We don’t explicitly teach the notion of a velocity at an instant in our introductory algebra-based course, although the term instantaneous appears in the text every now and again. The concept is, of course, implicit in everything we do beyond chapter two. I have noticed this past year, the negative consequences for students’ understanding of velocity, acceleration, forces, and energy.
When I have mentioned to several colleagues this lacking in our introductory curriculum, a common question that arises is how to think about teaching velocity at an instant (in an intellectually honest way) without calculus. In this post, I don’t intend to describe how to teach the concept, but to illustrate about how clear the ideas of constant, average, instantaneous, and change in velocity were in Galileo’s mind even without with the mathematical machinery of calculus. Here is a excerpt from a translation of Galileo’s Two New Sciences.
“When I think of a heavy body falling from rest, that is, starting with zero speed and gaining speed in proportion to the time from the beginning of the motion; such a motion as would, for instance, in eight beats of the pulse acquire eight degrees of speed; having at the end of the fourth beat acquired four degrees; at the end of the second, two; at the end of the first, one: and since time is divisible without limit, it follows from all these considerations that if the earlier speed of a body is less than its present speed in a constant ratio, then there is no degree of speed however small (or, one may say, no degree of slowness however great) with which we may not find this body travelling after starting from infinite slowness, i. e., from rest. So that if that speed which it had at the end of the fourth beat was such that, if kept uniform, the body would traverse two miles in an hour, and if keeping the speed which it had at the end of the second beat, it would traverse one mile an hour, we must infer that, as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it would not traverse a mile in an hour, or in a day, or in a year or in a thousand years; indeed, it would not traverse a span in an even greater time; a phenomenon which baffles the imagination, while our senses show us that a heavy falling body suddenly acquires great speed.”
OK. So while Galileo didn’t have the mathematical machinery of calculus, he certainly had many of the ideas:
“Time is divisible without limit”
“There is no degree of speed however small with which we may not find after starting from infinite slowness”
“So that if that speed which it had at the fourth beat was such that, if kept uniform, the body would..”
“As the instant of starting is more and more approached, the body moves so slowly that, if kept moving at this rate, it would not…”
A key idea is the hypothetical, “So that if that speed which it had was such that, if kept uniform, the body would…” Another words, Galileo is specifically thinking about speed at an instant by considering how far it would go in a measure of time if the speed it had at that moment was not allowed to change. Of course, this idea is really the same as slope of the tangent line idea. We zoom in on a moment, fix the rate of change, hypothetically extend a line with that rate, and measure how far that line extends vertically in a fixed measure of horizontal change.
A Second Excerpt
Here Galileo describes ideas about the relationship between velocity at instant, displacement, and average velocity:
Let the line AI represent the lapse of time measured from the initial instant A; through A draw the straight line AF making any angle whatever; join the terminal points I and F; divide the time AI in half at C; draw CB parallel to IF. Let us consider CB as the maximum value of the velocity which increases from zero at the beginning, in simple proportionality to the intercepts on the triangle ABC of lines drawn parallel to BC; or what is the same thing, let us suppose the velocity to increase in proportion to the time; then I admit without question, in view of the preceding argument, that the space described by a body falling in the aforesaid manner will be equal to the space traversed by the same body during the same length of time travelling with a uniform speed equal to EC, the half of BC.
Here Galileo has essentially constructed a velocity vs. time graph for an accelerated object (turned on its head), and is arguing that distance traveled during an interval of time is equal to the distance an object with uniform speed would cover if the speed was half the speed the accelerating object acquired at the end of that same interval. Not only did Galileo have in place many of the ideas for thinking about limits and rates of changes, but also the beginnings of integral calculus.
The claim I want to make here is that Galileo made sense of instantaneous rates of change and accumulation via two different hypothetical constant velocities he had to imagine.
Instantaneous velocity at a given moment of time was construed as the distance an object would travel in a measure of time if the velocity it had at that moment were kept constant.
Accumulation was conceived as distance traveled by an object with a hypothetical constant velocity, here described as the velocity half of the velocity obtained at the end of the time interval.
And this is what makes instantaneous velocity so difficult to comprehend… it is a discussion of a hypothetical moving object and its relationship to a real moving object. Instantaneous velocity is thus a huge imaginative leap of faith–one in which we imagine a differently moving object and aim to establish some relationship between the imagined object and our real one. Instantaneous velocity requires a suspension of reality–an acknowledgement that you aren’t going to talk about what actually happened, but to make an explicit analogy between reality and an imagined one.
If I’m right, that instantaneous and average velocity are merely analogies to imagined hypothetical objects moving at constant velocity, then it seems that pinning down the meaning of constant velocity becomes even more important. Galileo, refined his definition of uniform motion over time:
By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal.
We must add to the old definition (which defined steady motion simply as one in which equal distances are traversed in equal times) the word “any,” meaning by this, all equal intervals of time; for it may happen that the moving body will traverse equal distances during some equal intervals of time and yet the distances traversed during some small portion of these time-intervals may not be equal, even though the time-intervals be equal.