The Physics of War (4 page)

The ballista.

A variation on the ballista came a little later in the form of the onager, which was used mainly by the Romans. It also used torsion, but was basically a type of catapult. It consisted of a large frame that was placed on the ground. A vertical frame of wood was rigidly attached to it. This vertical frame contained an axis that had a spoke or arm projecting out of it. This spoke was attached to ropes (or a spring) that could be twisted; the arm was pulled back, or armed, against the buildup of torsion in the ropes. Again, there was a pin to release it, and when the pin was struck with a hammer the projectile was launched toward its target. Large stones were usually used as the projectiles.

Soldiers arming an onager.

The third type of new weapon, the trebuchet, was actually the most powerful of the three. It was invented by the Romans and had three main characteristics:

• It was not powered by torsion. Its power came from gravity acting on a counterweight.
  
• It used what is called the “fulcrum principle,” where one arm was much longer than the other. The throwing arm was usually four to six times longer than the counterweight arm.
  
• A sling with a pouch was attached to the end of the throwing arm to increase the speed of the projectile.
  

The device was loaded by placing a large and usually very heavy stone in the pouch. The throwing arm was then pulled down against the weight of the counterweight. It was tied down until ready. When it was triggered, it could throw rocks of three hundred pounds and more a distance of several hundred yards, but it was not nearly as accurate as the ballista or onager.
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The trebuchet and the onager were both a form of catapult. A catapult is a device that usually has an arm that is pulled back against a force and released. Several other types of catapults were also used, but the major ones were the two above. The physics of the above devices will be discussed in the
next chapter
.

ALEXANDER THE GREAT

One person who made extensive use of the new weapons was Alexander the Great. Born in Pella, the capital of Macedonia in 356 BCE, Alexander became the greatest military leader of his time, conquering most of the known world. Taught by Aristotle beginning at the age of sixteen, he developed a strong interest in science and physics. When he turned nineteen he began accompanying his father, Philip II, on some of his campaigns. Shortly thereafter, however, his father was assassinated, and because his father had multiple wives, and Alexander's mother was only one of them, his chances of inheriting the throne were not good. But he was determined to get it, and he took the necessary steps, killing several people in the process.
⁸

When he became leader he quickly undertook a series of campaigns that lasted for almost ten years. In the end he had conquered Egypt, Mesopotamia, Persia, Central Asia, and even India. And by the time he was thirty he was considered one of the greatest military leaders the world had ever produced.

Aristotle had instilled in him a love of knowledge, and it remained even after he became king. As a result, he created one of the greatest learning centers the world had ever seen. After conquering Egypt, he founded Alexandria in 331 BCE, setting it up as a scientific research center. Although he only stayed in the city for a few days, he left with his viceroy and a general named Ptolemy an outline of the work he wanted done.
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At Alexandria what was called a “mouseion” was set up for the study of engineering, astronomy, navigation, physics, and the machines of war. The best scientists in the country and in surrounding countries were then invited to study there, including Eratosthenes and Hipparchus.

Perhaps the greatest feature of the new Mouseion at Alexandria was its library. It eventually became the largest library in the world, housing over seven hundred thousand manuscripts. It thrived for centuries, but much of it was eventually destroyed by fire.

ARCHIMEDES

One of the men who studied at Alexandria was Archimedes, who was born into 87 BCE in Syracuse, Sicily. He made a large number of contributions to physics, one of the most important of which was a principle now referred to as Archimedes' principle. It states that
a body immersed in a fluid experiences a buoyant force that is equal to the weight of the fluid it displaces
. He also designed what is now called Archimedes' screw. According to early accounts,
the king of Syracuse commissioned Archimedes to design a large ship, but it was soon discovered that a considerable amount of water was leaking into the hull, and it was difficult to bail it out. Archimedes designed a machine with a revolving screw-shaped blade inside a cylinder that raised the water from the bottom of the hull as it was turned.
¹⁰

Archimedes was also one of the first to explain the principle of the lever. And he is reported to have helped the people of Syracuse when they were attacked in 14 BCE. Presumably, he set up large curved mirrors that reflected the rays of the sun upon the attacking ships, causing them to catch fire. Most modern scientists doubt that this was possible at the time.

Physics is related to the early weapons of war just as it is to the more sophisticated later weapons. So far we have talked mostly about chariots, men on horses, bows and arrows, spears, and such things as the ballista, the onager, catapults, and trebuchets. Physics is involved in all of these things, but we haven't shown how it is involved. In this chapter we will do this, but first we will discuss the basic concepts of physics, beginning with the most basic ones, such as speed and acceleration, and proceeding through to more complicated ones, such as energy and momentum.

VELOCITY AND ACCELERATION

Everyone knows that if you shoot an arrow into the air it rises to a certain point before falling back to earth. It's also known that its speed as it leaves the bow depends on how hard the string pushes on it, and it's easy to see that its speed throughout its flight is not the same. After all, if you shoot it straight upward, it stops at some point before it starts to fall back to earth.

We have a slight problem in relation to motion on earth, however. Every object that moves has to pass through air, and this air has an effect on its speed as well as the path it takes. Dealing with the effect of air, however, is rather complicated, so for now we will ignore it.

The first thing we can say about an object in motion is that it has a certain speed relative to the surface of the earth. Speed is a useful concept, but even better (as far as physics is concerned) is
velocity
. Speed is defined as the distance something travels in a unit of time, say, a second, or even in hour. An arrow, for example, can have a speed of fifty feet per second. The problem with this is that it doesn't tell us anything about the direction that the arrow is traveling. If we specify both speed and direction, we have velocity. The velocity of the above arrow, for example, might be fifty feet per second in a northern direction.

If we look at this arrow a little closer, however, it's easy to see that it doesn't
have a constant velocity. Its velocity is continually changing, and the biggest change will occur when it is shot directly upward. After all, it stops at its highest point. We refer to this change in velocity as
acceleration
. The arrow might leave the bow with a velocity of fifty feet per second, but a few seconds later it will be going only ten feet per second. Acceleration is clearly different than velocity, and it therefore needs a different unit. The unit in this case is feet per second squared (in the metric system it is meters per second squared). Velocity and acceleration are related by a simple formula: velocity (v) equals acceleration (a) × time (t), or more simply v = at.
¹

FORCE AND INERTIA

Closely related to velocity and acceleration is another important physics concept called
force
. For an arrow to gain speed—in other words, to accelerate—it must undergo a force, and as I mentioned earlier, it is the string of the bow that applies the force to the arrow. A force is simply defined as a push or a pull. And force is like velocity in that it has both magnitude and direction (we refer to such a quantity as a vector).
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We can relate force to acceleration, but before we do, let's introduce another important concept from physics. Everyone knows about weight, and how it seems to creep up on you when you eat too many chocolates. What we're interested in is closely related to weight, but it's not exactly the same. We refer to it as
mass
, and we abbreviate it as m. The mass of an object is its weight divided by the acceleration of gravity, which is usually abbreviated as g. I'll explain a little later why we need mass rather than weight.

The relationship between force and acceleration was given by the English physicist Isaac Newton. He included it in three laws of motion that he published in his
Principia
in 1687. He explained that an acceleration created by a force acting on a body is directly proportional to the magnitude of the force and inversely proportional to the mass of the body. We can write this an algebraic form as a = F/m. As it turns out, it is more convenient to use metric units in this formula (instead of the units you are probably more familiar with, namely feet, miles, and so on, which are units in what is called the British System). Within the metric system, however, we have two systems, referred to as
cgs
(centimeter, gram, second) and
mks
(meter, kilogram, second). In the mks system, acceleration is measured in meters per second squared, mass is measured in kilograms, and the unit of force is the Newton. In the cgs system, acceleration is measured in centimeters per second squared, mass is measured in grams, and the unit of
force is the dyne, which is the force required to cause a mass of one gram to accelerate at a rate of one centimeter per second squared.

The above formula is usually written as F = ma. So the force on an object is the product of mass and acceleration. For example, if you want to create an acceleration of 25 km/sec
²
in an arrow with a mass of .01 kg you would need a force of .01 × 25 = .25 Newtons.

Closely related to the concept of a force is what is called
inertia
. We encounter inertia every day; when you push on an object or lift it, you have to exert a force to get it going. If an object is not moving—in other words, it's just sitting there—it tends to resist motion, and it takes a force to get it going. Indeed, the heavier it is, the greater the force that is needed. This “resistance” to a change in motion is called inertia, and Newton described it in his first law of motion:
a body will continue in a state of rest or uniform motion in a straight line unless acted upon by a force
. Note that this applies not only to something at rest, but also an object in uniform motion.

This means we need a force to overcome inertia, and this force produces acceleration according to the above formula. Furthermore, a force is always associated with two bodies. If one body is being pushed, the other body has to do the pushing. This also applies to an object sitting on the floor; because of its weight it pushes down on the floor. But according to Newton, the floor pushes back with an equal force in the opposite direction. Newton postulated this in his third law of motion:
whenever a body exerts a force on a second body, the second body exert
s
a force on the first body
. These forces are equal in magnitude and opposite in direction. They are frequently called the “action” and “reaction” forces. A good example of these forces can be seen when you hold a garden hose with water pouring out of it. You feel a backward force on your hands; this is a reaction force, and it's why rockets work: explosive gases shoot out the back of the rocket, giving the rocket its forward thrust.

MOMENTUM AND IMPULSE

Another important concept in physics is
momentum
; it is the product of mass and velocity (m × v). It is particularly important when one object collides with another. As you no doubt know, when a massive object collides with a smaller, lighter one, it's the smaller one that suffers the most. To understand this more fully we have to introduce the concept of
impulse
. Assume a soldier hits the shield of another soldier with his sword; he's obviously applying a force to it, but this force is only exerted over a short period of time. The product of this force, and
the time it acts, is defined as impulse. Furthermore, it's obvious that this impulse is going to cause the shield to move with a certain velocity, and this velocity will depend on the mass of the shield. So impulse is somehow also related to momentum. Indeed, the impulse creates momentum; or, to be more precise, since the momentum of the shield was zero before the impulse, the impulse creates a
change
in momentum. So impulse is equal to change in momentum.
³

Now let's go back to our discussion of the collision of two bodies. Of particular importance in relation to such a collision is what is called
the principle of conservation of momentum
. It states that the total momentum of any isolated system remains constant. This means that the total momentum before the collision will be equal to the total momentum after it, assuming there are no outside influences. Let's assume that the collision is a head-on collision, and that both objects have the same (but opposite) momentum. It's pretty obvious that they will stop dead. It almost seems as if their momentum has disappeared—but it hasn't. Before the collision they had equal but opposite momenta, and the sum of two equal and opposite numbers was zero. After the collision it's still zero. As the collision occurred, each object imparted an impulse on the other, but the impulses were equal and opposite, so the objects stopped.

It's easy to see from this that if one of the objects has a greater momentum than the other, it will generate a greater impulse on the second, and if the two objects stuck together when they collided they would continue with a certain velocity in the direction of the one that had the greatest momentum.

THE EFFECT OF GRAVITY

Everyone knows that when you shoot an arrow at some angle in an upward direction, it doesn't travel in a straight path. It travels upward for a while then heads back toward earth, eventually landing. This is because of the gravitational pull of the earth on the arrow. In reality, the two objects are attracted to one another, but because the earth is so much more massive than the arrow, it appears to us that the earth is attracting the arrow. Again, it was Newton who explained what is going on. He postulated that all objects in the universe attract one another. Indeed, he even gave a formula for the force between any two objects.

Let's begin by considering a stone held at some distance above the ground. It is attracted by the earth with a certain force, and if we let it go, it accelerates downward until it strikes the ground. With a relatively simple apparatus we can measure this acceleration, and we find it to be 32 ft/sec
²
, or in metric units, 9.8 m/sec
²
.

Gravity is particularly important in relation to warfare because all objects, such as arrows, cannonballs, bullets, and so on, are affected by it. Such projectiles trace out trajectories that depend on several things, such as their mass and speed, and also the air pressure. (We'll talk about trajectories in detail later in the book.)

The acceleration of gravity is not the same everywhere, however. It depends on the mass of the planet you happen to live on. So if you were to travel to Mars or Jupiter, it would be different. As a result, your weight would also be different. On Jupiter for example, you would weigh 2.34 times your weight on earth. What is constant is your mass; it doesn't depend on the gravitational field you happen to be in, and that's why it is used in most of the basic physics equations. The relationship between mass (m) and weight (W) is given by W = mg, where g is the acceleration of gravity.

ENERGY AND POWER

If you lift something in an upward direction through a certain distance you do work. To perform this work it takes
energy
, and, as it turns out, there are several different forms of energy. Two of the most common forms are energy associated with motion, and energy associated with position. Energy associated with motion is called kinetic energy, and since it depends on motion, it will also have to depend on velocity. Furthermore, an object with a greater mass will have more kinetic energy than one with less mass, so kinetic energy also depends on mass. It is therefore defined as kinetic energy = 1/2mv
²
, where m is mass and v is velocity. Its units are foot-pounds in the British System and Newton-meters in mks units.

Energy of position is called potential energy. It also has the ability to do work. Consider a stone held at some position above the ground. If you drop it, it does work on that dirt it strikes; it compresses and heats it slightly. We define it as potential energy = mgh, where m is mass, g is acceleration of gravity, and h is the distance above the ground from which it was dropped.
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Like momentum, energy is also conserved. In short, the conservation of energy states that energy can neither be created nor destroyed; it can only change from one type of energy to another. This can be demonstrated quite nicely if you take a ball and throw it upward. When first thrown, the ball has a high velocity, and its energy is therefore mostly kinetic energy. As it continues to rise, however, it gradually decelerates because of the pull of gravity. Finally, it stops, and at this point it has zero velocity, and therefore it has no kinetic energy. In essence, all its kinetic energy has been converted to potential energy, so at this
point it only has potential energy. As it begins to fall again, however, its speed increases, and its kinetic energy also increases. At the same time, its potential energy decreases, and by the time it is just about to hit the ground all its potential energy has been converted back to kinetic energy.

The two above types of energy are not the only two types of energy. Other forms are deformational energy, heat energy, sound energy, electrical energy, chemical energy, and nuclear energy. You might, for example, ask what happens to the kinetic energy of the ball when it hits the ground. It appears to be lost, but it isn't. It is converted to deformational and heat energy.