Chapter 1 Basic Shooting Science The science of shooting is mostly the science of motion. That is to say, most of this book is about basic physics. There will be a bit of chemistry and some optics (which is physics, again) but physics covers most of it. The basic science is also the easiest but it is, for all that, also definitely the most important. Thus, I will begin with the basic science and then progress to some advanced topics where the motions or details are somewhat more complicate, as, for example, the odd spiral motion of the tip of a bullet in flight which very few shooters are even aware of. Lastly I will tackle the science of important devices, like telescopic sights, that shooters commonly use to assist them in the task of shooting accurately and effectively even at long distances. As a physics professor, I confess I am sometimes overcome with the desire to write equations as explanations. Although I have been unable to completely suppress this ingrained impulse, I have placed all such stuff in the appendices where only readers as nerdy as myself will likely venture. The rest of you are safe as anyone can be who undertakes to learn a little applied physics. Projectile motion The most important thing for a shooter to know about bullets and shot is that, while in flight, they are all projectiles. Projectile motion is one of the most familiar types of motion. Thrown rocks and sticks, balls hit or shot, a thrown javelin, and a stream of water are all projectiles. Projectiles have been studied since the time of Aristotle but a useful understanding of their motion had to wait until the Renaissance when cannon balls became important. As everyone knows, the Renaissance was a time of renewed learning but it was also a time of political unrest and warfare. The newly discovered cannon was busy making tall castles obsolete and the defense industry of the day was into designing better, and lower, fortifications. The offense industry of the day was of course busy trying to understand the path of a cannon ball so they could take castles apart more quickly and efficiently. A fellow by the name of Nicolo Fontana spent many hours observing and drawing the paths of cannon balls. Fontana was a mathematics professor best known now for being the first to discover the general solution for cubic equations. As a lad of twelve he had been given an horrific saber cut in the face when the French invaded his home town near Venice. The cut left him with a speech impediment, a serious handicap for a professor (and also for his students), but Fontana was widely known by his nickname, Tartaglia, Italian for “the stammerer.” He was so closely associated, in fact, with his nickname that for a long time his real name remained unknown. Many sources still refer to him as Nicolo Tartaglia or just Tartaglia. It seems obvious now but no one at the time realized that cannon balls, and all projectiles generally, follow smoothly curved paths. Aristotle had mislead everyone, himself included, into thinking that projectiles travel upward and outward in a straight line until they “run out of gas” and then fall pretty much straight down. Seemingly, Tartaglia was the first to realize the path is a smooth curve. The most important feature of projectile motion was then discovered by another Venetian, Galileo Galilei. It occurred to him that a projectile is doing two things at the same time. It is traveling horizontally, parallel to the ground, and it is moving vertically, first up and then down. The two motions are independent of each other but they happen simultaneously. This simple idea is the key to predicting the motion of a projectile. In fact, predicting that motion is exactly where Galileo went next. Since the two motions are independent, they can be studied separately; so Galileo first investigated the vertical motion and showed that it was just a case of free fall, that is, constant acceleration under gravity. He measured that acceleration. We now give it in meters or feet (neither unit was available to Galileo) as 32.2 ft/s2 or 9.8 m/s2. Use whichever unit makes you happier. Happily, when Galileo then studied the horizontal motion he found it is even simpler; it’s just motion at constant speed. With both motions well understood, Galileo then was easily able to prove that Tartaglia’s smooth curve was a parabola. Parabolas are familiar to most of us as the curves generated by plotting quadratic equations in high school algebra. The parabolas that describe projectile paths are the upside-down ones, cups that will not hold water, with the highest point in the middle of the flight. To see the relevant equation, check out Appendix 1. What part of this infinite curve a particular object follows depends on starting and ending conditions. A cannon ball shot uphill will end at a higher point than it started from and the highest point of the path will be closer to the end than to the beginning, an asymmetric trajectory. Water shooting horizontally from a spout will begin at the high point and fall out and down from there. But, in all cases, the entire path is some part of a parabola. Like all other projectiles, bullets fall continuously from the time they leave the muzzle. Simultaneously they move very quickly parallel to the ground. Because of the great speed horizontally, they arrive at the target very quickly and thus are falling for very short times. The short time of fall means a short distance of fall so the parabolic path of a bullet is very flat, a greatly stretched out parabola but still a parabola. The conclusion: bullets never travel in exactly straight lines, there is always some drop. It would be nice to be able to leave things in this simple state. Bullets travel on parabolic paths, no complications. But there are complications. A major one that worried even Galileo is air resistance. The supposedly constant horizontal speed of a projectile is not really constant because the air slows it down. The longer the time of travel, the greater is the slowing from the air. Air resistance also affects the vertical motion so the supposedly constant vertical acceleration is not truly constant either but also drops off over time. Additionally, Isaac Newton later found that the acceleration due to gravity decreases with altitude and is weaker at the top of a projectile’s path than at the bottom. Poor Galileo! Wrong on both counts. In Galileo’s defense, let me note immediately that this altitude effect is miniscule for anything but long range rockets or modern cannon that shoot 25 miles or more. But it does imply that the path of a projectile is not truly parabolic. The actual path is part of an ellipse but the difference is too small to measure for our projectiles. On the other hand, air resistance is the primary and dominant force acting on a projectile. For sporting arms bullets traveling at modern speeds, it is fifty to one hundred times stronger than gravity! Thus, how important each of these factors is depends mostly on the type of projectile and the distance involved. The lighter the projectile and the greater the distance, the more important air resistance becomes. For example, air resistance to moving size 6 shot is so great that 40 yards is about the useful limit of a shotgun. The rifleman or hunter, shooting out to perhaps 150 yards, will scarcely notice air resistance effects. Shooting greater distances, the varmint hunter may be affected by air resistance and the military or police sniper will definitely need to take it into account. Shooting even further, pity the poor artillery lieutenant of yesteryear who was very lucky if air resistance was the only factor he had to consider. Appendix 1 contrasts the equation of such a path with a parabolic path just to show how much more complicated this one factor makes the path. The modern lieutenant, of course, has a lot of computer help and it is the computer programmer who has to think of all these factors plus quite a few more. Figure 2 shows the difference air resistance can make in the actual motion of a projectile compared with the expected parabolic path. Figure 2. How Air Resistance Alters Projectile Paths One of the reasons Galileo got it wrong is that he was pretending, for simplicity, that projectiles are points when, in reality, they are three dimensional objects. A point-sized object will not be affected by air resistance but a real object will. Also, a point can travel on a parabolic path but a real object makes for, at best, a very thick path. Where in the thick line is the parabola then? The answer is that the center (center of mass) of the projectile is the point on the projectile that follows Galileo’s parabolic path. For most sports hunters and shooters, then, Galileo’s parabolic path works just fine. Only when you push the envelope do you get into serious complications. Shotguns push the envelope in the direction of small, light projectiles that are strongly affected by air resistance even though their speed is relatively low. On the other hand, distance shooters (even those with very large and heavy projectiles like 16” shells from a battleship) are up against both air resistance and air motion (wind) problems and, to a lesser extent, the weakening of gravity with increased altitude of the path. Muzzle Velocity and Recoil In the study of projectiles, we assume the projectile starts out with a horizontal speed and a vertical speed (usually upward but it can be downward). A combination of these speeds (the “vector” combination) is called the “initial velocity.” This initial velocity is stated in terms of a speed plus a direction. Note that the word “velocity” refers to more than just speed. For example, we might say the initial velocity of a thrown baseball was 38 m/s (the speed) at 350 up from the horizontal. Both the speed plus the direction must be specified for a full, useful description. Then, given the initial velocity, one can quite easily predict the precise parabolic path of the projectile. For bullets and shot packets the speed (in m/s or fps) part of the initial velocity is called the “muzzle velocity.” (See Appendix 2 for a technical correction). I suppose everyone knows that muzzle velocity is the speed at which a bullet or a shot packet leaves the end of the barrel of a gun, the muzzle. When talking about projectile motion, we don’t care how this muzzle velocity arises. However, we must take a look at the subject because you cannot understand the effects of different cartridges and loads without understanding the basic science behind muzzle velocity. That’s easy, you think. The pin strikes the primer and it detonates. That ignites the powder which burns and creates hot gases that expand rapidly, and push the bullet out of the barrel. True. But does the barrel length affect muzzle velocity? What about different powders or loads? What about different bullet weights? The bottom line is that all of these factors affect muzzle velocity. And there is a big bottom line here. Manufacturers of ammunition spend large amounts of research and development money massaging their ammo lines to give us the best rounds for the huge range of hunting and shooting needs. Detailed and pains-taking science goes into this area of the shooting sports. Let’s look at the process in two stages, ignition and expansion. First, in ignition, the pin strikes the percussion cap or primer and ignites it and then the powder. The currently used powders explode without air either because they contain oxygen in chemical form or detonate rather than burn. Once the powder is completely, or almost completely, expended the second stage is expansion as the hot gases of the explosion push the projectile down and out of the barrel. Of course, the two stages merge smoothly from one to the other, the exploding does not all occur before the gases start to expand. Both stages have associated times. Manufacturers keep a close eye on the time from ignition to the end of burning the powder. Powder recipes and processing are continually being adjusted and tested to control this time. The time of expansion of gases overlaps the burn time just as the ignition and expansion stages overlap. Those of us who hand load can choose powders types and amounts to control burning time but the ordinary shooter has control only over the expansion time. The longer the barrel of the gun selected, the longer the expansion time will be for any choice of ammunition. The most critical thing here is that the total time down the barrel must be longer than the burn time. If it isn’t, the powder will come out of the barrel still burning, a spectacular but dangerous situation. The sciences involved here are the chemistry of the percussion cap and powder and the physics of how the expanding gases accelerate the projectile. Once we get past the idea that the powder explodes and becomes a hot gas, the chemistry part is very much another subject and I will give more of the details at the end of this chapter. All I will say about it now is that some powders burn quickly and others slowly. For example, black powder is notoriously slow burning. Powder for pistol ammunition needs to burn faster than the comparable rifle powder and so on. In all situations, burn times must match the gun barrel. Don’t use rifle ammo in a pistol. Don’t shorten a barrel without checking up on the burn time of the ammo you plan to use. The physics is easier and more familiar. Most everyone knows Newton’s second law of motion: F = ma. It tells us that a force (F) applied to a mass (m) accelerates (a) that mass. Larger masses then must accelerate more slowly (for a given force). In a gun barrel, the important force is the pressure of the expanding gas times the area of the back end of the projectile (we’re neglecting friction with the barrel and the effects of the outside air). The mass is of course that of the projectile. The acceleration down the length of the barrel determines the muzzle velocity. Obviously, the longer the barrel, the more opportunity the acceleration has to create speed in the projectile. Hence, muzzle velocity increases as barrel length increase. On the other hand, the greater the projectile mass, the lower the acceleration and, therefore, the lower the muzzle velocity. To get the same muzzle velocity from a 180 grain bullet will take more powder than required by a 150 grain bullet in the same gun. Obviously, long barrels make the most efficient use of powder. Ideally, the barrel should be long enough for the hot gases to expand to exactly atmospheric pressure, the pressure on the front of the bullet, just before it exits the barrel. In calculating maximum possible muzzle velocity, that is precisely the criterion used. Oddly, it has another benefit; it leads to a silent gun (if the muzzle velocity is subsonic). The barrel gases match the outside air pressure and do not create a shock wave (muzzle blast) as they enter the outer air. So, a long barrel makes for a quieter gun as well as a more efficient one. Unfortunately, there is an obvious, major problem with a longer barrel. Long barrels get to be too big to fit in bags, cars, and, in the extreme, spaces between trees. Also (more physics) they are too hard to hold steady and, at least with shotguns, to swing easily and smoothly. In case my comments inspire you to silence your rifle by stretching out the barrel, the necessary barrel length is typically about 7 or 8 feet. The exact value depends on the load and propellant used, of course. Before you put down this book and jump up to tamper with your guns, please note that this type of silencer only quiets the sound of the hot gases hitting the atmosphere. Most rounds, however, make another, additional noise. Any projectile with a muzzle velocity above the speed of sound creates a sonic boom. The sonic boom is not as loud as that of a jet fighter but it cannot be silenced. No type of silencer can damp a sonic boom. The sonic boom is a shock wave made by and traveling with the bullet so any silencer for that would likewise have to travel with the bullet. Hopeless! Hence, silencers are used mostly with the subsonic rounds of pistols. The speed of sounds varies a bit with atmospheric conditions (and altitude) but is typically around 1100 fps or 345 mps. Check an ammo catalog and you will be very hard pressed to find muzzle velocities that low. The “cowboy-action” ammo that is designed to simulate the slower ammo of the old West is not nearly that slow. Even shotgun shot packets typically exit the barrel at speeds above the speed of sound. So the only guns you can totally silence with a long barrel would be air rifles. But then, they’re not all that loud to start with. Newton’s third law of motion introduces us to an important complication related to muzzle velocity. You have probably heard this law expressed in the remark, “for every action there is an equal and opposite reaction.” In this case the action is the force pushing the projectile (and hot gases) out the barrel, producing the muzzle velocity and muzzle blast. The reaction is then in the opposite direction and equal in strength, the recoil of the gun. As the projectile is pushed forward, the gun is pushed backward by an equal force. Now Newton’s second law, F = ma, takes over. The comparatively light bullet (m is small) and gases are strongly accelerated to the high muzzle velocity, in the case of the bullet. The gun, however, is much more massive than the projectile (its mass is big) and it is weakly accelerated to a very much lower recoil velocity. If you want to calculate the recoil velocity, the formula is in Appendix 2. Thus, the infamous recoil of a gun is the flip side of muzzle velocity. You can’t have one without the other. Of course, there are ways to reduce the effects of recoil. Recoil pads for the end of the gun stock and muzzle breaks are some of the inventions intended to help soften or control recoil effects. We will look at them later in the chapter on advanced shooting science. The easiest and cheapest way to reduce recoil is to increase the effective mass of a gun. Certain precision competition rifles are designed to be quite heavy for just this reason. But the easiest way to add mass to your gun is to add some of your own mass. That is why all the shotgunning instructors in the world tell their pupils to snug the butt of the shotgun firmly into their shoulders. The recoil is then shared between the gun and a substantial part of the body of the gunner. It’s good advice for all shooters. Done right, you will simply never bruise your shoulder. You can get a sore shoulder from proper shotgunning if you do a lot of shooting in a day but there’s no need to get a purple shoulder. Recoil has another effect worth mentioning: it makes the gun turn upward when fired. This effect is most problematic for an automatic weapon, like a machinegun, with a high firing rate. The muzzle rises and aiming is hard to maintain. It is also apparent when you shoot a pistol single-handed. The pistol jumps upward when fire. The reason for this is that the recoil produces an upward torque on the gun. The recoil is applied along the barrel but the gun is held at a lower point by the hand or at the shoulder. The lower point is the natural pivot of the gun but the recoil is applied at a higher point and backward toward the shooter. Thus, the barrel moves back, turning up and around the pivot. The end of the barrel rises and the gun is now pointing high and off target. It is possible to eliminate this effect by redesigning the stock or handle of the gun to be in line with the barrel but then aiming along the barrel would require mirrors or some other optical device. On the whole it seems best to learn to live with the effect. A related problem is one I encountered when I bought my second-hand .30-30 Winchester ’94 years ago. That model is famously top- ejecting and any scope mounted on it must be side- mounted. I went to the range to zero the gun and was having a terrible time. I’d make corrections and they’d be wrong each time I checked. The problem was that the scope mounting screws were not gripping. Once I put lock-tite on them, everything settled down. Without the screws firmly tightened down and with the side-mounted scope, every recoil applied a more or less random torque to the scope mounting and moved the scope around erratically. Rotation and Stability When an object rotates, the axis of the rotation acquires a special status. Get a top spinning very fast and upright. Tap it sideways. The axis of rotation will wobble briefly and then return to a steady upright position. Take a bicycle wheel with bolts along the axis as handles. When it is not rotating, the whole wheel falls over quickly if you hold it loosely by one handle. But get it spinning fast, however, and the wheel will refuse to drop. Instead, the axis will rotate around the vertical as the wheel rotates around the axis itself. If you let gravity act on a spinning body, the body responds to the force of gravity by letting the spin axis rotate around the vertical direction of the gravitational force. We can deduce a useful rule from this observation. The spin axis of the wheel, in rotating around the vertical direction of gravity is turning into the axis gravity would ordinarily make the wheel turn about as it falls. It is in fact a fully correct conclusion that when you apply a turning force to a spinning body, the spin axis rotates around the direction of the applied force. Physicists call this continuous rotation of the spin axis “precession.” What is going on here is actually just a form of inertia, going back to Newton’s first law of motion, the rule that the motion of an object doesn’t change unless a force is applied to make it change. The plane of rotation of the wheel is, in a sense, the sum of directions of travel of all the parts of the wheel. By Newton’s first law, it wants to stay pointed in the same direction. Since the axis of rotation is a line that is always perpendicular to the plane of rotation, it is simpler to think of the axis than of the plane. So we focus on the axis of rotation although the physics is really generated by the plane of rotation. Thus, a spinning body has an amazing stability; it tries very hard to keep its rotation axis always pointing in the same direction in space. This is very important for each of us in a very personal sense. That is, our home planet rotates on its axis once a day. Although the spin rate is slow, the mass of the Earth is enormous. Hence, the spin axis of the Earth is very stable and has been (with the very important assistance of our large, close Moon) for billions of years. Figure 3. Motions of a Precessing Top How is that personally important? Well, if the Earth did not have rotational stability, the North Pole would sometimes point towards the Sun, Alaska and Greenland would be hot while the South Pole would be even colder than it is now. Some land now at the equator would be freezing and other parts of the equator would be in the new temperate zone. Just as soon as we thought we were used to that, the axis would slip somewhere else. If you think the weather changes too often now, just think how miserable that would be. In fact, it would often be simply fatal for life. I kid you not, this could happen. Poor Mars with its pitiful excuses for moons has been doing it for billions of years. Back to projectiles. Air resistance on real, 3-D, projectiles not only slows them down but can also have random and unpredictable effects on their motion. Perhaps the most familiar such situation is the “knuckleball” pitch in baseball. Baseball fans know that knuckleballs are so unpredictable that catchers use oversized gloves to get an extra margin for catching them. The knuckleball is thrown without spin or with a very small amount of spin and has none of the directional stability of a pitch with a good amount of spin on it. Consequently, it is at the mercy of air drag disturbances created by the seams of the ball. Seams reduce drag so the air pushes the ball toward the seam. If the ball rotates slowly, the push rotates too and the ball seems to flutter around its expected path. As one player complained, trying to hit a knuckleball is like “trying to eat soup with a fork.” Cannonballs are much denser than baseballs but they travel further. Without much spin, a not truly spherical cannonball is affected by this knuckleball effect but this is not apparent over short distances. Gunners who needed to make accurate use of early versions of the cannon soon realized, however, that it was hard to get two shots in a row onto the same spot. Of course, they were not certain the variations between cannonballs or differences of powder loads or aiming were not the cause of this. As improvements in manufacturing techniques came along, it become apparent that the lack of spin was a factor in the lack of accuracy. Muskets were a much more obvious problem. They fired much lighter balls with very little spin on them. Consequently, they were totally unreliable beyond about 50 yards. Brigades of early musketeers were prized by their military commanders for the terror engendered by the loud, noisy discharge of their collective muskets more than any supposed havoc they could wreak on the enemy. Indeed, during the American Revolutionary War British officers had no “aim” command for infantry. They just said, “Ready … fire.” The infantry were expected just to load, point in the general direction of enemy lines and shoot. Of course, massed general firing at close quarters against masses of enemy produced many wounded men and dead men. But no one knew who shot whom; no one could be regarded as a marksman. Here again variations in loads and musket balls confused efforts to improve the weapon. The main improvement, spinning the projectiles by rifling the insides of the barrels, began in Germany around 1460. Then as now, spiral grooves were cut in the bore of the gun leaving high spirals called lands. The lands grip and spin any projectile traveling down the barrel. As early as 1520, sporting rifles were being made both in Germany and in Italy. It is not certain that improved accuracy was the only objective at the time. Some weapons historians suggest that rifling was an effort to help clean the barrel. True, powders of the day produced a great deal of fouling but I find the idea doubtful and unpersuasive. Straight grooves inside a barrel are easier to make and would seem far better suited to keeping a barrel clean than the twisting grooves of rifling. There has to have been a reason for the rotation. I have encountered claims that medieval crossbowmen had learned that angling the fletching to spin their bolts improved the accuracy of their weapon, so perhaps the gun makers understood from the outset that the rifling would produce a more reliable instrument. I am a bit skeptical of this information, however, since modern finned projectiles have no need of spin stabilization. It would seem medieval crossbow bolts would have profited little by being spun. Rifling was an expensive machining process and it did not come into its own until Pennsylvania gun smiths began to make the Pennsylvania long rifles (later called Kentucky rifles). These guns were later made famous by frontiersmen like Daniel Boone. By around 1750, Pennsylvania Dutch gun smiths in the Lancaster area, my people and my family home, began to make rifles with almost four foot long barrels but light stocks and major improvements in accuracy. Sharpshooters using these rifles claimed a standard ability to hit a man in the head at 200 yards. Very likely they did not exaggerate. British officers were both dismayed by and impressed with the rifles Americans used against them in the Revolutionary War. Indeed, the breech-loading Ferguson rifle (invented and patented by Major Patrick Ferguson) was used by his riflemen at the Battle of Brandywine, as in part, a British answer to the American weapon. A few decades later, the British used companies of green-coated riflemen in the Napoleonic Wars. Like the rifles of the American Revolution, they were used to pick off officers in advancing troop formations. Keep in mind that most of these guns were still flintlock muzzle-loaders that fired a ball and patch. They were basically hunting tools, ill-adapted to military use. The military deficiency was their slow loading speed, a failing that the breech-loading Ferguson rifle was designed to overcome. Ferguson himself could fire his rifle with very high accuracy at a rate of one shot in ten seconds. In muskets, the ball drops easily and quickly down the barrel for a high rate of firing. In a rifle, the lands of the rifling must grip the ball so there must be a tight fit down the barrel or the ball will not spin. Loading the muzzle-loading Pennsylvania long rifle was a chore involving time and effort and a goodly force applied to the ramrod. The close fit down the barrel meant that the propulsive force of the expanding gases was finally being used efficiently. This, however, was purely an accidental improvement lost at the time in the over-riding interest in the greater accuracy of the spinning projectile. By that time, hunting had become mostly a sport in Europe but it remained a necessity in much of America where rifling was evidently seen as an initial investment that would soon be paid back in more game bagged per round of ammunition. Thus, American rifle-making evolved rapidly over the next century. The exigencies of the Civil War provided the impetus that finally permanently drove both handheld weapons and cannons into the camp of rifled barrels and spinning projectiles. Rifles and rifled cannon like the Parrott gun became dominate weapons. By the end of the Civil War the smooth-bore musket was obsolete. With it went the ball projectile, replaced by the more bullet shaped “minie ball”. Rifling can be either clockwise or counter-clockwise (seen as the projectile goes away from you) and typically involves 4 to 8 turns down the length of the barrel although a wide range of twists is available. Since bullets are made slightly bigger than the barrel diameter before grooves are cut in it, bullets fit quite tightly and must twist with the lands and grooves as they move down the barrel. As fans of detective fiction know, the bullet is scored by this process in ways characteristic of the barrel. The end result is a bullet spinning tens of thousands of times a second as it begins its brief life as a projectile. The exact spin rate depends on the twist rate of the lands and grooves and also on the speed of the projectile. An object spinning at twenty thousand times a second has great directional stability. If it is a fast bullet, it also moves on a very flat trajectory. These two effects combine to keep a bullet basically pointing forward throughout its entire flight. This means it presents the same aspect or face to the air as it travels. Air resistance is then very constant in direction as well as strength. No knuckleball effects here; the bullet follows a true, predictable path. The importance of spin can be easily seen if you ever find an old, much used rifle with the lands badly worn down. Bullets fired from such guns often “keyhole.” That is, they hit a target turned sideways rather than front first, making a slot-like hole rather than a round hole. Two consecutive shots are likely to end up in very different places. After you test such a gun and take pictures to show how bad it is, give it to the Cracker Barrel Restaurants to hang over a fireplace somewhere. You certainly don’t want to keep on shooting with it. There is a down side to the high spin rates. They can tear a bullet apart. The inertial tendency of each part of the bullet to travel on a straight line produces internal strains in the bullet (often incorrectly called centrifugal “forces”) that can destroy it. Bullets made of a single piece of metal can be spun very fast before trouble develops. Many high speed modern bullets, however, are composites with coatings, jackets, and tips added to the main body. Composites bodies help prevent bullet deformation under the tremendous acceleration needed to go to high muzzle velocity but they do not have the integrity of solid bodies and tend to break apart at spin rates that would not be a problem for a single piece of metal. High spin rates below those needed to tear a projectile apart can still be a problem because they can cause a projectile to be “over-stabilized.” That is, with a very high rate of spin a projectile is very determined to keep its spin axis direction constant. Since the projectile path curves and, hence, does not maintain a constant direction the spin axis and the path direction increasingly come in conflict. We will look more closely at this problem shortly. The spin rate is primarily controlled by the rifling of the barrel. The other major factor is the muzzle velocity of the particular cartridge. The same cartridge can be fine in one gun but, because of different rifling, unstable or even over-stabilized in another gun. A “hotter” cartridge in your old reliable gun may become unstable with a higher muzzle velocity and a composite bullet. It is even more likely to become over-stabilized. Air Resistance It’s time now for a closer look at the effects of air resistance on a projectile. Air resistance is usually called the drag force (drag, for short) and it continuously slows the projectile down. It is directed exactly opposite to the projectile path at all points along the path. As I have already noted, drag is far and away the greatest force acting on a projectile. Surprisingly, while it is typically 50 to 100 times the force of gravity, it is not the dominant force for determining the projectile’s path. Gravity controls projectile motion. Air resistance only modifies gravitational effects. How can a force that is far and away the largest force not be dominant? Like all frictional forces, air resistance is a reactive force. The projectile pushes through the air, applying an action force to it. By Newton’s third law of motion, the air then applies an equal and opposite reaction force to the projectile to retard its motion. Without the motion of the projectile, there is no air resistance. In effect, the projectile creates the air resistance. With its direction always opposing that of the projectile’s motion, the air can only continually reduce the speed of the projectile. Thus, drag only deforms the parabolic path gravity would otherwise impose on the trajectory. While drag is the most important way air affects a projectile, it is certainly not the only way. Air resistance also creates spin damping, that is, it not only slows down the forward and up and down motions of a projectile but it also slows down the spinning of a projectile. By good fortune, this effect is so small that we can ignore it. There are other ways air affects a projectile. Basically there are two divisions of aerodynamic forces: those that directly oppose motion (the drag and spin damping forces) and those applied at right angles to the motion (the path). These forces are perpendicular to the path and hence are called cross-path forces. Here again, these forces are not usually important and we can reserve them for later consideration under advanced science topics. The force of gravity is applied all over an object and the forces on all parts of the body add in such a way that we can regard them all as a single force (the weight of the object) acting at the center of gravity of the body. For most bodies, the center of gravity is the same as the geometric center of the body (the centroid). The center of gravity of a rigid body, like a bullet, does not change position in the bullet so long as the bullet retains its shape. The orientation with respect to the Earth, the speed of the body, or whatever factors you might suggest, will not affect the center of gravity. Of all the aerodynamic forces, only drag acts on the center of gravity. If you try to imagine the interaction of the bullet and the air on a microscopic level where the bullet is plowing through an enormous collection of air molecules, you can see how complicated air resistance is. The molecules in front of the bullet collide with it and bounce away forward or off to the side. The molecules at the side strike glancing blows and air somehow has to get in behind the bullet as it moves. Parts of the air will be pushed forward and parts aside. Some parts will swirl around behind. All of these actions take different amounts of time to execute and the bullet passes in another time which too is different from the others. These considerations tell us that drag will depend on things like the bullet speed, bullet shape, air density, air temperature (molecular speed) and other factors like the speed of sound. Thus, when we measure drag at different bullet speeds for different bullets we find a messy picture. To simplify a little, it turns out there are four major speed regimes. In each regime we can describe the drag quite accurately with an equation. The equations are unique to each regime and quite unlike the others. The speed regimes are as shown in Figure 4. where K, the drag divided by the speed squared, is graphed against the speed. They are: 1) subsonic –below the speed of sound, up to about 1000 ft/s where drag is lowest, 2) sound barrier or transonic – just below to above the speed of sound, 1000 to 1200 ft/s where drag increases very rapidly with increasing speed, 3) peak drag – 1200 to 1400 ft/s where the greatest K value occurs and 4) supersonic – 1400 ft/s and above. The regimes do not matter much if your bullet stays in just one throughout its flight. Dropping into a lower regime makes calculations of bullet drop extremely complicated and, hence, accurate shooting becomes a serious mathematical challenge. Dropping into the sound barrier regime, or worse, dropping through it, can really mess you up. Ammo manufacturers are well aware of this and keep muzzle velocities well above 1400 ft/s. The regimes mostly matter for long range shooting where bullets drop into the peak regime from the supersonic. Figure 4. K (= Drag Force/velocity2) vs. Velocity The curve here is for a highly streamlined Spitzer type bullet. It is modeled after data given by Arthur Pejsa in his classic work Modern Practical Ballistics. The curve would be different for other bullet shapes but would still have the same four regions. Zeroing In Given all that happens or can happen to a projectile in flight, hitting a target exactly where you intend to is not a trivial accomplishment. Indeed, accurately hitting a target is no accident but the result of very careful preparation. The most important preparation is the zeroing in of the gun. This exercise basically tells the shooter how the gun behaves in shooting situations and gives the shooter a chance to make adjustments as needed. As we begin, a word of warning: zeroing is for very specific conditions. A gun zeroed for one type of ammunition is not zeroed for another. Once your gun is zeroed for a 150 grain bullet it quite definitely is not zeroed for a 180 grain bullet. Nor is it zeroed for your buddy. There is no substitute for practice so zero constantly, always noting the conditions. Zeroing in is an operation conducted with the target on the same level as the gun so that shooting is supposedly on a flat, horizontal line. But that never actually happens. A projectile fired horizontally, as in Figure 1, is at its high point, its zenith, as it leaves the muzzle. It’s all downhill from there. In order to hit a target on the same level, a gun must be fired with its muzzle axis slightly elevated. The projectile then reaches its zenith at the mid-point of its path to the target (assuming air resistance is negligible). For a given gun and cartridge, the further away the target is the greater this angle of muzzle axis elevation must be. It follows that zeroing in is only good for a particular distance. Zeroing in, then, is just the beginning of accurate shooting. So, what’s the best distance to zero in your gun? That depends on the shooting situations you expect to encounter. Maybe 30 yards for a shotgun, 50 yards for a pistol, 100 yards for a deer rifle in the eastern US and 200 yards for varmints and deer in the west are distances you might chose. An air rifle for the 10 meter air rifle events should obviously be zeroed at 10 meters. Personally, I always zero my deer rifle for 50 yards because I seldom get a shot at a deer further away than that. Whatever distance you choose should be a sort of judgment call average of the maximum shots you are likely to have to make. And once the gun is zeroed in, remember that distance! Without it, all your zeroing effort goes down the drain. Perhaps it has occurred to you that manufacturers could solve this problem by selling guns stamped with a known zeroed distance. Some of you may be way ahead of the curve, already planning to make and sell your own brand of pre-zeroed gun to a clamoring public. Forget it. Zeroing in is about the shooter as much as, perhaps more than, it is about the gun. A zeroed gun for you will not necessarily be zeroed for your buddy. Furthermore, a gun clamped into a solid rest will probably shoot better than one fired free-style but zeroing in should replicate the conditions of firing. Even competition shooting doesn’t come close to perfect conditions. Zero in under shooting conditions, not ideal conditions. You’re not just zeroing the gun; you’re zeroing yourself. Once you have settled on the zeroing distance, zeroing in has two simple steps: first you find out how far off the center of the bulls-eye you are and then you make corrections. The steps may need to be repeated until you reach the point where you decide you’ve zeroed enough. Zeroing in is not one of those things with a clear and obvious end. Years ago I heard the poet W. H. Auden say a poem is never finished, just abandoned. That also describes zeroing in. At some point, you “jest gotta quit.” Finding out how far off the gun is is a matter of shooting it. The more shots you take, the better your information but taking many shots makes zeroing in expensive. Most of us shoot three shots (although 4 or 5 is better if your wallet can stand it) and then go to step two. Of course, you will be shooting at a good, sharply define target, solidly set it up against a safe background. There are many ways one can scientifically and mathematically evaluate the data consisting of those holes in your target but let’s not get carried away. Used sensibly, the human brain is a great calculating machine, so use it. Eyeball the shot cluster and put a dot (real or imaginary, it doesn’t matter) at what your sensible brain tells you is the “center” of the cluster. Now, measure how far away from the target the dot is both vertically and horizontally. Using a target with a vertical and horizontal grid (a large scale graph paper) is a good idea here. You must now correct the direction you first pointed the barrel axis, both horizontally and vertically, to zero in the gun. And here’s where things get just a bit tricky because the necessary adjustments are made to the aiming device rather than directly to the barrel axis. I will cover aiming aids, including telescopic sights, in a section of the chapter applied shooting science. For now just we’ll assume we are dealing with front and rear iron sights. For vertical corrections (also called elevation corrections): Shooting low – raise the rear sight or lower the front sight Shooting high – lower the rear sight or raise the front sight For horizontal (windage) corrections: Shooting left – move the rear sight right or the front sight left Shooting right – move the rear sight left or the front sight right Note the rule is to move the front sight in the direction of the error or the rear sight in the direction of the required correction. Most iron sights systems only allow rear sight corrections. This is not so much physics as pure geometry, not science but math. How much should you move the sight? Most iron sights just have clicks adjustments. If that’s all you have and you do not yet know the amount a click changes aiming by, all you can do is move the sight in the required direction by a click and shoot another cluster. If that moves you a ½” in the right direction and you needed 2” then move 3 more clicks and shoot a third cluster and so forth. If you happen to know the value of a click in M.O.A. (minutes of arc), your task is a bit easier. A minute of arc is a 60th of a degree of angle. For an object at 100 yards, a minute of arc amounts to just over an inch change in pointing direction. For me, zeroing at 50 yards, it is a half inch. If a click is ½ M.O.A. and you need to adjust 2” on a target at 100 yards, then 4 clicks will do it. To give you a formula that is good for either horizontal or vertical corrections: required adjustment in inches # of clicks required = click value in M.O.A. x hundreds of yards zeroing distance For example, to adjust 2” on a target at 200 yards with a click value of ½ M.O.A. will take (2)/ (½ x 2) = 2 clicks. For zeroing at 50 yards (½ of 100 yards), a 3” adjustment and a click value of ½ M.O.A. this is 3/(½ x ½) = 12 clicks. Collisions with Obstructions and Targets We have looked at the firing of a projectile (called internal ballistics in books on ballistics) and the flight of a projectile (external ballistics). What about the final phase of the trajectory (terminal ballistics)? What happens when a bullet or a shot packet strikes an object? If you are simply a competition shooter, you probably don’t care about tree branches or shrubs in the way of a shot and, as long as you can put a hole in the target or knock it over, you don’t care about damage done to a target. You shoot in artificially clear environments at targets showing only minimal damage or response. Skip this section, it’s not for you. This section is for hunters and others who shoot in the great and messy outdoors. Whenever a projectile encounters another object, whether a dry grass stalk, a twig, or a target, something happens. Its trajectory changes. It may change only slightly but it changes. This is because contact always applies a force to the projectile. However small it is this force will change the velocity of the bullet. A force applied back along the bullet path merely slows it. But most forces will be applied in more random directions and that makes good predictions impossible. Once the trajectory is changed, the projectile follows a new path. If the remaining time of flight to the target is very small, the change in trajectory cannot change the point of impact very much. Thus, grass or twigs close to the target are less of a concern than twigs half way to the target. Hence, hunters wait for a shot through a “hole” in the branches and bushes where they hunt and cut “shooting lanes” around their deer stands to create obstruction free paths for good shooting. Hunters sometimes say that a gun or cartridge is good because it “cuts through” the “thick stuff.” The remark is usually tied into other remarks about a heavy bullet. It is true that the more massive a projectile is, the less it will be deflected by an interaction with any particular obstacle. The .30-30 cartridge is among those I have heard credited with such abilities, but the .30-30 bullet in 150 gr. is no more a “bush buster” than any other 150 gr. bullet. If you want a real “bush buster,” you will have to upgrade to a cannon. Physicists have two ways of viewing or understanding how and why things change their motion. One way is the force picture. In this view, things happen because forces act to make them happen. Forces are causes of actions in the world. I have been using this force picture, so far, in talking about the behavior of projectiles. In the force picture, objects apply equal and opposite forces to each other (remember Newton’s third law). Even a stalk of dry grass applies a force to any bullet touching it. The details of the force cannot be determined without an enormous amount of information. Thus, in practice, talking about forces here is useless. The other view is the energy picture. Energies are not causes of actions so much as constraints on what actions are possible. In many ways the energy picture is the easier of the two to use because it is one of the basic beliefs of physicists that energy is never created or destroyed. Thus, all actions in the world involve only transformation of energy from one form to another. Physics then becomes accounting. All you must do is add up the amounts of energy distributed over the different forms and you must always have the same sum. Energy, we say, is constant. Using the energy picture to describe a bullet’s motion, the chemical energy stored in the powder of the cartridge is transformed into heat and gas pressure. This in turn is transformed into kinetic energy (energy of motion) of the bullet (and some sound energy). Air resistance does work to transform some of this kinetic energy into air motion and heating of the air. The sonic boom, when present, also turns some of the kinetic energy into sound energy. It is when the projectile strikes the target that the energy picture is most useful and interesting. Ammo catalogs typically list expected muzzle velocity (in fps or m/s) and muzzle energy (in ft- lbs or ergs or Joules) for each cartridge on sale. Data is available from ammo manufacturers on kinetic energy at various distances traveled. What is the value of this information? The kinetic energy a bullet has when it strikes a target is the maximum energy it has available to do damage to the target. If you want the target to swing over or fall over, there is an easily calculated minimum energy required to accomplish this. If the bullet does not have that minimum energy, you need to shoot from less distance or perhaps use a more powerful cartridge. However, deliver too much energy and you may destroy the target. Hunters need to know how much kinetic energy their chosen ammo delivers at typical distances. Too little energy means wounded but not downed game. Too much energy can also be a problem. The first time I shot at a quail I centered it at 20 yards with my 12 gauge and a load of 6’s (I was hunting pheasants). It looked like an acid-eaten rag when I picked it up. Too much energy and a waste of a shell and a nice quail. I did wait a bit to give it distance before I shot but it wasn’t enough. Having said all that, I must hasten to add that a great deal of nonsense is being and has been written about kinetic energy of bullets. Such terms as “killing power” and “stopping power” are usually tied to the kinetic energy of a bullet. Additionally, there is the phrase “knockdown power.” This latter phrase has recently been elevated to pseudoscience with the invention of a knockdown power rating for ammunition. It is generally associated with the momentum of the bullet. These terms are essentially meaningless. Let me explain. The kinetic energy of a bullet is half the mass times the square of the speed and the momentum is just the mass times the velocity. (recall that velocity is speed plus direction of travel). The physics of these quantities is well established. When a bullet strikes a stationary target, it transfers both kinetic energy and momentum to the target. If it stops in the target, it transfers all its momentum and all its kinetic energy to the target. If it passes clean through the target it transfers less than all the momentum and kinetic energy to the target. If it bounces back from the target it transfers part of its kinetic energy to the target but the target actually receives more momentum than the bullet had initially! So much for the simple physics. Please note that we cannot specify the details of how these two transfers affect the target. To get an idea of the possibilities, let’s consider some extreme cases. Compare a 150 grain bullet traveling at 1400 fps (955 mph) and a one quart (2-lb) water balloon traveling 150 fps (99 mph). They have the same kinetic energy but the slower water balloon has ten times the momentum of the bullet! Suppose you have on a Kevlar vest to stop the bullet. Which would you rather be hit by? The bullet will impart its energy over a smaller area and would bruise you badly. The water balloon, at 99 mph, might cause some bruising and would be ten times as likely to knock you down. It would impart a backward speed of 2 fps to a 150-lb person, assuming all parts of the body moved together, an unlikely prospect. Obviously, the water balloon is the better choice. Being knocked down seems of little concern compared with a bad bruise. Thus, “knockdown power” corresponds with momentum transferred but energy transfer does the real damage. We’ve all seen people shot in movies. They are almost always blown backward by the impact. This is rubbish. The recoil of the gun is the same thing as the impact of the bullet, a momentum transfer. Because of the muzzle blast component of the recoil, the bullet transfers more momentum at the beginning of its flight than at the end. A bullet capable of blowing its target backward would blow its shooter backward even harder! So, next time you see such a movie scene, notice that nothing happens to the shooter. You are being humbugged. The rubber bullets used for crowd control are designed to impart maximum “knockdown” to a target. Theoretically, a bullet that bounces back from a target can impart as much as twice its initial momentum to the target. That is, the target is hit with up to twice the recoil felt by the shooter. Of course, in real life, the bullet never imparts that much momentum. The problem is that the momentum involved is not really great. A lucky shot might knock down the right target. Then too, an unlucky shot will penetrate the target and do serious harm. Overall, rubber bullets are a very imperfect type of crowd control. The problem with terms like “knockdown power” or “stopping power” is that what actually happens to a target depends mostly on the details of the collision. Bullets shape and behavior and shot placement are far and away the controlling factors. Bullet momentum and kinetic energy merely set the outer limits of what can happen. For example, I have a single pump air pistol that shoots a BB so slowly the shooter can easily watch it in flight. I have hit squirrels with it. It makes them jump and run away. I doubt it could kill a squirrel even if I hit it in the eye. The extremely low momentum and kinetic energy of the BB limit the effects it can produce. The most famous extreme example of how details of a collision can seemingly defy common sense and elementary physics is the shot that killed President John F. Kennedy. Film of the event clearly shows Kennedy’s head snap backwards toward the shooter. It also shows a jet of blood, flesh, and bone flying forward from his head (it is indeed a grim, grisly film). Many people argued that the head snap was proof of a second shooter in front of the presidential procession. The law of conservation of momentum, they said, did not allow any other conclusion. If a bullet is shot into a solid, rigid body, it is true the body must move in the direction of the bullet. But a human head is neither solid nor rigid in the sense required. Thirteen years after the assassination, Luis Alvarez, a Nobel Laureate and President of the American Physical Society at the time, did a “back of the envelop” calculation and then persuaded colleagues to carry out experiments on the problem. They shot 150 grain, 30-06 bullets at about 3000 fps into melons wrapped in tape. The wrapped melons were to simulate a human head. Almost all of the melons clearly moved toward the shot rather than away from it (one “just rolled around [the support] a bit”). His report is in the American Journal of Physics of September 1976. The forward jet of material is vital to a physical explanation. Instead of the bullet and the head, we must consider the jet as well. With three objects to consider, the analysis and the possibilities both become more complicated. The jet absorbs more of the kinetic energy of the bullet than the head and consequently it also carries off several times more momentum than the bullet ever had! To compensate, the head snaps backward. Unlike kinetic energy which is always positive, momentum can be positive or negative. Treating the momentum of bullet and jet as positive, the head then has negative momentum. The momentum of the three objects still sums to exactly that of the incoming bullet and momentum is conserved. The laws of physics hold and explain the unexpected behavior of the target. On the other hand, conspiracy buffs like to cite tests done for the House Select Committee on Assassinations at the Edgewood Arsenal in 1978 which tried to replicate the Alvarez results with jacketed 6.5 mm bullets like those used by the assassin, Lee Oswald, on human skulls filled with material to simulate a human brain. All of the ten skulls tested reportedly moved forward. The lesson here is that “the devil is in the details.” What targets do when hit depends much more on shot placement and the type of projectile than on the kinetic energy or momentum of the bullet. If knowledge of precise quantities like kinetic energy and momentum tell us little about how targets respond to hits, then less precise concepts like “knockdown power” or “stopping power” are basically useless. Even similar placement on similar targets is unpredictable. Data from the experience of deer hunting clubs show that half of all deer shot in the chest initially show no sign of being shot while others jump and run and some seem to be knocked down. I say “seem to be” because the impact alone does not have enough impulse to knock down a deer. Moving Targets Target Tracking and Depth Perception Far and away the most difficult type of shooting is shooting at moving targets. It is, in fact, so difficult that we basically decline the challenge of using a single projectile and go with a gun that fires a large number of projectiles simultaneously– the shotgun. The idea is to use the power of statistics, of large numbers, in accomplishing the task at hand. Although one projectile alone has a rather low probability of hitting the target, several hundred projectiles together increase the chance of at least one hit by a factor of several hundred times. There is, of course, no guarantee that even one of these hundreds of projectiles will hit a moving target. A shotgun must be well handled to have the desired effect. The challenge is twofold. There is no hitting a target you cannot see. Because we track targets “naturally,” we are apt to overlook the fact that the shooter must first of all be able to track the moving target visually. It is the second part of the task, hitting the target, that we see as the challenge. Not so fast. Visually tracking moving targets is quite difficult. Very few animals can do it. Because visual tracking requires specialized eyes and eye muscles, and, of course, a good nervous system to coordinate and process it all, almost no invertebrates track prey visually. The octopus can and the preying mantis; perhaps that’s it. I have heard rumors some hover flies also do it. If that’s true it’s also very strange. Why hoverflies!? Even in the vertebrates the skill is uncommon. Chameleons, birds of prey and primates about exhaust the list. Most predators cannot do it. The insect catching bats do track with ultrasound but they cannot do it visually. I have seen scissor-tailed flycatchers go after and capture moths with seeming easy but I really don’t know how they manage it. Perhaps, like hawks and owls, they track by turning the head rather than the eyes. It may be a rare talent but, for whatever reason, we humans have it and that’s a start. Being able to visually track a target is, as the mathematicians say, a necessary but not a sufficient condition for shooting at and hitting a moving target. One needs not only to track but to also to hit the target. In tracking, your mind creates a record of the three dimensional path of the object with information on its direction and distance from yourself. Hitting the object additionally requires your mind to predict where the object will be fractions of a second into the future and to operate the shotgun so its load will arrive at the same place as the object at the same time. See how smart you? Did you know you could do all that? Well, don’t think about it too much or you’ll freeze up when you should instead be slapping the trigger. A rule that grows out of all these thoughts is that shotgunners must keep both eyes open as they shoot. You can track the direction of a bird with one eye but you cannot get good distance information unless you track with both eyes. Good depth perception demands two eyes set slightly apart (the further the better but don’t go getting plastic surgery to move your eyeballs around). As my Uncle Earl found when he lost the use of one eye through a work accident, being a one eyed hunter makes you pretty much a rifleman. Earl kept his shotguns but the only use he got from them thereafter was in turkey hunting, which became his passion. Yes, turkeys have wings but turkey hunting is not bird hunting, it’s big game hunting; more like deer hunting than pheasant hunting or grouse hunting. When you try to shoot a turkey, you get to aim the shotgun like you do a rifle. For that, one eye will do just fine. The basic idea is then that you mount the shotgun so your master eye looks along the barrel but you keep both eyes open and on the target. You can figure out which eye is your master eye by looking at a fairly distant object with both eyes and then, still with both eyes open, stretch out your arm with the thumb up and place the thumb directly under or over the distant object. Without moving your arm or thumb, now close one eye. If the object is still lined up with your thumb, the open eye is your master eye. If the object seems to change position, you have just closed your master eye. Either way, you have discovered your master eye. Aligning your master eye down the shotgun barrel makes for the most reliable and consistent pointing of the gun. You need to practice the movement of mounting the shotgun to your shoulder and aligning your master eye down the barrel. You should be mounting the gun smoothly so it immediately lines up with the master eye. Try mounting with your eyes closed then opening your eyes to check alignment. If problems continue, it may caused by features of the gun shape like, the “drop at the comb” or the “pitch.” You may need a different shotgun. The mounting movement must become natural and precise. If you mount the shotgun in a slightly different position each time, you will never be able to point the shotgun reliably and consistently. This will then make it impossible for you to predict consistently where your shot load is going to go. In other words, you’re going to miss a lot of targets. Given that we are “naturals” at tracking, keeping both eyes open and consistently mounting the shotgun with the master eye looking down the barrel are all you need to do on the tracking end of things. Your body will do the rest without you having to think about. Your head, shoulders and body will turn as you track the target. Don’t think about it or you’ll mess up. Yes, planting the feet correctly helps too but most of us do this well enough that we most often get it right, or right enough. Hitting the target is another kettle of fish entirely although, even here, you want to practice so often that you do the shooting pretty much automatically without (much) thinking. As Prince Hamlet discovered, thinking too much inhibits action and that’s really bad when it’s action and not thought that’s needed. Let me use a specific example to illustrate what hitting the target involves. Consider a bird flying past you on a straight level path so that, at closest approach it is exactly 20 yards from you. Let the bird loaf along at a constant 25 fps and suppose the shot from your shotgun also moves at a constant speed of 1250 fps, 50 times faster than the bird. A short bit algebra and trigonometry get us the following results. θ d in feet Δθ 0O 1.20 1.15O 15O 1.25 1.10O 30O 1.40 0.99O 45O 1.72 0.81O The bird starts at the position of closest approach where by definition the bird is flying perpendicular to your line of sight and sight angle θ = 0O. It then quarters away from you until the sight angle is θ = 45O. The distance the bird flies from the θ values in the table to where the shot hits it is d and the angle it travels from θ to the hit position is Δθ. That is, at each angle you must lead the bird by d feet or by Δθ degrees For a bird that has seen you and is flying flat out to get away, the speed might double to about 50 fps. This will also double the lead distance needed, d, and will also almost precisely double the lead angle, Δθ. Note that both d and Δθ are relatively constant. This means that, over most of its path, the angle by which your shot must lead the bird is small and almost constant. Really, this is what makes hitting birds at all possible for us. If Δθ, the lead angle, changed rapidly with sight angle we would never learn to keep up with all the possible paths birds take with respect to our position. The task in hitting a target, then, is getting that lead angle right as you shoot. Leading the Target There are two main methods of leading the target: you either swing the shotgun to point ahead of the target by a constant “distance” or you swing it from behind the target through the target and slap the trigger just as the master eye sees the barrel go ahead of the target (hit the beak of the bird). The first method can be called constant or sustained leading and the second is then swing-through leading. Each method has its advocates. I’ve used both and I suppose I prefer the constant lead method, perhaps because I think I understand it. In all candor, I do not understand how or why swing leading works. I can tell you that, for me, in some situations it has been the only way to start hitting targets. Not understanding it doesn’t keep me from using it. There is a third method. In pull-away leading the gun is first pointed at and then pull forward of the target as the trigger is pulled. This is another method I don’t fully understand but I have used it, once, by accident. I was on the skeet range and messed up my timing. The clay was getting well around me so I quickly point at it and jerked the gun forward. As luck would have it, I powdered it! Notice that, whatever method you use, the shotgun is always swung to track the target. In fact, you must continue the swing even after you have fired in order to be sure the swing doesn’t stop before the shot load leaves the muzzle. There is, after all, a time delay between when you form the intent to pull the trigger and when the load leaves the barrel. Stop too early and you’ll be behind when the shot leaves the muzzle. Keep the barrel swinging! Stationary gunning is rifle shooting, not shotgunning. You seldom get a shot at a bird going straight away from you. Actually, grouse always seem to fly straight away from me but they also always manage to first put a tree in the way. Thus, as I said, you seldom get a shot at a bird going straight away from you. If you prefer to lead by a constant amount, look at the lead distance, d, in the table and think of it in terms of bird body length. A quail is about 8” long so you would need to give it a constant lead of about two body lengths. A grouse is about 1.15’ long so lead it by a body length and pheasants should be led by about 2/3 of a body length. Of course, if the bird is flying faster than 25 fps you must increase the lead proportionally. Gauging speed is best learned through much practice. What if the bird is flying a path closer to you or further away than the 20 yards of the example? The answer is that if, for example, the closest approach is 30 yards, then d in the table increase by the factor 30/20 = 1.5 but, and here’s the surprise, Δθ does not change! Thus, leading by a constant angle rather than a constant distance should always get the job done. If we only had a mental protractor in our heads to reference, leading a target would be pretty easy. Of course, real shooting is not a simple as the example. You never know the exact distance of closest approach; the speed of the bird and even its direction of flight are all apt to change abruptly and without notice. Shot never travels at a constant speed and tends to slow down significantly over a 40 or 50 yard flight. The fight path may not be level or it may pass directly over you. All of these differences matter to some degree or other. Fortunately, the shot string is there to hide our ignorance and errors of judgment. The “Shot String” It is no accident that the shot packet from a shotgun cartridge spreads out into a “shot string.” If all the mass of the composite projectile from a shotgun stayed together as a unit, we would never gain the advantage of firing multiple projectiles. The whole point of having a projectile that quickly falls apart into a loose mass of many projectiles is to increase the chances that some part of the projectile mass will hit the target. The term “shot string” is a good description of what the collection of scattering shot looks like down field. Usually the shot string has a fairly compact mass of shot at the front with a stretched out tail of slower shot behind it. The overall effect is that of a much enlarged projectile mass going out to meet the target. The shot in a shell do not all have the same velocity leaving the muzzle. This is because of random collisions with each other and the inside of the barrel. These collisions also deform some of the shot, especially if they are lead shot. The hard shot types of course are not as affected by deformation. These differences leaving the muzzle are increased by time of flight and, in the case of deformed shot, by air resistance forces. Obviously, deformed shot can be deflected in ways that move it across the direction of the shot string. All of these causes make the shot string open up and stretch out as it moves down field. Do keep in mind, however, that this mass is mostly elongated; the shot do not move up or down nearly as much as they spread out horizontally so the shot string is far more forgiving of lead errors than of vertical, up or down, position errors. That is why the swing lead methods are so important. The inertia of the swing keeps the gun and, hence, the shot string in the plane of the motion of the target and thus it minimizes vertical pointing errors. Shot Patterns When the shot string hits a flat vertical target like a big piece of cardboard or a large spread of paper on a wall, the holes in the target create what is called a shot pattern. All the shot in the shot string contribute to the shot pattern (assuming the paper was big enough and in the shot was “on target”). It is, in fact, a collapsed, two dimensional picture of the three dimensional shot string. Knowing details of the shot string gives you much better information about the behavior of the shot but pictures in three dimensions are note easy to create. The shot pattern is, then, a still useful piece of information about your shot string. Making shot patterns for your shotguns and the loads you use in them is a very good idea. The standard procedure is to set up a target 40 yards away. It should be a good bit more than thirty inches wide and of material that will readily show where shot strikes it. A four foot square is a good size. Then shoot one shell at the center of the target and see what pattern the shot holes create on the target. Draw a thirty inch in diameter circle around the heaviest concentration of shot. Count the shot inside the circle and outside it. Hold on to the numbers, I’ll come back to them shortly. Also note how close the center of the circle is to the center of the target. This tells you if you are aiming the shotgun well for that distance. You will find the shot makes a somewhat random pattern on the target. Note especially that the pattern has “holes” in it where no shot arrived. Remember as you think about this that the shot do not arrive at once. Some of the regions without apparent holes may be from 3-D holes in the string that were filled in by late arriving shot that would not have been there to hit a flying bird. Choke Shotgun choke ought to be mentioned somewhere and this is perhaps as good a place as any. Shotgun bores are basically cylindrical holes of constant diameter down the length of the barrel. Someone got the idea to try narrowing down the bore in the last few inches of the barrel as a way of squeezing the shot string together and thus controlling its expansion once out of the barrel. Well, it worked. So now you can buy shotgun barrels with about a dozen different sizes of constrictions or chokes at the end of the barrel. You can about drive yourself mad deciding what is the best choke for you or for you shooting pheasants vs. you shooting doves and on and on. You can even buy insert chokes if you have the misfortune of owning a shotgun that lacks the choke du jour. Roughly speaking, the more you choke the bore, the more you concentrate the shot string 40 yards out. For example, here are the expected percentages of shot in the 30 inch circle for various chokes: cylinder (no choke), 40%; skeet, 50%; improved, 55%; modified, 60%; improved modified, 65% and full, 70%. Your skeet choke may give you 53% or 48% with some loads so don’t fixate on these expected numbers. Different loads and different types of shot all affect the string as much and perhaps more than the choke size. By the way, choke sizes are graded in increments of 0.005 inch changes in the inner diameter of the bore. Since this is a book about shooting science, let me get right to the science. Actually, you have heard about all of it already. There are no good and complete explanations of how chokes work. Science, applied to this problem, has not been of much help. The only science involved is then experimental science. That is, you gotta’ try out all the chokes with all the loads to find the best one for you. Sounds expensive? You bet! It may make you feel better to know that I own several shotguns and I yet don’t know the choke of any of them. Okay, I’m pretty sure the 12 gauge is “modified choke.” I don’t worry about such stuff. I just buy boxes of shells and shoot them at a target to find out the shot pattern. Beyond that, I buy boxes of clays and we shoot them to learn to lead targets. There’s no substitute for practice. Haven’t I said that before? The Chemistry of Shooting It’s not just my prejudices as a physicist that have made me put physics first in the basic science of shooting. Physics is just, well, basic. Nonetheless, shooting is not just physics. There is some chemistry, maybe a lot of chemistry, involved in shooting science. Two areas where chemistry is, in fact, absolutely dominant are propulsion of the projectiles and the fouling of gun parts that follow from the propulsion. A shooter needs to know a bit about both. Propulsion Propulsion is the result of expanding gases pushing on the rear of the projectile. These gases are generated by either deflagration (which is a fancy name for burning) or detonation of powders of various types. Burning is oxidation which usually means it is an actual chemical reaction of oxygen with some flammable material. In deflagration, the boundary between burned and burning powder moves at speeds less than the speed of sound. That is to say, burning is a relatively slow process and hence produces lesser explosions compared with those produced by detonation processes. Gunpowder, now usually called black powder, is the model material for burning powder. It is made of about 10% sulfur, 15% carbon such as powdered charcoal and 75% potassium nitrate (saltpeter). I can attest that these percentages need not be precise. The bombs I made as an ignorant kid did, more or less, explode with mixtures of about equal amounts of these three ingredients. The hot, propellant gases produced in these explosions are mostly carbon dioxide and nitrogen. Since the material begins as low volume powder and turns fairly rapidly into a lot of high volume hot gas, the process creates high pressures which are, of course, the source of the propulsive properties of black powder. The oxygen for the deflagration of black powder, which, by the way, is black because of the charcoal in it, comes from the potassium nitrate. The sulfur helps slow the burning and thereby cools the flame. The potassium nitrate has at times been replaced with sodium nitrate. This works but has the undesirable side effect of sucking up water from the air because, unlike saltpeter, sodium nitrate is hydroscopic (which is just a fancy way of saying it sucks up water from the air). Sodium nitrate black powders therefore deteriorate unless sealed from the air. On the other hand, saltpeter black powders remain dry in air for years and even centuries. That old loaded old muzzle-loader hanging over a fireplace mantle will probably still fire normally. Black powders can be made to burn more quickly by processing grain sizes, coating the grains and perhaps other techniques. Most of these treatments are proprietary and they are certainly advanced science so I merely mention them here because the revival of interest in black powder guns has led to a revival of interest in and improvements of these techniques. Burning black powder produces much more than propellant gases; it also produces many solid compounds which mean that black powder produces a great deal of fouling. I will discuss fouling shortly but at the moment I merely want to note that this by product of black powder explosions is a serious drawback to using black powder. It was so significant a problem that, after a while people initiated efforts to get around it. At length this led to the development of so- called smokeless powders. In order to avoid fouling, a propellant powder should ideally create nothing but gases. It would be nice too if the gases were all non-corrosive of barrels, etc. For example, powders that produce steam are undesirable because hot steam is highly corrosive of metals and especially of steel. Also ideally, one would like a material that breaks up quickly into non-corrosive gases. Stuff that breaks apart quickly tends to prefer breaking up; it’s always just on the edge of doing so, so to speak. That is, we need to look at unstable compounds. Detonation basically means the molecules fall apart, turning back into some of the stuff used to make the compound in the first place. There’s no adding another atom or so as there is in oxidation. But unstable compounds are, well, dangerous. Black powder can be set off with even a stray electric spark but the stuff of smokeless powders can go off either shaken or stirred; sorry, James Bond. Some of these materials you don’t even want to look at cross-eyed. The earliest smokeless powders were based on nitrocellulose (guncotton). This is highly amusing stuff easily (and often) made by even high school age chemists. I have had any number of students make it over the years. You may see it classified or listed as single-base powder. Nitroglycerine mixed with nitrocellulose, (double-base powder) came into use later and triple- base powder, which combines nitrocellulose, nitroglycerine and nitroguanidine, was even later. Guncotton is the basis of some of the most famous early propellants such as cordite and ballistite as well as poudre B. All of this stuff becomes highly unstable with time. Nitroglycerine is particularly famous for this and, in fact, old, unstable nitroglycerine left by a railroad line blew my great-great grandfather, Philip Cramer, and his son, James, literally to smithereens on Christmas Eve of 1874, an unexplained event that still lives in family history. Another son, John, was also killed but his body was left more or less intact. The 90 lb can of nitroglycerine blew a hole in the ground that the Lancaster (PA) Examiner and Herald newspaper of Dec. 30, 1874 said could hide “a four horse team.” The hole, somewhat filled in by erosion, is there to this day. These smokeless powder materials transform almost entirely into gases, mostly carbon monoxide, carbon dioxide and nitrogen with some hydrogen and water vapor as well. There are other trace gases like methane and ammonia present in the propulsive collection. Smokeless powders detonate, sort of. Careful studies show that there is often an early deflagration that rapidly turns into a supersonic wave of detonation in the powder. The pressure change alone detonates the explosive. The heat of the burning moves much more slowly (at the speed of heat conduction) in the powder. Generally, the chemical change here from powder into gases produces much greater volumes of gas than the deflagration of black powder. That means smokeless powders generate much greater propulsive pressures than does black powder. This fact has a very important safety implication: never use smokeless powder in a gun designed for black powder. Like black powder, smokeless powders are processed to control burn rates. I have already pointed out that powders used in pistols must burn more quickly than those for rifles and shotguns because the shorter barrels make for shorter times down the barrel for the projectiles. Manufacturers largely control burn time by forming the powder into grains of varying sizes. As one might expect, larger grains burn more slowly. So, the ticket is smaller grains for pistol ammo and larger grains for shotguns and rifles. Beyond that, details become quite secret, proprietary information. Fouling Fouling from black powder is relatively severe. It consists primarily of two salts, potassium carbonate and potassium sulfate although there may be carbon powder, sulfur and potassium sulfide in the mix as well. The salts are the main problem because, allowed to sit on the inside of the barrel and absorb water from the air, they reverse the reaction that created them. One of the most familiar and important types of chemical reactions every chemistry student learns is the acid/base reaction that creates salts and water. Most of these reactions highly favor going to the salt and water side of the process but the salt and water always have a non-zero probability of going back to being the acid and base that created them. So salt fouling where there is contact with humidity of the air, given time, will coat the inside of the barrel with some very nasty, corrosive stuff, particularly sulfuric acid and potassium hydroxide, a strong acid and a strong base both of which happily attack metals like the barrel of a gun. Over time, this fouling will badly pit the interior surface. Given more time it will corrode holes through the barrel. All in all, not a desirable situation. Black powder fouling also creates serious more short term problems. It fills in the grooves of rifling so the lands lose some of their grip on the projectile. It changes the “choke” of a shotgun in a random and irregular way so that the shot pattern is altered, usually in undesirable ways. It will also fill in any critical small holes in the barrel like vents or the touch-holes of earlier muzzle-loaders. For the same reason, you should not use black powder in autoloaders. Autoloaders were not possible before the advent of smokeless powders. Smokeless powders are, as you would expect, much kinder to guns. With most of there mass converted to hot gases, they leave much less solid material behind to gum up the works. The fouling from smokeless powders appears to come mostly from some of the proprietary additives like stabilizers added to inhibit deterioration of the unstable propellants and from metal particulates that sometimes appear and are perhaps used as stabilizers. (Remember, Alfred Nobel made most of his money by inventing dynamite, nitroglygerine stabilized with a very particulate material, diatomaceous earth.) Even though smokeless powders are kinder to barrels, kind enough to make autoloaders feasible, they still foul barrels. Shoot even a few shots with modern cartridges and your formerly clean barrel will have a noticeably dirty interior. Open the breach to sunlight or a bright indoor light and look down the muzzle. Not a pretty sight. You need to clean that barrel. Every shooter needs to know about cleaning the bore. There are lots of cleaning kits available. Some are shotgun only, some are shotgun and rifles kits, some are just for pistols and so on. Get one for your gun (or guns) and use it throughout the shooting season. Don’t put a gun away at the end of season without giving it one last cleaning.
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