.
Figuring Trajectories with Different Bullets and MVs
If the bullet travels slower, or if you shoot a bullet with a lower BC, the angle elevation required for a 3000-yard impact goes up, but the principle is the same. Let’s say you have a 168gr HPBT MatchKing launched at 2750 fps MV from a .308 Winchester. (That’s a typical tactical load.) With a 100-yard zero, the total drop is 440.1 MOA, or 7.335 degrees. That’s more up-tilt than our example above, but seven degrees is still not that much, when you consider how a rifle might be handled during a negligent discharge. Think about a hunter getting into position for a prone shot. If careless, he could easily touch off the trigger with a muzzle up-angle of 10 degrees or more. Even when shooting from the bench, there is the possibility of discharging a rifle before the gun is leveled, sending the shot over the berm and, potentially, thousands of yards down-range.
Hopefully this article has shown folks that a very small amount of barrel elevation can make a huge difference in your bullet’s trajectory, and where it eventually lands. Nobody wants to put holes in a distant neighbor’s house, or worse yet, have the shot cause injury. Let’s go back to our original example of a 185gr bullet with a MV of 2850 fps. According to JBM, this projectile will still be traveling 687 fps at 3000 yards, with 193.7 ft/lbs of retained energy at that distance. That’s more than enough energy to be deadly.
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Shoot 101 Quiz
One of our readers asked “What effect does altitude have on the flight of a bullet?” The simplistic answer is that, at higher altitudes, the air is thinner (lower density), so there is less drag on the bullet. This means that the amount of bullet drop is less at any given flight distance from the muzzle. Since the force of gravity is essentially constant on the earth’s surface (for practical purposes), the bullet’s downward acceleration doesn’t change, but a bullet launched at a higher altitude is able to fly slightly farther (in the thinner air) for every increment of downward movement. Effectively, the bullet behaves as if it has a higher ballistic coefficient.
Part 2 of this Volume is focused on various aspects of advanced hand-loading. Modern Advancements (Vol. II) employs live fire testing to answer the important questions that precision hand loaders are asking. What are the best ways to achieve MVs with low ES and SD? Do flash hole deburring, neck tension, primer selection, and fill ratio and powder scales sensitivity make a difference and how much? All of these questions are explored in detail with a clear explanation of test results.
When you’re out at the range, the Earth seems very stable. But it is actually a big sphere zooming through space while spinning around its axis, one complete turn every 24 hours. The rotation of the earth can create problems for extreme long-range shooters. During extended bullet flight times, the rotation of the planet causes an apparent deflection of the bullet path over very long distances. This is the ballistics manifestation of the 
In our Shooters’ Forum, there was an discussion about a range that was threatened with closure because rifle over-shoots were hitting a farm building over two miles from the firing line. One reader was skeptical of this, asking “how’s that possible — were these guys aiming at the stars?” Actually, you may be surprised. It doesn’t take much up-angle on a rifle to have a bullet land miles down-range. That’s why it’s so important that hunters and target shooters always orient their barrels in a safe direction (and angle). Shooters may not realize how much a small tilt of the barrel (above horizontal) can alter a bullet’s trajectory.
