1. From Surely You’re Joking, Mr. Feynman!:
    “Then I held up the elementary physics textbook they were using. “There are no experimental results mentioned anywhere in this book, except in one place where there is a ball, rolling down an inclined plane, in which it says how far the ball got after one second, two seconds, three seconds, and so on. The numbers have ‘errors’ in them – that is, if you look at them, you think you’re looking at experimental results, because the numbers are a little above, or a little below, the theoretical values. The book even talks about having to correct the experimental errors – very fine. The trouble is, when you calculate the value of the acceleration constant from these values, you get the right answer. But a ball rolling down an inclined plane, if it is actually done, has an inertia to get it to turn, and will, if you do the experiment, produce five-sevenths of the right answer, because of the extra energy needed to go into the rotation of the ball. Therefore this single example of experimental ‘results’ is obtained from a fake experiment. Nobody had rolled such a ball, or they would never have gotten those results!”

    The only quibble is that the textbook he is referring to is a Brazilian textbook, and it’s a textbook, not a school, but otherwise everything in the story indicates that they “got it wrong” — very wrong!

    I’ll add that what he is describing at length (beyond the quote above) about what he saw in the Brazilian teaching of Physics in the 1950s was exactly the same problem I encountered in the US in high school in the 1980s.

  2. @ Arthur – I would have had an easier time comprehending the gravitational acceleration law if I had first encountered it in precise metric units: 9.8 m/s^2 is clearly an experimental result. Unfortunately, in the elementary school science book that I first had, it was presented as 32 feet/s^2; with the “squared” on one end, I figured that the integer power of two on the other end must be the result of some fundamental principle that they had not bothered to explain.

  3. Kilby, that’s a good point. I never noticed the nice, round number and that it was a power of two. And, btw, it turns out it’s *not* a nice, round number, it’s actually 32.174049 feet/second/second, or, with the same accuracy as 9.8, 32.2 ft/s/s.

    If we hadn’t had this conversation on CIDU, I might never have noticed that 32 wasn’t exact, though it should have been obviously likely.

  4. and just to pile on the opinions for how to teach a formula, so that it is comprehended not just memorized, I think “per second per second” is actually far more cognitively sensible than “per second squared”, no matter how much the latter may seem more efficient, and strictly correct.
    (*Eventually* the “squared” as a notation by superscript will match up to the notation for taking a second derivative — which is not quite what is happening here but relates)

  5. @ Mitch4 – I would agree that the concatenation is definitely better for understanding what is physically happening, but once you get past that and start slicing and dicing constants (and their corresponding units) in equations, it is much easier to adjust and cancel things when there is only one denominator line.
    P.S. I remember seeing an equation that had sequential exponents, which I cannot reproduce here, but effectively is was something like “X^Y^Z”. I was never able to determine which operation should be evaluated first. Should that mean “(X^Y)^Z”, or was is supposed to be “X^(Y^Z)”? Mathematicians (and physicists) have a tendency to reduce their theories to the briefest (prettiest) form, but a lot of important details get swept under the rug in the process.

  6. And there may be in some contexts good practical reason for using together units of the same dimension but different sizes. If your car goes from zero to sixty in six seconds, that would be 10 mi/hr/sec, “miles per hour per second” — but it’s still acceleration and the abstract dimensions could still be called x/t^2. But it’s easier to see “metres per second per second” as entirely parallel to this, than for “metres per second squared”.

    BTW, James Joyce had some fascination with the value of g in English units. It shows up in Ulysses with Bloom sometimes thinking “thirty two feet per second per second” as a kind of mantra. And in Finnegans Wake the numerical value 32 stands in for The Fall — while 11 (since it starts fresh on a new decade after 10 closes stuff out) is a rise or resurrection. So a combined 3211 or 1132 can in condensed shorthand stand in for the doctrine of the Fortunate Fall.

  7. Speaking of “units of the same dimension but different sizes”, this was recently in the news: the U.S. uses two different definitions of “foot”, and they’re trying to ditch one of them:

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