Since the text editor will not support subscripts and superscripts, I define my own notation to represent them: (1) All kinematic variables are represented by capital letters. For example velocity is represented by V and acceleration by A. (2) Subscripts are designated by lower case letters. For example, initial velocity “V naught” is Vo. (3) Powers are represented by **. For example, “time squared” is T**2.

The first topic most authors cover in their introductory physics textbooks is kinematics. They make this choice because students must have a firm grasp of position, velocity, and acceleration before they can study the various topics in Newtonian mechanics. Unfortunately, kinematics seems to encourage students to formula hunt and plug. While learning to solve kinematics problems, students usually have far too many equations bouncing around inside their heads. Their problem solutions include equations such as V = Vo + AT, X = VT, the range formula, the maximum height formula, and so forth. But physics is not a study of equations — it’s a study of fundamental principles, most of which happen to be expressed as equations. Even when teaching kinematics, I tell students they will never understand physics if they approach it as a bunch of equations to memorize. I emphasize that they have to learn to think in terms of principles. Unfortunately, that message is difficult to get across when students see equation after equation in their textbooks. It is especially difficult because in high school most students have learned the old-fashioned problem-solving technique of “identify the knowns, then plug these into the right equations to find the unknowns”.

Far too many students see all of physics as nothing more than an exercise in hunting for the right formula. I have had some success combating this unfortunate difficulty by treating a set of basic equations for constant-acceleration kinematics like they were fundamental principles. I acknowledge that they are not basic principles; nevertheless, I ask the students to treat these equations as if if they were fundamental. I teach the students to start all kinematics problems (both one-and two-dimensional) in terms of the same basic equations.Then the logical thought processes they use for problem solutions in kinematics are just like those they will employ later when they actually encounter the fundamental principles. All textbooks derive the important equations for motion with constant acceleration. All problems can be solved in terms of only three of these. In covering problem solutions, I never stray from these three equations. With Xo and Vo the position and velocity at T= 0, the three equations are the obvious ones:

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In a previous article (Teaching Kinematics), I argued that all free fall problems should be solved using the same three basic equations. These equations relate position to time, velocity to time, and velocity to position. I also argued that the three equations should be treated like they were fundamental principles. I claimed that this approach discourages the formula hunting and plugging that far too many students employ when solving free-fall and other one-dimensional kinematics problems. The purpose of this article is to argue that this same approach should be encouraged when teaching students how to solve projectile problems. Let’s look at an example that would be described as a difficult problem in an introductory physics course.

Problem. A football is kicked from the ground at 25° to the horizontal, as illustrated in the sketch (I did not include the sketch.). It has to clear a crossbar 10 ft above the ground and 150 ft away. What is the minimum speed Vo at which the football must leave the kicker’s toe if it is to clear the crossbar? (Use g = 32 ft/s**2).

Analysis. We use an (x,y) reference system whose origin is at the original position of the football. As usual, Xo = 0, Yo = 0, Vox = VoCOS(25° ), Voy = VoSIN(25° ), Ax = 0, and Ay = -g. At the time when the ball is 150 ft away from where it is kicked, it must be a minimum of 10 ft above the ground. Said another way, at the same time T that X = 150 ft, Y = 10 ft. This statement virtually screams at us to relate horizontal and vertical positions to to their common time (with X = 150 ft when Y = 10 ft). When we do, we find

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Honestly, I am sick to death of the eternal squabbling between science and religion.
Religion is a methodology that has been constructed to enable humans to
understand their place in the universe. Science is a methodology that has been
constructed to enable humans to understand their place in the universe. There are
similarities in the limitations of both systems of thought. In some sense, science
and religion are the same. In my opinion, the most important discovery in science is
that there are things that we cannot ever know about the universe. This is neatly
paralleled in most theistic thinking, that we cannot truly know God. He is a mystery
that cannot be fully comprehended.

Let’s start with science. The central tenets of mathematics, physics, cosmology,
biology etc., are not invented by human endeavour. They are discovered. They exist
independently of humans. The universal layers of opacity have, over hundreds if not
thousands of years, been slowly and painstakingly removed by scientists in all
fields, to reveal ever more clearer views of not only what our universe is, but how it
is. The problem is that these systems do not attempt to explain why it is.

Some people think the whys of the universe are inappropriate even irrelevant
questions for scientists to try and answer, but this is just disingenuousness. The
most important thing people want to know about the universe, about their lives,
about themselves, about their place in the world is, why?

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