The Physics of War (23 page)

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 were 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/sec
²
, 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.

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/velocity
²
) 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.