The Barron's book, attempting to do Physics the Easy Way, simply presented the formula for centripetal acceleration without giving any account of its origin. But since I have adopted, late in my life, the habit of mathematical and scientific curiosity, I unnaturally wanted to know, how did this formula come about? Now that I am working through the more mature Schaum's book, it has been revealed to me. It took me a couple of days to work through the complications of the book's explanation. Finally I feel I have understood it well enough.

Or have I? The derivation for the formula which finds the centripetal acceleration of an orbiting body, whether it's a tether ball or a moon, is not something that just springs out of observation. You can tell, if you were to spin an object on a tether, that it takes more effort to hold it back, the faster you spin it. And you can also feel the difference in your effort and the "pull" of the object if you lengthen or shorten your tether. But translating that into mathematical language is something that isn't easy to do. Someone must have done it, though, for how could these formulas otherwise exist?

The formula is based on comparing vector calculations between two equal velocities going in two subsequent directions around a circular orbit. To get the acceleration formula, the book (or physics history in general) leads you through a series of algebraic manipulations and substitutions, eventually resulting in solving for the quantity you want. It is not like spinning and measuring. It seems quite abstract and removed from the original experience.

What perplexed, and fascinated me, was that algebraic manipulations of known quantities resulted in the specification of an unknown quantity. It was not through measurement but through pure calculation. As long as the relationship between the quantities is known, or even theorized, the math will take you there. I was not able to visualize or perceive the relationships or the observed experience behind these numbers and variables. Perhaps others more talented in physics, such as that proverbial geniusboy I am always mentioning, can do this. Or perhaps such an ability is acquired; I am hoping to acquire it, then. But faced with deriving formulas, I must walk in obscurity and let the bright red book lead me.

And now I have lots of problems to solve; I am going to war. I hear the guns of August. "A cannon is fired from ground level with a muzzle velocity of 2000 ft/second, at a 40 degree angle above the horizontal….A bomber is flying horizontally at a speed of 800 ft/second and at an altitude of 1000 ft. when it drops a bomb…"
The problems, tiny battles for me to fight, remind me that physics and war have always been close companions.

Posted at 2:57 am | link