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Teaching Projectile Motion
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
Albert Einstein, the Dishonest Newton?
Newton’s black-box physics.
From laws of force and motion in mechanics, Isaac Newton developed laws of orbital motion around ‘centres of force’ and saw gravity as governing the motions of the celestial bodies as well as apples falling from trees. But despite using the ‘force’ terminology perhaps more readily associated with mechanics, Newton concluded that gravity might be due to unseen signals acting across empty space in line with William Gilbert’s physics theory, or might be caused by the impact force of unseen ether matter in line with Rene Descartes physics.
But Newton saw his laws of science as correctly predicting natural events without the need to know why things happened, in the manner of ‘black box’ behaviour laws that related only inputs or stimuli to outputs or responses. Newton considered hypotheses regarding unseens like ‘gravity signals crossing space’ or ‘ether matter filling space’ as matters of philosophy or logic and not science, but as not easily disproven by science. Newton’s physics was about what the universe did, but explicitly excluded explanations of why.
Why
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?
