by Barry Parker
Let's look at the air pressure around the bullet in more detail. It creates a force called drag, which acts in a direction directly opposite to the direction the bullet is traveling. And interestingly, it is a much greater force than gravity (fifty to one hundred times greater), but gravity is still the main force that determines the bullet's trajectory. In practice, the shape of the bullet has some effect on its path, but in a first approximation we can assume gravity is acting on the bullet at its center of gravity. Basically this is just the “balance point” of the bullet.
The drag caused by air resistance actually depends on several things, such as the bullet's speed, its shape, the density of the air that it is passing through, and the air temperature. In practice, calculating drag is usually a difficult problem. Furthermore, there's a serious problem at the speed of sound, or, more specifically, when the bullet passes through the speed of sound. For this reason it is best to deal with four separate regions:
Below the speed of sound, up to about 1,000 ft/sec.
Just below the speed of sound, from 1,000 ft/sec to 1,200 ft/sec.
The peak drag region, which is 1,200 ft/sec to 1,400 ft/sec.
The supersonic region, which is above 1,400 ft/sec.
If we refer to drag as D we can see how it varies in the three regions by plotting it against the velocity of the bullet.
A plot of k (drag force/velocity2) versus velocity.
The ballistic coefficient (BC) is a term that denotes the rate at which the bullet slows down. In conjunction with the muzzle velocity of the bullet, it gives us a good approximation of the bullet's trajectory. The ballistic coefficient (BC) is defined in terms of what is called the sectional density (SD) and a form factor (FF). The sectional density is the mass of the bullet divided by its caliber squared. The form factor is a measure of the aerodynamic efficiency of the bullet, which depends on it shape, so it is more difficult to determine. In terms of SD and FF, the ballistic coefficient is BC = SD/FF. So if we know the ballistic coefficient, the muzzle velocity, and the angle at which the gun was aimed, we can plot the bullet's trajectory. In practice, however, you need tables giving information about the bullet, so I won't get into it in detail. But we can state the following:
Bullets with high BCs are the most aerodynamic and bullets with low BCs are the least aerodynamic.
High BCs are desirable because they give a flatter trajectory for a given distance.
Bullets with high BCs get to the target faster, so their trajectories are less likely to be altered by wind or other variables.
There are other things that also affect the flight of the bullet. Wind velocity can have a serious effect, particularly if it is at right angles to the direction of flight. In addition, the wind velocity frequently varies over the distance of the flight. Yaw, which is a consequence of the spin of the bullet, can also be a problem; it is a rotation of the nose of the bullet away from the line of flight. A similar effect, called precession, also occurs in the case of a spinning object such as a bullet. It is a rotation around the center of gravity of the bullet. It is easily seen in a gyroscope. Finally, there is something that is only important in very long-range shells. It is referred to as the Coriolis force, and it is created by the rotation of the earth. In effect, the earth rotates under the shell as it moves in flight, but from the perspective of an observer on the ground it appears as if the shell is moving away from its intended trajectory.
Another thing that is particularly important in the case of guns is their maximum range. In other words, at what angle do you aim a gun for maximum range? Galileo showed that in an ideal case, where there is no air resistance, maximum range is achieved when the gun is aimed at an angle of 45 degrees to the ground. But, of course, air pressure changes things quite dramatically. We now know that rifle bullets achieve the greatest range for an angle of 30 to 35 degrees. High-velocity, large-caliber artillery, on the other hand, achieve the greatest range at an angle of 55 degrees. The maximum range of a bullet, however, is not equal to its effective range. The effective range is the range that produces reasonable damage. In general, the greater its mass, the closer the effective range is to the bullet's maximum range. Light bullets, like .22 caliber bullets, for example, have a maximum range of nearly a mile but an effective range of only about a hundred yards.
Trajectory of a bullet.
STABILITY OF THE BULLET
As we saw earlier, the main thing that stabilizes a bullet is spin, and this spin is created by the “rifled” interior of the gun's barrel. A rifled barrel has spiraled or helical grooves down its length. The bullet is forced into these grooves, which creates spin along its long axis. When the bullet emerges from the barrel, it behaves like a gyroscope. In particular, it has the stability of a gyroscope. If you have ever played with a gyroscope, you know that it takes considerable force to move it out of the direction it is spinning. This is what gives the bullet its stability and its increased range; without stability the bullet would tumble while in flight, and air pressure would act on it much more strongly.
Spiraling grooves in the barrel of a rifle.
Rifling is generally quantified by twist rate, which expresses the distance the bullet travels down the barrel while it makes one revolution, or one complete turn. The shorter the twist distance, the greater the spin rate. If you look closely at the spiraling inside the barrel, you will see that it is a series of grooves with relatively sharp edges. The spaces that are cut out along the barrel are, indeed, called grooves, and the regions that are left are called lands. This type of rifling is usually referred to as conventional riffling, but there is also another type. In this type the entire barrel is cut in the shape of a polygon (e.g., a hexagon), with the polygon shape given a twist as it goes down the barrel. It is referred to as polygonal rifling. In the case of larger shells, such as those shot from ship guns and tanks, the shell is equipped with fins that ride in grooves as they pass through the barrel.
In most cases the spiral rifling is encountered by the shell almost immediately after it leaves the chamber; in some cases, however, the spin is increased gradually. This is referred to as gain-twist. In this case there is no rifling from the chamber to the “throat” of the gun. When the bullet first leaves its cartridge, therefore, it is not spinning. However, it encounters rifling after it has traveled a short distance, and in most cases the rifling increases its twist rate gradually. This allows the projectile to spread out its increase in torque over a larger distance.
The number of grooves in the barrel can vary, as can their shape and depth. Furthermore, the twist direction can be either clockwise or counterclockwise, and the twist rate can vary depending on the bullet shape, weight, and length. In breach-loaded guns the projectile is placed in the chamber. When it is fired, “seating” occurs in the throat. The throat is usually slightly larger than the bullet, so when the bullet is fired it expands under the pressure of the gas behind it until its diameter matches that of the interior of the barrel. The enlarged bullet then travels down the throat to the rifling, where it is “ingrained”; in other words, grooves are cut into it. As a result of these grooves, the bullet begins to spin.
The twist rate given to a particular bullet is critical. First of all, it has to be sufficient to stabilize the bullet, but it should be no greater than this, and the ideal twist rate depends on the bullet's weight, length, and overall shape. Twist rates can vary considerably; older guns, for example, frequently had twist rates as low as one in seventy inches. More modern guns have much higher twist rates, such as one in twelve inches, or even one in ten inches. In general, rifles have much greater twist rates than pistols. The twist rate (T) is frequently expressed as T = L/D, where L is the length for one revolution, and D is the barrel diameter.
If the twist rate is too small, the bullet will yaw (move back and forth along the direction of flight), and if this happens, it will eventually begin to tumble and lose its accuracy. A twist rate that is too low can also cause a bullet to precess around its cent
er of gravity. As we saw earlier, this is a motion you can easily see in a gyroscope.
On the other hand, the twist rate can also be too high. In many spinning objects there is an outside force usually called the centrifugal force (this is actually an erroneous term), and the faster the spin, the greater this force. Countering it is a cohesive force that is holding the bullet together. When the centrifugal force becomes greater than the cohesive force, the shell disintegrates and flies apart. In theory, a bullet can have a spin rate up to about three hundred thousand revolutions per minute. Most bullets, however, have spin rates much smaller than this, typically in the range twenty to thirty thousand revolutions per minute.
TERMINAL BALLISTICS
Terminal ballistics is a study of what happens to the bullet or projectile after it hits the target. It's obvious that its velocity will change rapidly. It may be stopped by the target, or it may pass through it. There are two ways in physics to deal with this case; they are referred to as the force or momentum picture, and the energy picture. In the force picture we deal with forces (or momenta), so we use Newton's third law, which says that for every action force there is an equal and opposite reaction force. In this case we are concerned with the force applied to the target, or the momentum delivered to it. In the other view, the energy picture, we are concerned with the kinetic, potential, and any other types of energy that might be involved. In most cases the energy picture is the easier of the two to use, and the major reason for this is that conservation of energy states that energy cannot be created nor destroyed; it can only be changed from one type into another. So for a given problem all you have to do is look at each type of energy involved to make sure that everything adds up. In the case of a bullet fired from a gun the chemical energy of the bullet is transformed immediately to gas pressure and heat energy in the barrel. This energy is then transferred to the kinetic energy of the bullet's motion plus sound energy. And some energy is lost to air resistance. It's the kinetic energy that the bullet finally has just before it hits a target that is important.6
Several things can happen when a bullet hits a target. If the bullet stops within the target, it transfers all its kinetic energy to the target, and at the same time it transfers its momentum to the target. It may also pass through the target and emerge at the other side. In this case the bullet transfers some of its kinetic energy and some of its momentum to the target. Finally, it is possible that in the case of a well-armored target, the bullet could bounce back. In this case it delivers all its kinetic energy to the target, but the target actually receives more momentum than the bullet initially had. Because of this, terms such as “knockdown power” and “stopping power,” which are frequently used in terminal ballistics, are actually meaningless. Knockdown power refers to the momentum transfer only, but in reality it is kinetic energy transfer that does the real damage. What happens to a target depends on the details of the collision and which of the above three cases applies, so you can't say a certain type of ammunition (or gun) has a certain knockdown power.
One of the major issues related to terminal ballistics is the penetration of the bullet. A measure of it is given by the impact depth of the bullet, which is the depth the bullet reaches before it is stopped. In some cases bullets are designed to achieve maximum penetration, in others they are designed to do maximum damage. Bullet design is quite different in the two cases. Bullets designed for maximum penetration are made so that they do not deform on impact (or at least, deform as little as possible). They are usually made of lead that is covered with a layer of copper, brass, or steel. The jacket usually covers only the front region of the bullet. In particular, armor-piercing bullets for small arms are usually made of copper jacketed with steel. For larger artillery such as tank guns, tungsten, aluminum, and magnesium are usually used in the shells.
Although some bullets are made to expand when they hit the target, this type of ammunition is now prohibited in warfare according to the Hague Convention of 1899, Declaration III.
Soon after the first airplanes were invented they became important weapons of war. Early on they were used mostly for observation and reconnaissance, but it soon became obvious that they could play a much more important role. They could be used to release bombs on the enemy. It was a mere ten years after the Wright brothers flew their first airplane that the First World War began, but by then airplanes had already been used in warfare. In 1911 the Italians used an airplane to drop grenades on the Turks in Libya. And as it became obvious that airplanes would be useful in war, the technology associated with them advanced rapidly, and soon after World War I started they began to be used extensively by both sides.
DISCOVERIES THAT LED TO THE AIRPLANE
Although the most eventful day in airplane history was December 17, 1903, when the Wright brothers made their first flight in a power-driven, heavier-than-air machine, it was not the first attempt that humans had made to fly. Many important developments led up to that day, and I will begin with them.
We saw earlier the Leonardo da Vinci was obsessed with flight. Not only did he observe birds in flight for hundreds of hours, but he also studied the flow of both air and water around objects of many different shapes under various conditions. He noticed that water sped up as it moved around a rock in a stream, and he assumed that air did the same thing. Much of his effort went into trying to develop a pair of wings, like those of a bird, which a man could use to fly. He wasn't successful, but he did design a helicopter and a parachute, and both of these designs would have worked. Furthermore, he stated that the fluid dynamics are the same for an object moving through a fluid as they are for a fluid moving past the object in the same way. And finally, he also made an extensive study of drag, the frictional force an object experiences when moving through a fluid.
It was Galileo, however, who showed that the drag exerted on a body moving through a fluid is directly proportional to the density of the fluid, where density is mass per unit volume. The French scientist Edme Mariotte took this a step further in 1673 when he showed that drag is also proportional to the velocity of the object squared (v2).
One of the most significant discoveries in relation to aeronautics, however, came in 1738 when Daniel Bernoulli of Holland showed that in a flowing fluid the pressure decreases as the velocity of the fluid increases. And of course this applies to all fluids, including air. It eventually became known as Bernoulli's principle. About the same time, the French chemist Henri Pitot demonstrated a device he called the pitot tube in which the change of velocity could easily be measured as the diameter of the tube changed.1
A further advance in the understanding of drag came in 1759 when the English engineer John Smeaton invented a device for measuring the drag produced on a paddlewheel moving through air. He showed that D = ksv2, where D is drag, s is surface area, v is the velocity of the paddle, and k is a constant that became known as Smeaton's coefficient.
One of the most important people in the history of aeronautics, however, was the engineer George Cayley of England. He is usually considered to be the first person to understand most of the basic underlying principles and forces involved in flight, and because of this he has frequently been referred to as the father of aerodynamics. In particular, he discovered and identified the four major forces associated with flight: lift, weight, thrust, and drag. We will look at each of them in detail later. He also showed that “cambered” or curved wings produced the best lift. His three-part treatise titled “On Aerial Navigation,” which was published in 1809 and 1810, was the most important early work on airplane flight. Most of the basic ideas associated with lift, drag, and thrust are discussed in it.
Although Cayley designed, made, and flew many gliders, it is Otto Lilienthal of Germany who is usually referred to as the “glider king.” He made several important advances in hang gliders, and over his lifetime he made over two thousand flights in gliders of his own design. In August 1896, however, while making a flight, his glider stalled. He tried to regain control by ad
justing the position of his body, but he failed. The glider fell to the earth from a height of fifty feet. He was conscious when help reached him, but he died soon thereafter.
The first American to make important contributions to aviation was Octave Chanute, a civil engineer from Chicago, Illinois. He published the book Progress in Flying Machines in 1894, which was the most complete survey of the research on heavier-than-air aviation up to that time. And although he designed many gliders and invented the “strut-wire” braced wing, he never flew any of his gliders himself. He is perhaps best remembered for the interest and encouragement he gave to the Wright brothers. Indeed, he visited their camp near Kitty Hawk, North Carolina, in 1901, 1902, and 1903—the critical years in the development of their first airplane.
THE WRIGHT BROTHERS
Although many men made important contributions, the Wright brothers of Dayton, Ohio, are credited with designing and building the first engine-powered heavier-than-air craft to successfully carry a man on an airborne flight. This occurred on December 17, 1903. Their major contribution to aeronautics is usually considered to be their invention of the three-axis control, which enabled the pilot to maintain equilibrium and to steer the aircraft effectively.2
Orville and Wilbur Wright spent their early years in Dayton, Ohio. They were the two youngest of eight children, and according to most biographers their interest in flying was sparked at a young age when their father bought them a toy helicopter that was powered by a rubber band. Wilbur was four years older than Orville. Neither man completed high school, but they became interested in newspaper publishing after they built a printing press. They started with the West Side News and later published other newspapers.3