# The problem of instant velocity

The concept of velocity is by definition the movement divided by the interval of time between initial position and final position.

If $f(t)$ is the position of a particle at time $t$; the velocity in the interval $[t_0;t_1]$ is $\dfrac{f(t_1)-f(t_0)}{t_1-t_0}$

The problem is that in a single instant there is no movement and the time is not changed; so no velocity.

I can consider $\lim_{t_1 \to t_0} \dfrac{f(t_1)-f(t_0)}{t_1-t_0}$, but mathematically it is only the limit of average velocity function and doesn't represent velocity at instant $t_0$

• Why the downvote? I think it's an excellent question for anyone new to calculus. – Ethan Bolker May 11 '16 at 23:17
• I downvoted the Zeno's Arrow paradox approach, because otherwise we shall discuss about what is time – G Cab May 11 '16 at 23:49
• This is philosophy, not calculus. $f$ gives the position at a given time $t$. If $f$ is differentiable then the velocity is just $f'$. – copper.hat May 12 '16 at 5:09
• It is an average over an interval that tends to $0$, i.e. virtually a single point. – Yves Daoust May 12 '16 at 12:33
• Feynman explains that in an easily understandable way (you can start at section 8-2) – Jasper May 14 '16 at 11:39

Your excellent question is as old as the invention of calculus. As you correctly point out, velocity makes no sense if all you know is what is happening at just that instant of time. Physicists and mathematicians take the limit of the average velocity as the very definition of instantaneous velocity.

That turns out to be a very good definition, since it leads to physics that accurately describes the behavior of the world and mathematics that's consistent and interesting and useful. So people no longer worry about the question in the form in which you've asked it.

Edits to respond to comments. Edited again (as @Polygnome suggests) to incorporate the sense of the comments as well

@pjs36 Yes indeed thanks. The question really does go back to Zeno's paradox of the arrow. On that wikipedia page you can read

Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not.[13] It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.

@Max says

In the Newtonian model of the universe, momentum/velocity is something that objects have at every instant of time

I didn't know that. Perhaps it's why he could develop calculus reasoning with infinitesimals without addressing the philosophical problem and without formal notion of limits. His assumptions were not universally accepted at the time. The philosopher George Berkeley argued that

... forces and gravity, as defined by Newton, constituted "occult qualities" that "expressed nothing distinctly". He held that those who posited "something unknown in a body of which they have no idea and which they call the principle of motion, are in fact simply stating that the principle of motion is unknown."

@leftaroundabout I agree that momentum is a better fundamental notion than velocity (certainly for quantum mechanics, possibly for Newtonian too). I don't think it's better to start calculus there, though.

@Hurkyl notes correctly that there are new mathematical structures - germs - that capture the idea of what happens near but not at a point. But I think the idea of the germ of a function is more technical and abstract than called for by the question.

• One might even say it's older than the calculus; it feels a bit like Zeno's arrow to me. – pjs36 May 11 '16 at 23:45
• "velocity makes no sense if all you know is what is happening at just that instant of time." Depends on what you mean by "happening". In the Newtonian model of the universe, momentum/velocity is something that objects have at every instant of time. – Max May 12 '16 at 9:45
• Also depends on what you mean by "instant" -- geometry is moving in a direction that allows individual points to nonetheless have some infinitesimal extent. e.g. at any instant, a particle could be said to have an entire germ of motion rather than merely a position, or a position and a momentum. – user14972 May 12 '16 at 13:43
• @Max +1 IMO many things in physics become more intuitive if you think that velocity doesn't result from movement in time, but rather movement is a consequence of velocity and time. – JiK May 12 '16 at 14:47
• @Max: it could be argued that momentum is defined at every instant of time, but indeed velocity is not. Quantum-mechanically, momentum is a spatial derivative, while velocity is always a time-derivative; only in the classical limit are these quantities linked in the way we're familiar with. – leftaroundabout May 12 '16 at 16:21

Do you have a prior notion of "instantaneous velocity"?

## No, I don't have a prior notion of instantaneous velocity

The quantity defined by the limit is very useful. Thus, it needs a name. "Instantaneous velocity" is an accurate enough phrase to make it a good choice of name.

## Yes, I do have a prior notion of instantaneous velocity

Then proceeds in three steps:

• Define it. (or realize it's a tricky concept to define)
• Realize that instantaneous velocity is 'close' to average velocity over short durations
• Formalize the meaning of the previous statement, concluding that the instantaneous velocity is equal to the stated limit.
• An excellent Socratic approach. – AakashM May 12 '16 at 14:00
• In the second case it seems to be quite likely that the prior (intuitive?) notion will not agree with the calculus concept in corner cases such as $x^2\sin(1/x)$. – Henning Makholm May 13 '16 at 13:03
• @HenningMakholm: Pondering on your comment, it strikes me that I have no clue whether the "intuitive" notion is supposed to be "zero", "meaningless", "everything between -1 and 1", or "too weird to guess without further analysis". I can see reasons to intuit all four, and imagine you'd find them all in the wild. – user14972 Sep 4 '16 at 22:31
• Instanteous v must be the v in p=mv and ke=mvv/2 or very bad things happen. That is, kinematics must work if acceleration is alternately zero and infinite. – Joshua Oct 31 '16 at 21:07

My thoughts:

This is something that is very common in mathematics. We have a concept that is natural and we are used to using, but when you actually try to define it carefully in all situations, the simple definition doesn't work in general.

Another example is area. The area of a rectangle is easily defined and understood (length times width). But what about the area of a circle or an ellipse, or between a parabola and a chord? How exactly do you define those areas? It isn't a case of just saying "the area of a circle is $\pi r^2$." After all, if we are just going to call a formula the definition, why use $\pi$? Why not just say "the area of a circle is $3r^2$"? The obvious reason is: $3$ doesn't work. $\pi$ does.

And that is the clue: we don't want just any definition of area. We want a definition that satifies certain useful properties, most particularly the property that if you divide a shape into parts, the sum of the areas of the parts should be the area of the whole, and the property that if one shape is contained inside another, its area is less than or equal to the area of the other. We combine this with a trick that Eudoxus taught us long ago: If there is only one number that works, that is the number you want! A circle of radius $r$ cannot have an area greater than $\pi r^2$ because for any larger value, we can cover the circle in a bunch of rectangles whose total area is smaller than that value. So the area of the circle must be smaller yet. And for any value less than $\pi r^2$, we can find a bunch of non-overlapping rectangles inside the circle whose total area is greater than that value, so the area of the circle has to be greater as well. $\pi r^2$ is the only value that works. So we define the area of the circle to be $\pi r^2$.

Similar remarks apply to instantaneous velocity. The simple definition of velocity breaks down at a single point. But if we assume that the concept makes sense, and decide that we want it to have the property that when the time interval gets shorter, the average velocity should approach the instantaneous velocity, then for most distance functions of interest, we discover that there is indeed only one value that is approached by average velocities over shrinking time intervals. Any other value will be approached for a while, but as the interval shrinks farther, the average velocity starts pulling away from those values instead. So we give a nod to Eudoxus again and define the instantaneous velocity to be the value that is always approached. (If our velocities don't approach a single value, then we don't define an instantaneous velocity for such distance functions at all.)

The definition we use for instantaneous velocity is the way it is because it is the only value that makes sense for the concept.

Your first equation is the average velocity, that's what we can really measure with physical instruments, the second one is the instant velocity which is an ideal concept (as everyrhing defined as a limit) and can not be really mesured in our natural world, so it is just a mathematical object (a limit, a derivative) in the same sense that spheres or any other geometrical objects does not exist in our physical world, we only can build "imperfect" (in a platonic sense) spheres.

As others have pointed out this is really a philosophical question.

As a physicist, I don't have any issue with there being no time difference in an "instant", because I accept that the ratio of two quantities equal to zero can be finite.

However to make the concept mathematically sound, we can take:

• Standard approach: Define the instantaneous velocity as the limit of average velocity as the time interval shrinks to zero.

• Smooth infinitesimal analysis approach: The continuum is not made of points, but rather infinitesimally small segments. So "instants" of time don't exist, there are only infinitesimally short time intervals, and the problem of the particle not moving goes away.

• May every downvoter give their reason? – u54112 May 13 '16 at 14:32
• I do not see the difference between instants and infinitesimals. – Frank Hubeny May 14 '16 at 2:39
• @FrankHubeny That's because smooth infinitesimal analysis doesn't operate under classical logic. In particular it rejects the law of excluded middle, so the set of nilsquare infinitesimals $\Delta = \{ \epsilon : \epsilon^2 = 0 \}$ does not necessarily coincide with $\{ 0\}$. – u54112 May 14 '16 at 4:30
• If you take the distance arbitrarily smaller than Planks constant you could have problems with quantum physics. – Frank Hubeny May 14 '16 at 8:57
• @FrankHubeny OP did not ask about quantum physics. We are dealing with the classical notion of instantaneous velocity. – u54112 May 14 '16 at 10:19

Instant velocity is best thought of a tangent to the continuous curve representing position over time or equivalently as a vector with a magnitude and direction at a specific instant in time. It is not really an average. The idea of limits in calculus is about what happens to the function as a specific value tends towards another value. In this case, what happens to dS/dt as dt=>0, where S is displacement and t is time.

The limit encodes information about the behavior of the average velocities on time intervals $(t_0-\epsilon, t_0+\epsilon)$, when $\epsilon>0$ gets small. It is therefore an approximation of the average velocity of the particle on an interval around $t_0$ so small, the function on this small interval could for all practical purposes be assumed to be linear. Let's call this interval $(t_0-\epsilon_0, t_0+\epsilon_0)$. Of course, for different functions $f$, $\epsilon_0$ will be different. The beauty of the definition of limit is that it does not matter how small $\epsilon_0>0$ is for a given function, what matters is that such an interval exists. So instantaneous velocity is the average velocity a particle has on some interval that is so small the particle can be assumed to proceed linearly on this interval, that is, if I choose $n$ equally spaced moments in this interval to measure the position of the particle, I will see that every moment the position will have increased by the same constant.

Note: my answer is quite informal, but I hope you understand what I mean.

If the path of a particle is "smooth" (differentiable, if you like this term) you can define the number $\lim_{t_1 \to t_0} \dfrac{f(t_1)-f(t_0)}{t_1-t_0}$ and that number represents what we call instant velocity or velocity at the moment $t_0$.

Your question about how can there be any movement and change of time at single instant is more of a philosophical nature and I really do not know does there exist velocity of some physical object at some instant of time or even if "instants of time" exist at all. Be aware that inside of classical physics this method of description of point particles was useful and that somehow serves its purpose.

You may get two limits $$\lim_{t_1 \to t_0^+} \dfrac{f(t_1)-f(t_0)}{t_1-t_0}$$ and $$\lim_{t_1 \to t_0^-} \dfrac{f(t_1)-f(t_0)}{t_1-t_0}$$ i.e. at one instance. Or you can have two velocities at one instance.

• Are $t_1$ different in the two formulae? – Karlo May 12 '16 at 8:09
• @Karlo It doesn't matter, in both limits $t_1$ is the dummy. – u54112 May 12 '16 at 8:13
• And is it a problem that $t1>t0$ in one case? Physically speaking, this would be acausal. – Karlo May 12 '16 at 8:43

For a smooth function, the Taylor development applies.

In particular, with just the constant term

$$v(t)=v(t_0)+R(t_0)$$ where $R$ is the remainder term, such that $\lim_{t\to t_0}R(t)=0.$

Now compute the average

$$\bar v(t_0,t)=\dfrac 1{t-t_0}\int_{t_0}^t v(t)\,dt=\dfrac 1{t-t_0}\int_{t_0}^t(v(t_0)+R(t_0))\,dt=v(t_0)+\bar R(t_0,t).$$

When you take the limit, the last term vanishes, so that the average and instantaneous speeds do coincide, contradicting the objection "it is only the limit of average velocity function".

The concept has a long history...

The definition of (average) speed dates back to Aristotle.

We can fiund it in Galileo's Discourses and Mathematical Demonstrations Relating to Two New Sciences (Italian: Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze), published in 1638.

See: Engl.transaltion by Henry Crew and Alfonso de Salvio, (1914):

[THIRD DAY - page 190] UNIFORM MOTION In dealing with steady or uniform motion, we need a single definition which I give as follows:

DEFINITION 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.

But things has progressed from Aristotle's times:

[THIRD DAY - page 198] NATURALLY ACCELERATED MOTION [...] a motion as uniformly and continuously accelerated when, during any equal intervals of time whatever, equal increments of speed are given to it. Thus if any equal intervals of time whatever have elapsed, counting from the time at which the moving body left its position of rest and began to descend, the amount of speed acquired during the first two time-intervals will be double that acquired during the first time-interval alone; so the amount added during three of these time-intervals will be treble; and that in four, quadruple that of the first time interval.

To put the matter more clearly, if a body were to continue its motion with the same speed which it had acquired during the first time-interval and were to retain this same uniform speed, then its motion would be twice as slow as that which it would have if its velocity had been acquired during two time intervals.

And thus, it seems, we shall not be far wrong if we put the increment of speed as proportional to the increment of time [emphasis added]; hence the definition of motion which we are about to discuss may be stated as follows: A motion is said to be uniformly accelerated, when starting from rest, it acquires, during equal time-intervals, equal increments of speed.

Here we have a little but significant "conceptual shift": increment of speed is proportional to the increment of time.

But "obviously" time is a continuous magnitude.

And hence it is, that in what follows, I consider Quantities as if they were generated by continual Increase, after the manner of a Space, which a Body or' Thing in Motion describes.

In conclusion, in every "point in time" we can consider the corresponding values of those magnitudes generated by continual increase (i.e. which are function of time):

space, speed, acceleration.

If there are two velocities calculated at T1=T0 depending on time direction/reference approaching T1=To, then there is some implied impulse. This would be described by a dirac delta function with magnitude equal to instantaneous acceleration change needed to shift between the observed velocities

You can look at it that way, the velocity something has at the moment is the velocity it will have when suddenly no more forces act upon it. Because something always has a velocity, even if it is 0, that notion is defined for every moment, because at a future time it will move, even if that movement is 0. So the current velocity is the constant velocity of it's future movement if no force were to act upon it.

In my idea it's difficult to see velocity as a scalar absolute and have much more directive when thought of as a relative inconsistency.

The idea of speed is simplified greatly, but with correlation to coordination I prefer to see velocity and speed as the measurement of physical changes in coordination.

This may help understand the idea as in an instant no change in coordination is possible so in a moment no velocity and no speed.

Terms are absolute, but how you see them affects the problem

"The concept of velocity is by definition the movement divided by the interval of time between initial position and final position."

Says who?

As with any other conceptual notion, the definition is whatever we deem it to a) answer a desired question while b) not contradicting itself or other connected definitions. For example, nobody says non-integer rational exponents are to be considered comparable to roots; that's simply a convenient discovery that syncs up nicely with the already-established (by general principles of the definition of multiplication) properties of integer exponents. It's the transitive property dropped right in our laps.

In this case, one might simply say that "velocity" has no inherent meaning other than that which we assign to it, and we've collectively decided to treat it as if it were "naturally" the definition we mean when we actually say "average" velocity (probably because we lack the biological ability to extract ourselves from the fourth dimension and thus escape the compulsion to measure that elapsed time as having an unavoidably non-zero amount). For all we know, we might well already have the instruments to measure instantaneous velocity; we just lack the ability to pull ourselves outside of time and actually do so.

EDIT: OK, since my answer has rubbed a few the wrong way, let us consider it this way: it is not true to say that in an instantaneous, non-elapsed-time sense, there is "no velocity."

"No" implies "zero."

The velocity is not zero in this case, it is the indeterminate form 0/0. (Despite several of my students' beliefs to the contrary, 0 ÷ 0 $\not =$ 0 automatically.) It is, at best, unclear; it may be zero, or infinitely large, or some finite non-zero amount. What limits (and, through them, L'Hospital's Rule) explores is how we can explore such singularity-esque quantities only by looking at them indirectly. My previous comment was phrased merely to suggest that we are as likely to understand instantaneous velocity as a limited case of average velocity as we are to understand average velocity as an extension of instantaneous velocity.

• edited to address the OP's original concern: that the velocity is zero because of a lack of "progress" – thebishopofcalc May 13 '16 at 0:31
• I did not vote on your answer, but it seems to me that if we are able to put a limit on an indeterminate form and find a specific value then that indeterminate form has been determined. It is no longer indeterminate. – Frank Hubeny May 13 '16 at 11:09