The Physics of War

by Barry Parker

  The Union was not very active in the production of submarines, but they made one that they called the Intelligent Whale, but it never saw action. Interestingly, several private ventures in both the North and the South also attempted to build submarines, but little is known about most of them. In all there may have been twenty submarines built during the Civil War, with most of them seeing no action. But the experimentation and innovation that went into them soon led to much better submarines. In particular, airlocks, compression air ballast tanks, electric motors, periscopes, and air purification systems were developed.

  BALLOONS

  Hot-air and hydrogen-filled balloons were used for surveillance by both the Union and Confederate armies during the war. But they were used more extensively and effectively by the Union army. In 1861 Lincoln gave orders to form a balloon corps with Thaddeus Love in command. And indeed, in several battles the information obtained using them was of considerable value. During the Seven Days Campaign of 1862, for example, Union balloons stationed seven miles from Richmond could easily observe troop movements within the city. The largest balloons (called the Integral and the Union) could carry five people and had a capacity of thirty-two thousand cubic feet. Hydrogen gas was used in most of the early balloons; it was generated from water using portable generators.15

  Almost all balloons were tethered to the ground by a long line, but they were able to climb to almost five thousand feet in the air. And although Union balloons were shot at extensively by Confederate cannons, they were generally too high to be in range, and none were ever knocked down. Most of the larger balloons also had telegraphic equipment to transmit information to the appropriate people below.

  How did these balloons work? For a balloon to rise, there has to be a force on it, and the only force readily available is a buoyant force. The Greek mathematician Archimedes was the first to understand this force, and as we saw earlier, it is now known as Archimedes principle. It states that any body completely or partially submerged in a fluid (in this case the fluid is air) is buoyed up by a force equal to the weight of the fluid displaced by the body.

  In the case of a balloon, a buoyant force (B), which is equal to weight of the air displaced, acts upward, and a gravitational force (W) acts downward. To see how this causes a balloon to rise, let's begin with the density of the gas () and the volume (V) of the balloon. The mass of the displaced air is going to be V, and for its weight we need to multiply by g (the force of gravity). So the buoyant force B is Vg. Now we need W, the weight of the balloon. For it we need the mass inside it, and this is the density of the gas inside it (D) times its volume (V), or DV, so W = DVg. The total upward force is therefore B – W = Vg – DVg. If the upward force is positive, the balloon rises. Since hydrogen is less dense than air, the balloon will rise if filled with hydrogen (or, for that matter, any gas that is less dense than air). Also, if we heat the air inside the balloon, the molecules move farther apart and its density decreases. And again the net force is upward. This is the principle of the hot-air balloon.

  In an earlier chapter we were introduced to the problems related to the accuracy of rifles and cannons. For many years, in fact, gunners had no idea what the trajectory of a bullet looked like. Tartaglia made some important advances, and Galileo clarified many of the problems related to gravity, but it was Newton who finally showed why and how gravity was involved. In this chapter we will look much more closely at the problem, and although we have discussed only muskets, rifles, and cannons up to the time of the Civil War, we will also deal with more modern rifles and cannons in this chapter.

  The study of ballistics is concerned with the motion of projectiles, but it is also concerned with what happens to the projectile, or bullet, inside the gun, and also what happens when it hits a target. In all, there are four basic areas of study within this topic:

  Interior ballistics

  Transitional ballistics

  Exterior ballistics

  Terminal ballistics

  Interior ballistics is concerned with what happens between the firing of the cartridge and the exit of the projectile from the muzzle. Transitional ballistics, which is also known as intermediate ballistics, is a study of the projectile from the time it leaves the muzzle until the pressure behind the projectile is equalized (in other words, when the air pressure behind the projectile is equal to the surrounding air pressure). Exterior ballistics deals with the projectile while it is in flight under the influence of gravity. Terminal ballistics is the study of what happens to the projectile after it hits the target. I will talk mostly about rifles, but much of what I say also applies to cannons.

  INTERNAL BALLISTICS

  Internal ballistics depends on what happens inside the breach and barrel of the gun, so it is the best place to begin our discussion. Of particular importance, the trajectory depends, to a large degree, on what is referred to as the muzzle velocity, which is the speed of the bullet as it leaves the end of the barrel. So let's look at how muzzle velocity develops. Two separate events are critical here: ignition of the gunpowder, and expansion of the gases that develop as a result of the initial explosion. Ignition occurs when the firing pin hits the percussion cap (or primer) and causes it to explode. This, in turn, ignites the gunpowder in the cartridge. As the gunpowder explodes, the gases produced by the explosion are trapped behind the projectile. They are at high temperature and therefore create pressure that accelerates the projectile down the barrel. It is particularly important that the burn time is less than the time it takes the projectile to reach the end of the barrel. If not, powder will come out of the barrel still burning, and this would create a dangerous situation.1

  Expanding gases behind a bullet in a barrel.

  As the gas expands, it cools according to the basic gas law discovered by Jacque Charles in the late 1800s, which is now referred to as Charles's law (it is sometimes referred to as Gay-Lussac's law, since it was discovered by Joseph Louis Gay-Lussac about the same time).2 This law tells us that pressure times volume is proportional to temperature, and therefore a sudden increase in temperature will create an increase in volume that will impart a higher pressure on the projectile. Inside the barrel the pressure does, indeed, increase initially, but as the projectile moves forward, the pressure decreases. A plot of pressure versus distance along the barrel is given below. We see that the pressure reaches a peak rather quickly before dropping rapidly. The maximum pressure is important because it occurs mainly in the breach area of the gun, and this area has to be able to withstand the pressure. This is why the steel of guns and cannons is thickest in this area. The rearward push on the gun's bolt, or breach, which results from the explosion, is referred to as the bolt thrust. It depends on both the chamber pressure and the diameter of the cartridge case. And it's important that the design of the gun is sufficient to withstand the bolt thrust. The pressure in this area (the chamber) is usually measured in pounds per square inch (psi), or in the metric system it is measured in kilograms per square centimeter (kg/cm2). Typical pressures in this area depend on the type of gun, but for rifles it is approximately fifty thousand pounds per square inch.

  A plot of pressure versus distance along the barrel.

  It's fairly obvious that the longer the projectile is accelerated by the explosive force, the faster it goes. The law controlling this is Newton's second law, which states that force is equal to mass times acceleration (F = ma). And this acceleration continues as long as the projectile is in the barrel (and slightly beyond, as we will see later). So the longer the barrel, the greater the velocity of the projectile when it exits; this is called the muzzle velocity. And although longer barrels provide greater velocity, there is a limit. Very long barrels are hard to handle, and the weight of the gun also increases with barrel length.

  The length of the barrel is important in another way. As the hot gas explodes down the barrel, it decreases in pressure, and ideally it should not be much higher than atmospheric pressure when it reaches the end of the gun. In practice, ho
wever, it is usually much higher. As a result, it creates a shockwave as it hits the air. The problems related to this will be discussed in the section on transitional ballistics.

  In addition to barrel length, the muzzle velocity also depends on the mass, or weight, of the projectile for a given amount of explosive; a lighter projectile will emerge with a greater muzzle velocity. The type of powder in the barrel is also an important factor; different types of powder have different propellant energies. How much powder should be used is also a factor, but for a given caliber, there is a limit on the amount of propellant.

  Another important question is: What is the maximum velocity that a projectile can be pushed to without extreme danger to the gunner? In rifles this is about four thousand feet per second. Large-caliber guns and cannons can safely be pushed to about six thousand feet per second.

  RECOIL

  It's important to note, however, that higher muzzle velocities have another problem. As we saw earlier, Newton's third law tells us that for every action there is an equal and opposite reaction. The action in this case is the force pushing the projectile and the hot gases out of the barrel, creating the muzzle velocity. The reaction is therefore opposite to this, and the gunner feels it as the gun's recoil. The direction of the recoil force is directly opposite to the force that pushes on the projectile. Anyone who has ever shot a gun has experienced recoil, and at times it can be quite strong. Directly related to the third law is the conservation of momentum. We can write it as mv = MV, where m and v are the mass and velocity of the projectile, and M is the mass of the gun (or the gun and gunner) and V is the recoil velocity. The mass of the gun is much greater than the mass of the projectile, but the muzzle velocity of the gun (v) is extremely large, so without constraint, the gun would achieve a relatively high velocity. But it is constrained by your shoulder, and this creates a relatively large force on it, which slows its velocity very rapidly. Gunners are, in fact, told to hold the gun snugly against their shoulders so that M includes not only the mass of the gun, but also the mass of the gunner.3

  Also, in movies and TV, you've likely noticed that the actors always hold their revolvers with two hands in an effort to steady them. One of the reasons for this is that recoil tends to cause the revolver to turn upward. This is caused by an upward torque that occurs because the recoil force is pointed along the barrel of the revolver, but the revolver is held by a lever, namely your arm and shoulder. Since this force is at an angle to the level arm, it creates it torque, causing the barrel to move back and turn up at the same time.

  One of the major ways to reduce the effects of recoil is to have a recoil pad at the end of the gunstock.

  TRANSITIONAL BALLISTICS AND THE SONIC BOOM

  Transitional ballistics is sometimes referred to as intermediate ballistics because it is concerned with the projectile behavior in the time between internal and external ballistics; in other words, the brief time from when the projectile leaves the barrel until the pressure behind it reaches the pressure of the surrounding air. When the bullet reaches the end of the barrel, the gases behind it are frequently at a pressure several hundred times that of atmospheric pressure. When the bullet flies free of the barrel, however, the gases behind it are free to expand and move outward in all directions. This sudden expansion causes a loud explosive sound, namely the boom that you hear when a gun is fired. It is also sometimes accompanied by a flash as the gases combine with the oxygen of air.4

  This is the first boom you hear when a gun is fired, but in many cases there is also a second boom, referred to as a sonic boom (it will be discussed a little later in the section). Earlier I mentioned that the bullet accelerates as it moves up the barrel because it is being pushed by the expanding gases, but once it leaves the barrel it has a uniform horizontal velocity. This is not exactly true. There is still a force on the bullet for a short time after it leaves the barrel due to the expanding gases behind it. This is one reason why the “muzzle velocity” of a rifle is not measured at the end of the barrel, but several feet in front of it.

  In the design of a gun it is important to make sure that the gases that expand behind the bullet as it leaves the gun do not disturb the bullet from its path. If it is somehow pushed off to one side, the accuracy of the rifle would be compromised. So the gun has to be designed so that this doesn't occur; in short, the designers have to make sure the gases expand symmetrically around the base of the bullet.

  In the case of military weapons—particularly sniper rifles—it is important to decrease the sound and flash from the rifle as much as possible so that the position of the sniper is not given away. This is done by flash and sound suppressors; in both cases devices are used to change the flow of the escaping gases. In the case of flash suppressors, turbulence in the escaping gas is introduced in an attempt to reduce the combustion efficiency of the flash. In the case of sound suppressors the gas is allowed to cool so the velocity at which it exits the barrel is reduced; this prevents a shockwave from forming. Unfortunately, suppressors are bulky and heavy, and they are not used extensively.

  But even if they reduce the shockwave from the exiting gases, there is still a sonic shockwave that can easily be heard. Let's look at this wave. It's well known that any object traveling through air at a velocity greater than that of sound will cause a sonic boom. This applies to most bullets, and, as it turns out, no type of silencer can dampen a sonic boom, since the shockwave that causes the boom travels along with the bullet.

  For a supersonic shockwave, the bullet has to exceed the speed of sound, which is about 1,100 feet per second (or approximately 750 miles per hour) depending on the air pressure and a number of other factors. About one-half of all pistol ammunition is supersonic, as is virtually all rifle ammunition, and most shells shot from cannons are also supersonic. Tank guns, as I mentioned earlier, have velocities up to six thousand feet per second, which is many times the velocity of sound.

  To understand the sonic boom better, let's look at how it is created. It's well known that when an object creates a sound, a wave travels outward from the object at the speed of sound. If you look closely at this wave you see that it consists of a series of compressions and rarefactions. The compressions occur because the molecules of air are pushed together in certain regions, and the rarefactions are caused because the waves spread out in other regions. This means that a wave that is uniform in all directions passes outward from the source. But when you move the object that is creating the wave, the wave pattern around it changes. The compressions get closer together in the direction that the object is traveling and farther apart in the opposite direction. Furthermore, as the object moves faster, the waves in the forward direction start merging into one another, and at the speed of sound they merge completely together.

  At this point the pressure on the nose of the bullet is much greater than it is on the rear of the bullet. But sound in air can only move at approximately 1,100 feet per second, and a bullet can move at any speed; in particular, it can move at speeds greater than the speed of sound. Because of this, when the bullet breaks, or passes through the sound barrier, it creates compressions faster than the compressions themselves can move away, so they just pile up on one another. As these compressions are brought together, they do not form a smooth progression from compression to rarefaction, as they do in ordinary sound waves. Instead, there is a sharp dividing line between a volume of strong compressions and the normal atmosphere around the wave. As a result, the strong compressions stream backward in a cone-shaped band. When this cone passes an observer on the ground, he or she experiences a sudden difference in pressure as it moves by, which he or she interprets as a sonic boom. In many ways it is quite similar to the crack a bullwhip makes.

  Cone that forms during a sonic boom.

  EXTERNAL BALLISTICS

  External ballistics deals with the behavior of the projectile in flight from the time it is just beyond the end of the barrel until it hits the target. Galileo realized that two separate motions wer
e involved: a horizontal motion parallel to the ground, and a vertical motion. And although there were two motions going on at the same time, they could be dealt with separately. The horizontal motion was the horizontal component of the muzzle velocity, and it had a constant velocity. The vertical motion was free fall due to gravity, and it was therefore a constant acceleration of 32 ft/sec2, which is the acceleration of gravity. Galileo also showed that the overall trajectory when you combined the two motions was a parabola. (As we saw earlier, the easiest way to visualize a parabola is to take a cone and slice it somewhere along the side so that the slice passes through the base.) As it turns out, this is only approximate because of air pressure. Air pressure slows the bullet and causes the trajectory to deviate from a parabola.5

  Path of a bullet with and without air resistance.

  One of the things that's easy to show is that a bullet drops in the same way that something drops if you hold it above the ground and let it go. A demonstration of this is frequently used in physics classes; it has a simple projector (a gun) that throws an object straight out from it, parallel to the ground, and releases another object at the same time that falls directly downward. The first object traces out a much longer path, but the two objects hit the ground at the same time.