panic at the thought of studying trigo- nometry. They hear a story of how difficult it is and that it is impossible to understand. This is not true. With a scientific calculator, most trig problems can be worked without understand- ing the theory by inserting the proper numbers in the equation and pressing the correct keys. Except for introduc- ing a few new words and using some strange looking letters from the Greek alphabet, trigonometry should create no problems. The computations are, for the most part, just simple algebra or arithmetic. The formulae are important and must be followed carefully in the proper order and with the proper units. While it would be nice if a few of
the readers who know nothing of trigo- nometry would become interested and want to learn more, the main idea is to be able to work the ballistic problems that we want an answer for. With a calculator and a little thought, this can be done.
Trigonometry is simply the study
of triangles and the relationship or ra- tios between the sides and the angles. It also can be applied to other problems in which angles or directions are involved.
For example, if a gun is pointed 15 de- grees east of north and the barrel is at an angle of 22 degrees with the horizon, trigonometry would be involved in any calculations. These ratios are repre- sented by the cosine (cos), sine (sin) and tangent (tan). These change in numerical value as the size of the angle changes. Angles are kept separated by us-
ing Greek letters to denote them. Lower case theta θ and beta β are examples. As the bullet arcs toward the
target, at any given moment the bul- let is moving in a definite direction. In respect to the ground, this will be an instantaneous slope and a change in the bullet’s altitude. This can be regarded as changes with respect to the elapsed time since the bullet left the muzzle. At any specific point, we can discuss the rela- tionship between the distance, altitude, and time. In calculus, this would be a derivative (differential coefficient) and written as simply dy / dt, where t = time. A derivative, or limiting value of
a ratio, is the fundamental process with calculus. The logic and mathematics of calculus can be used on problems of time, points on a curve, and many other ballistic problems. It can be described as
the developing difference between the situation at one moment and the situa- tion at the next moment. This gives clues or evidence of how a situation is shaping up. If the ratio of the net changes that take place is judged as a limit neared as the space between the moments ap- proaches zero, then the limit shows how rapid the situation is developing. This rule is applied to the classic
equation for gravity/acceleration of a falling object. Calculus permits us to stop motion and break down movement into points that can be traced through space and time. We can calculate the ve- locity/acceleration at a specific moment. Students who have not studied
calculus think it is the ultimate math, and something to be feared. Don’t worry. If you find a problem that has a calculus problem and the formula is given, it is no harder to work out the answer than with most other formulas. For most people, getting over the
fear of physics and mathematics is most of the battle. And even if some parts of this article didn’t make sense, if you got this far, I bet you have learned a lot.
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Page 86 Winter 2012
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