Ballistics Physics And Math Robert A. Rinker
grown men and turns them into little kids. I was the same way in school, even when studying engineering in college. Because of a few problems, I walked into my fi rst college math class about an hour late on the third or fourth day. The professor had the blackboard covered with a long complicated problem and I didn’t have the slightest idea what was going on, and to make it worse, I was scared to death. I obviously survived, but I understand why people can be scared of the subject. I will try to make this as simple and easy as possible, and even if you don’t understand every detail, I believe you will learn a few valuable points. The laws of physics and mechan-
Y
ics govern every aspect of ballistics. We discuss trajectory and drag in terms of physics. Rifling deals with spin and moments of inertia. Almost every single area of discussion is involved in one or many fi elds of basic physics and mathematics. This article will touch on a few of them. This is not intended to be a course
in physics. This is basic, simple, and, I hope, easy to understand. Most people, if they are honest with themselves, will need to learn some of it. Relax. It isn’t that hard. Much of the theory behind fi rearm
ballistics depends on the discoveries and physical formulations of English math- ematician, physicist, and astronomer, Sir Isaac Newton (1642-1727). This is the basis for much of our
study: Sir Isaac Newton’s three basic laws of motion, which were listed in the last half of the 17th century. 1st. Law: “A body at rest tends to
remain at rest, and a body in motion tends to remain in motion at the same speed and in the same direction unless acted upon by a force.” This means that nothing stops or
starts moving unless some force makes it do so. For example, the bullet won’t start in motion until the expanding gas pushes it, and once moving it will do so until something overcomes its inertia.
es, I know that just the men- tion of this subject scares many
2nd. Law: “When a body is acted
upon by a constant force, the resulting acceleration is inversely proportional to the mass of the body and is directly pro- portional to the applied force.” This is not as diffi cult as it sounds. It deals with overcoming the 1st. Law. Acceleration, in this case, can be both positive, as in starting up and negative as in decelera- tion. In effect, it says that force = mass times acceleration. The result may be in a straight line, a curve if acted upon by another force as gravity affects a bullet’s trajectory, or, as with torque, in a rota- tion. An easier way of wording it would be, “... change of motion is proportional to the force applied, and takes place along the straight line in which the force acts.” The applied force is a total of all the forces acting on the object. 3rd. Law: “Whenever one body
exerts a force on another, the second body always exerts on the fi rst a force which is equal in measure but opposite in direction.” This is stated more commonly and
briefl y as “for every action there is an equal and opposite reaction,” which is the fundamental principle behind the movement of a jet airplane or how a gun may recoil and give you a kick. Energy is the capability to do
work. A body may possess this capa- bility through its position or condition. When a body is held so that it can do work if released, it is said to possess energy of position or potential energy. When a body is moving with some velocity, it is said to possess energy of motion or kinetic energy. The funda- mental law of Conservation of Energy states that energy can be neither created nor destroyed. There are different types of energy and energy can, however, be transferred from one type to another and from one body to another. Briefl y, a foot-pound (ft. lb.) is a
unit of kinetic energy but that informa- tion helps little at this point. Let’s look at it from the basic facts. The foot-pound was originated by
Italian physicist and astronomer Galileo Galilei in 1585 with a simple equation, y = 16t2
. This shows how gravity acts on
a free falling object, a rock, a bullet, or a cannon ball. Y is the distance fallen in feet and t is the elapsed time in seconds from the start of the fall. Newton, by differentiating the
equation once, determined that the speed an object is falling at any mo- ment equals about 32 times the number of seconds which it has been falling. Differentiating the equation a second time, he determined that the object’s ac- celeration is always 32 feet per second, every second. This does not change and represents a law of nature. When an object falls, starting
from a dead stop, it accelerates, that is gathers speed, at the rate of 32.17 feet a second for each second. Therefore, after one second, the falling object (what it is doesn’t matter) is dropping at 32.17 feet per second. After 2 seconds, 64.34 f.p.s. After t seconds it would be t times 32.17 or 32.17t. Of course the object will not fall 32.17 feet the fi rst second as it starts at 0 and has to accelerate, but it will increase in speed at 32.17 feet for each second. Velocities are given in feet per sec-
ond (f.p.s.) and accelerations are given in feet per second per second, a small but important difference. The sum of the kinetic and po-
tential energies of a body acted on by gravity alone is constant. If a body is at a certain height above the ground and is motionless, its potential energy is equal to the work done in raising it to that height. If that body falls to the ground, its velocity v will be the square root of 2 gh, and its kinetic energy will be wv2
/2g. The potential energy before
starting to drop is, therefore, the same as its kinetic energy when it reaches the earth. At any point during its fall, its total energy, obtained by adding the kinetic and potential energies, will be the same wv2
/2g, where v is the velocity
that the body would have if it fell freely to earth from the original height. The final velocity of a falling
body is proportional only to the verti- cal distance through which it falls, and is completely independent of the path it follows. In other words, it does not
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