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What time interval did Newton consider in F=ma?

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What time interval did Newton consider in F=ma?. Are You mam own that kind of problem?, If yes then please check the best feedback below this line:

He was not specific about the time interval … and for good reason, as I will try to explain.

In Philosophiae Naturalis Principia Mathematica, Newton doesn’t actually state the second law in terms of acceleration. His statement of the law is given in terms of words, not equations, and it is quite subtle.

Time is not mentioned explicitly in Newton’s statement of the second law: instead time is implicit in the statement. The modern abbreviated formulation in terms of force, mass and acceleration [math]F=ma[/math] taught in physics courses today came about somewhat later on.

It’s really worth reading some of Newton, at least once in your life, and there are now various good English translations of Principia. He is struggling to define the concepts there and to give an axiomatic presentation, along the lines of Euclid’s Geometry, of a systematic method of calculating motion that he has already completely developed for himself, and it’s illuminating what he says in support of his system.

Newton undoubtedly knew and would have understood and recognized that formulation, [math]F=ma,[/math] once the terms were defined, and I would lay long odds, if it were possible to settle the bet, that he also wrote it down for himself, since he had fully developed the methods of differential and integral calculus long before the time when he wrote Principia.

But that is not the way that Newton actually states the second law.

Here is one English translation of Newton’s statement of the second law. (The original is in Latin.)

The change of motion [momentum] is proportional to the motive force impressed, and is made in the direction of the straight line in which that force is impressed. (Newton 1934, 13)  [1]

(Translation from: Newton, Isaac. Sir Isaac Newton’s Mathematical Principles of Natural Philosophy. Translated by Andrew Motte, revised translation by Florian Cajori. Berkeley: Univ. of California Press, 1934.)

Note that Newton makes use of the notion of “motive force” here, and that earlier discussion in Principia makes it clear that by the “motion” or “quantity of motion” of a body, Newton means what we now call the momentum of a body [math]p,,(equiv mv)[/math].

One could write, using modern notation, “change of motion” as [math]Delta p.[/math]

But time is implicit in the statement of the law, in the idea of “change of motion”, which Newton referred to in terms of absolute space and time, and a straight line for Newton means a straight line in absolute space, which for Newton is of course a Euclidean space, which Newton thought of in terms of a Cartesian coordinate system. By “motive force” and “impressed” he actually means what we now call “impulse”, or [math]FDelta t[/math], where [math]F[/math] is the “motive force” and [math]Delta t[/math] the time interval over which it acts, and it’s clear that the time interval is simply not specified.

But the time interval implied would certainly appear to be the time over which the “motive force” is “impressed”, and the “change of motion” occurs, since in the absence of “motive force”, Newton makes it clear that his first law of motion applies.

Every body remains in a state resting or moving uniformly in a straight line except insofar as forces on it compel it to change its state.

(Translation from: Wolfson, Richard, and Pasachoff, Jay M. Physics. Boston: Little, Brown & Co., 1987)

It appears that Newton has in mind that a motive force, or an impressed force, acts on a body for some time and in some direction and that this is what causes a change in the motion. Then the motive force ends, and the change in motion is done. This is what distinguishes Newton’s notion of force from Aristotle’s notion of force.

So one can translate Newton’s statement of the second law into an equation, thus:

[math]Delta p = F Delta t,[/math]

interpreting [math]Delta p,,[/math] and [math]F[/math] as vector quantities, acting, as in the statement of the law, along a straight line.

But the time interval [math]Delta t [/math] over which the motive force is impressed is not specified, and, as the law is stated, it could be arbitrarily short or long.

It’s clear that if [math]F=F(t)[/math], and [math]F(t), [/math]varies significantly over the interval [math](t,t+Delta t) [/math] as one would say in modern terms, there would need to be an averaging process over this time interval, to define what the value of [math]F[/math] was. It’s clear that there could be more than one force [math]F(t)[/math] that could produce the same change in motion (although not the same path in absolute space), if [math]Delta t[/math] is arbitrary.

It’s clear also that if the law is true for arbitrary [math]Delta t[/math], then it’s true also in the limit [math]Delta t rightarrow 0,,[/math] and from this the differential form of the law in terms of acceleration, mass, and force will follow, given only that the mass of the body is constant over time. Newton understood this limiting process in terms of what he called “fluxions”, not in the later rigorous mathematical sense that the word “limit” was given by Cauchy.

Note also that it’s often said, incorrectly, that Newton’s first law is a consequence of his second law, and/or that the first law is circular or tautological.

This is false.

For certain singular kinds of forces, the differential form of the second law can be shown to have solutions which do not satisfy the first law. Additional assumptions on the nature of the forces are required to guarantee the uniqueness of the solutions of the differential equation form of the second law.  [2]

So both the first and second laws are required, and they are logically independent of one another.

Finally, I should say explicitly – the second law today is understood as a differential equation:


So the time period [math]Delta t[/math] is considered to be extremely short – so short that any variation in [math]F[/math] with time produces a correction that is higher order in [math]Delta t[/math], and may be neglected. Given the right conditions on [math]F(t)[/math] it will always be possible to choose [math]Delta t[/math] short enough that this is satisfied.

Newton would have written that same equation:

[math]dot p = F[/math]

And this equation not only doesn’t refer to any specific time interval but also applies at all times.

In this one case and perhaps a couple of others, Newton’s dot notation for the time derivative is actually superior to Leibniz’ notation, I think, in that it gives a more concise rendition of his statement in words of the second law of motion, with [math]dotp[/math] representing the “change of motion” and [math]F[/math] representing the “motive force”.

Now, as to your special case given in the details, that of constant acceleration, constant force: it’s very clear that any time interval will work equally well, since constant acceleration means that the velocity will increase in direct proportion to the time interval, the constant of proportionality being the acceleration of course.


[1] http://Newton, Isaac. Sir Isaac …

[2] Nonuniqueness in the solutions of Newton’s equation of motion

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