To the editors:
The many-brained and almost-all knowing Cecil Adams, who recently discussed correctly a question involving a somewhat subtle point in the general theory of relativity [June 30], has now flubbed what he (a bit hastily) describes as a problem “straight out of Physics 101” [Letters, July 21; Straight Dope, December 16]. I refer to the matter of the comparative times before hitting the ground of a plummeting and a (horizontally) fired bullet.
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Now, that’s plumb wrong. The air is not smart enough to take separate account, directly, of the horizontal and vertical components of the motion. What it “knows” is merely that it is being penetrated, at a certain rate, in a certain direction. Its resistance is a force, in the direction opposite to that of the moving body, whose magnitude is some function of the speed of that motion.
An easy bit of elementary vector algebra (or geometry) leads to the conclusion that the actual effect produced by such a force will be the same as “independent action upon the two components” (that is, considering only the vertical component as the one of interest, that the vertical resistance will be the same as if only the vertical component of the total motion were in fact present), if and only if the law of dependence of the magnitude of the resistance upon the speed is that of direct proportionality. A more delicate (but still fairly elementary) analytical argument shows that if the resistance increases more rapidly, with increasing speed, than in direct proportion to the speed, the effect of the resistance will increase the time of fall more for a projected object moving obliquely downwards than for the same object plummeting straight down.
Cecil Adams replies:
I am not going to abuse you about this, doc, though you richly deserve it, because your mistake is a common one. Even Jearl Walker, author of the Amateur Scientist column in Scientific American and a professor of physics at Cleveland State University, initially agreed with you when I called to discuss the matter with him. I am pleased to report, however, that he eventually came around. With luck, so will you.
Let’s now consider an extreme example. Suppose we simultaneously shoot a gun and drop a bullet from an elevation of 300 meters above the ground. Suppose further that the fired bullet has an initial horizontal velocity of 300 meters per second. No one disputes that in a vacuum both bullets would hit the ground at the same time. Using other formulas in McGraw, we calculate that the time of flight for both bullets in a vacuum would be 7.8 seconds, that they would achieve a maximum vertical vector of velocity of 76.68 meters per second, and that the shot bullet would strike the ground at an angle of slightly more than 14 degrees relative to the horizon.
It should now be clear that the fact that drag increases with the square of velocity is irrelevant to the problem. The dropped bullet and the shot bullet are traveling downward at identical speeds; the vertical component of drag on them is the same. Your argument is tantamount to saying that some of the horizontal component of drag slops over into the vertical. It is tempting to believe this occurs, but the equations above indicate otherwise.
