# How can there be really any instantaneous velocity?

I have read about Zeno's arrow paradox that tells us there is no motion of the arrow at a particular instant of its flight. It can be inferred that there can be no velocity at any instant. Moreover we cannot calculate velocity at any instant in the real world (of course it can be done by using calculus) but how can this be possible? What is the intuition behind this concept?

• @ user36790: You are absolutely right to trust your own reason. There is no movement without time flow. However, many people do not think it is a problem and for instance postulate freezing time to get rid of acceleration, but still maintain velocity - check the EDIT to my answer here to see how time is played with. Commented Jun 5, 2014 at 8:39
• The intuition is that mathematics is a tool and experiment is above all. Commented Jun 5, 2014 at 11:34
• Instantaneous only exists if you accept that you can slice time into an infinitely small slice. At that point, there are infinite slices, so you have two infinities you're dealing with, and while infinity is useful as a concept, once the theory meets the practicality you can't actually slice time so small that there is an instant where there's no movement, you can always slice it more finely than that, and each slice still has movement. So the concept is useful as a concept and may clarify certain other concepts, but the actual condition doesn't exist as a practical matter. Commented Jun 5, 2014 at 17:03
• What about Zeno?Acc. to him,the arrow can move if it comes to the point at that instant or goes to the other point but since it is already at the point and cannot go to the other point as it is an instant,and since time is made up of many,many,instants,there can be no motion at all.So what is instantaneous velocity?
– user36790
Commented Jun 6, 2014 at 2:09
• @ adam davis: sir,can i tell instantaneous rate is abstract? If not, how can i explain this or what is the intuition? How can time be sliced ?
– user36790
Commented Jul 14, 2014 at 11:39

Zeno used his paradoxes to proof movement was impossible. But of course he knew movement existed! If you were going to punch him, he will not trust your fist would have to get infinite times half of the way before reaching him; he would try to avoid it. His philosophical motivation was to "stirr" the reason, show that by logical arguments we can fall into wrong conclusions, ergo incurring in fallacies. This was a big thing, because in the age of the reason, someone shown that logic may take us nowhere.

I imagine it was the equivalent in modern times to Heisenberg Principle in Quantum Mechanics, where in the age where science was believed to be infinitely precise, it was shown that there were unmeasurable things in Physics (the flagship of science!); or Gödel's theorem, where he shown that there are things in Mathematics that cannot be proven right or wrong.

From a conceptual point of view, instantaneous velocity is a limit: if you compute the average velocity ($\Delta x / \Delta t$) for every smaller values of $\Delta t$, you will see that it nicely converges to a value: this is the instantaneous velocity.

From an experimental point of view, this is unreachable. You cannot measure arbitrarily small periods of time just because your equipment has a limit. Also, your measurement of the position has a precision too. Error analysis will show you that, assuming the velocity is constant along an interval, and for noisy measurements of $x$ and $t$, the bigger the step, the more accurate the measurement will be. For a real movement, where the velocity is not constant, expanding the interval will increase the error introduced by variability. If you have an idea of the evolution of the velocity (for example, the second derivative computed from the last few points), one could estimate the optimal interval for maximum precision.

From a theoretical physics point of view, things get weirder. You cannot trully define the position, or the momentum, with infinite precision. And if you go to very small scales, and very short times, you will hit the weirdness of Plank foam: where time and space are predicted to not behave in any intuitive way.

• If there are absolute barriers (in the form of a constant or something else) to assuming ever smaller values of time, than we should accept them in our calculations of velocity and not pretend we are going down to zero. Because any particular "infinitesimal" is considered small only by comparison - show me any small number, and I can always produce another number billions of times smaller than that. (Obviously 1s is as good unit of time as 1 billionth of second.) Commented Jun 5, 2014 at 8:59
• That means i am dealing with such a concept which has actually no intuition & can't be felt in the real world???
– user36790
Commented Jun 5, 2014 at 9:24
• Velocity is "as real as" height. They are concepts, no solid objects, but they "exist". If you want more intuition, throw a rock fast to your face (just kidding, but there is my point). Commented Jun 5, 2014 at 11:41
• 2 user36790: this concept is exactly how you feel it in the real world. In fact, classical description of particles moving in space with a certain coordinates, velocity and acceleration is based on how we experience world around us. Velocity and acceleration are the derivatives of coordinates that agree well with our everyday experience and make much sense in the model they come from, which is classical mechanics. And the model is not right or wrong per se, it is applicable or not so much. For example, any classical model is not well fit to describe very small chunks of matter. Commented Jun 5, 2014 at 11:43
• @brightmagus good luck explaining how a car works from QM principles. All theory is an approximation, valid in a certain range. Classical mechanics work very well in the macroscopic world, and is intuitive (and this is what the question is about). QM is wrong, it is not fully compatible with GR; but we don't trash either because when they are applicable, they work. Commented Jun 5, 2014 at 12:01

At a "frozen" instant of time, the arrow may not be moving - but this is a tautology, since movement is something that requires time. However, even in that frozen instant the arrow does have a velocity (instantaneous velocity, if you will). Imagine that time is a series of huge number of discrete frames (or instead imagine that it is continuous, and that we are taking finer and finer discrete approximations). The position of the arrow jumps to the right from frame to frame. How does the arrow "know" how far to travel from one frame to the next? If the only piece of information "stored" in one frame were its position, then the arrow wouldn't be able to determine this! The necessary information, which is the instantaneous velocity of the arrow, must be as much a part of this frozen frame as all the information related to the arrow's position.

More formally, one says that the configuration space of a physical system, which is the set of all information needed to predict its future (and thus all the information associated with a point in time) includes not only the list of positions of all objects, but also their velocities.

• +1, I think this answer includes an important philosophical idea: Velocity is not a consequence of movement and flow of time, but rather movement is a consequence of velocity and flow of time. (Although being philosophical, it might be off topic in SE, but so would the question be.)
– JiK
Commented Aug 20, 2014 at 21:00
• But how is this velocity "stored" as a property of a physical system in the real world? Commented Aug 12, 2019 at 11:58
• @rinspy That's a philosophical question. To turn it around, how is the position "stored"? Any answer you have will boil down to you being able to see an object's position, but it's fallacy to assign more reality to something just because the human brain may model it as a first class "real thing." What we can confidently say is that classical mechanics, a very good model of macroscopic reality, insists that both the position and velocity are "stored" in the same way in reality at any given instant of time (whatever that means, ontologically). Commented Aug 12, 2019 at 15:44
• @rinspy Although I didn't state it, the explanation in my answer is equivalent to Newton's second law, that force is equal to mass times acceleration. This implies, acceleration being the second derivative of position, that the information required to predict future configurations from current ones requires all positions and velocities as initial values. If we lived in a universe where Newton's second law was "F=mv", then configuration space would only consist of positions, while if "F=mj", j meaning jerk (the derivative of acceleration), configuration space would also include accelerations. Commented Aug 14, 2019 at 12:49
• @rinspy And quantum mechanics, depending on one's interpretation of it, actually might describe how an object's position and velocity are simultaneously "stored," at whatever shared level of (dis)reality: the complex-valued wavefunction of a particle gives its expected position via a weighted average of its positions, and velocity through a weighted average of its gradient. Commented Aug 29, 2019 at 0:48

bright magus puts his finger on the problem when he says in a comment: There is no movement without time flow.

Physicists describe the universe as a four dimensional manifold, in which points are identified by their position $(t, x, y, z)$. The time coordinate $t$ is just a coordinate like $x$, $y$ and $z$, and there is no sense in which time is flowing. We are used to differentiating with respect to time e.g. velocity = $dx/dt$, but there is nothing special about time. For example if you have every seen a road sign warning you about a steep hill, the gradient on the sign is $dz/dx$. As far as I know Zeno never complained that you can't have hills because you can't calculate $dz/dx$, and he should not have made the corresponding complaint about $dx/dt$.

At this point you're going to object that everyone knows time flows. After all we can sit at constant spatial coordinates just by staying still, but there is no way to stay at a constant time coordinate. True, but the perception of time flowing is likely an artefact of human consciousness rather than a fundamental principle of physics.

If you search this site for flow of time you'll find several questions exploring exactly this issue.

• On a purely historical note, Plato attributes the motivation for Zeno's arguments as a partial defense of Zeno's teacher Parmenides, who in turn argued any kind of change or difference is an illusion. So while Zeno might not have complained that you can't have hills, it's rather probable that he would have agreed with non-reality of hills and with the claim that "there is nothing special about time". Commented Jun 5, 2014 at 10:26
• @JohnRennie: On the contrary, the difference between dz/dx and dx/dt is fundamental: relation between points in the first one is fixed, while in the other it changes. The first one does not describe any interaction between objects. If you "freeze" x or y or z, interaction (change) is still possible (assuming matter can be flat), but if you freeze t, the world comes to a standstill. There is no physics anymore. (There is only mathematics left, but nobody to play with it.) And that's why showing time as orthogonal to other dimensions and equivalent to them is necessarily not real. Commented Jun 5, 2014 at 10:36
• Actually time makes all the difference between physics and maths. If you treat time on equal terms with other dimensions you are in an abstract world of maths; it's not the real world anymore. Physics itself admits time is different than spatial dimensions, which can be seen in equations like this one: $ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$. If $t$ was equivalent to $x$, $y$, and $z$, it wouldn't have to be multiplied by $c$. Seems that treating time like a spatial dimension is the reason why people get into trouble like I showed in the link in my comment above to the question by user36790. Commented Jun 5, 2014 at 11:56
• @brightmagus: As you say, with no time (but with space), there are no interactions and physics is left meaningless. However, one can make the dual observation: with no space (but with time), there are also no interactions and physics has similarly dubious meaning. Sure, time is in some fundamental ways different from space, but I don't see how your objection connects to the conceptual problem of defining a rate of change of something with respect to something else. Commented Jun 6, 2014 at 0:00
• @StanLiou: The difference between dx/dz and dx/dt is such that without time world does not need to be uniform (although the question is how it was created at all) but is necessarily unchanging. Commented Jun 6, 2014 at 5:14

I'd like to add something to these answers. In the classical mechanics, we cannot distinguish a moving body from the body at rest, if we look at it at any particular instant. So, we have to add some hidden information to the picture, that is instantaneous velocity. But that's what physics only knew in the 19th century. In 20th century physics, there have several more deep and precise pictures appeared.

First, special relativity. It tells that the moving body contracts by the Lorentz factor. So if we know the size of the body at rest, we can calculate its speed, and in 3D - the line of motion. Only the direction of motion remains unknown, but motion and rest are clearly distinct.

Second, the theory of field, for example electrodynamics (in fact, it arose in the end of 19th century). It tells that a charged particle is accompanied with electric and magnetic fields, and they show the velocity of the particle, $\vec{B}=\tfrac{1}{c}\vec{v}\times\vec{E}$ in Gaussian units. Electrically neutral bodies consist of many charged particles (electrons and nuclei), and their microscopical fields contain the same information.

And at last, quantum mechanics. It tells that any particle is represented at any instant by its wave function. And the wave function is not only the probability distribution - it does have phase, and its phase shows the motion. Namely, there is the velocity operator which is $\hat{\vec{v}}=-\tfrac{i\hbar}{m}\nabla$, and applying this operator to the wave function, one can get expectation value of velocity and some more detailed information. For example, if the wave function at some instant is $\Psi=\Psi_0\exp(ikx)$, then the probability density is flat for any $k$, but $\hat{\vec{v}}\Psi=\tfrac{k\hbar}{m}\vec{\imath}\Psi$ shows that the particle has velocity $k\hbar/m$ along the $x$ axis.

I think Zeno would be happy to know these theories.