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The velocity is the variation rate of the position correct? So does it make sense to talk about velocity without time?

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    $\begingroup$ I think this question would generate much better answers if you explained in a bit more detail what caused this question: What situation did you encounter where someone tried to talk about "velocity without time"? Why did you become unsure about the definition of velocity? (For instance, the "generalized velocities" of Lagrangian mechanics are not time derivatives in all contexts, but it is entirely unclear if that has anything to do with what you're trying to ask about here) $\endgroup$
    – ACuriousMind
    May 25 at 14:40
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    $\begingroup$ Does this answer your question? What's the difference between average velocity and instantaneous velocity? $\endgroup$ May 25 at 15:14

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Average speed is defined as passed-distance-over-passed-time:

$$v_\text{average}=\frac{\Delta s}{\Delta t}.$$

In other words, choose a point on your path. Then choose one more point. Plug in the difference in distance and time between them.

If your speed varies in-between those two chosen points, then the average speed will not show it clearly. Any variation - any information - is "lost", so to say, in-between the chosen points. For a more accurate speed that more accurately follows the actual speed variation during the trip, we can thus simply move the two points closer (make measurements more quickly after one another).

Soon we have moved the points so close together that we basically can't distinguish them - so close that they to all practical purposes overlap so that the two points appear as just one point. The average speed between them can then be thought of as the speed in just that point.

In that specific case we might then rename the average speed to instantaneous speed. We might even invent new notation for this specific situation:

$$v_\text{instantaneous}=\frac{\mathrm ds}{\mathrm dt}.$$

When we say instantaneous, we are idealising this scenario. We are imagining that the "average" indeed is found in just a point. Mathematically we might say that we let the two points go towards each other (in fact we let the moments in time go towards each other which means that their difference goes towards zero) so that the value of average speed goes towards the value of instantaneous speed at its limit:

$$v_\text{instantaneous}=\lim_{\Delta t\to 0}\left(\frac{\Delta s}{\Delta t}\right).$$

When considering what your speedometer shows in your car, you are right that we in reality never can measure instantaneous speed with perfect precision. But as long as the measurements happen over a very short period of time, then it counts as instantaneous to all practical purposes.

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    $\begingroup$ And to make OP feel better, it took humanity some good 2000 years to solve this conundrum... $\endgroup$
    – AnoE
    May 25 at 9:58
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    $\begingroup$ The mechanical speedometers in older cars directly measured speed. I won't call it instantaneous speed because the signal was inherently low-pass filtered, but my point is, those old speedos did not compute speed from time and distance measurements. $\endgroup$ May 25 at 15:44
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There is the universe, which contains things like electrons, light, and rocks. It has repeatable patterns of behavior. For example, if you put a rock near Earth, they attract each other.

Then there is physics, a mathematical description of the behavior. It is a set of laws like $d = vt$.

Actually physics has a number of different descriptions, each suitable for different circumstances. No description is perfect. We don't know the full behavior of the universe, and we don't have a perfect description for the part we do know.

So we have classical mechanics, which gives good answers for big objects at slow speeds, quantum mechanics for small objects, special relativity for high speeds, and several more theories.

Sometimes good enough answers are better than perfect answers. They can be simpler. Given a room full of air, the perfect classical description would trace the trajectory of every atom. A perfect quantum mechanical description would be a wave function for every atom. A simpler description just gives the temperature and pressure of the air.

Sometimes we idealize the universe to get a simpler description. Instead of describing the position of a rock, we describe the position of a point particle. You can talk about the position of a rock by averaging over the spatial region occupied by the rock. For a point particle, you don't have to. You don't have to consider rotation. It allows you to focus on the important parts of the description.

Instantaneous velocity is an idealization like this. As the other answers have pointed out, it is the limit of an average velocity. In the universe, what happened just before determines what happens now. But if you can separate time into independent instances, you can do math with it to get a description of the important patterns.

This is all useful because you want to know how rocks behave. A first step is learning how point particles behave. This leads to laws like $d = vt$ and $F = ma$.

Then you have to figure out how to apply that to a rock. You can say a rock is a collection of atoms, which are point particles that exert forces on each other. The rock is rigid, which means the atoms maintain their positions inside the rock. Many of these forces are equal and opposite. They cancel in the sense that they do not affect the motion of the rock as a whole. This leads to the idea of center of mass. This is a point that represents the position of the rock. It allows you to treat the entire rock like a point particle, affected only by external forces like gravity.

Physicists get so used to the math and idealizations that they forget there is a difference. This is useful. If you understand the behavior of the math, you understand the important behavior of the universe.

So it is good to question idealizations and approximations. You need to understand the difference between behavior of the universe and math. You need to understand the limits of where you can apply a law. But once you know that the law is valid for a situation, you can forget the difference and just think about the math.

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Instant speed is defined as : $$ v = \lim_{\Delta t \to 0} \frac {\Delta s}{\Delta t} = \frac {ds}{dt}$$

So that it's not that instant speed is about not having a time duration, it just infinitesimally small.

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Instantaneous speed is just the speed something has at some instance in time. Time is still involved, but in an infinitesimal fashion.

You are correct in that if you took a picture of a moving object at some time, the picture would be still and you cannot tell anything about the motion of the object from the photo. Only with multiple photos at different times motion can be measured.

But that does not mean that motion does not exist. This is the classic Zeno paradox.

Instantaneous motion is an intrinsic property of object relevant to many physical calculations, but one that cannot directly measure directly, but rather from the displacement over time and thus actually measuring average speed and not instantaneous speed.

The only way to measure the instantaneous speed of object from a photo is if the object was of known length and a ruler was present in the photo while the body is moving with relativistic speeds. From the length contraction measured once can estimate the instantaneous speed without needed multiple photos.

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Velocity is indeed the rate of change of displacement w.r.t time. You are right that it is paradoxical to consider velocity at a point, as at one point there is no change in time or a change in displacement.

For velocity for any two points in time:

$$ v = \frac {\Delta s}{\Delta t} $$ where s is displacement, and $\Delta t = t_2 - t_1 $.

If velocity is constant, the graph for displacement-to-time is a straight line, so slope is equal for any 2 points on the line (and slope = velocity) So, $ \frac {\Delta s}{\Delta t} $ gives us the accurate velocity.

However, if the velocity is not constant, then, as @Steeven said, we lose the information about the changes in velocity between the two points in time $ t_1 $ and $ t_2 $.

As we consider smaller and smaller choices of $ \Delta t $, the information lost between the two points in time reduces more and more (because the time between the two points, i.e. $t_2 - t_1 $ reduces).

When we want to find "instantaneous" velocity, we let $\Delta t$ go to zero.

The derivative of displacement with respect to time, or $\frac{ds}{dt}$ is what we call "instantaneous velocity". The $dt$ represents a tiny, tiny change in time, $ \Delta t $ ($ t_1 $ and $ t_2 $ are very close to each other).

Thus, we can say that $$ v = lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} $$

In conclusion, the time derivative of displacement does not actually give us instantaneous velocity (there is no velocity if there is no time). Rather, the $dt$ represents smaller and smaller changes in time, until the stage where the two points in time are almost (but not exactly) at the same point. This method to find "instantaneous" velocity is only an approximation, using very small changes in time. But there is still a change in time, without that velocity has no meaning.

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Let's say your position is given by a function $x : I \to \mathbb{R}^3$, where $I \subset \mathbb{R}$ is an interval of time. Then the velocity is defined by $$x'(t) = \lim_{h \to 0}\frac{x(t+h) - x(t)}{h}.$$ If $I$ is just a single point, this limit does not really make sense, because there is only one $t$ we are allowed to put into $x$, we can't approach it using a limit.

So, no, velocities do not make sense without time. (This doesn't mean that instantaneous velocities don't make sense. You just need time to exist in a neighbourhood around them for them to work.)

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    $\begingroup$ If I'm not mistaken, the denominator in your equation should be h $\endgroup$
    – AVS
    May 25 at 11:47
  • $\begingroup$ @AVS Correct – fixed. $\endgroup$ May 25 at 14:04
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I think math has more to do with your question than physics. Passage of time does not disappear at infinitesimal periods. There is a difference between infinitesimal and zero.

Infinite-small in practice means that no matter how much smaller you make it - some property will be preserved. Speed is preserved in arbitrary small valued periods of time, it also has a statistical tendency of becoming more precise (closer to the idealistic tangent one) as periods get smaller, because overall set of “time x position” choices is getting reduced on every step. Note: Although, it could depend on how those sets are defined - naive notion of precision may not always hold: error can grow for smaller periods if there were elements of a set of possible times and positions with existence conditioned on inclusion of other elements - sounds strange but actually quite common when you consider that measurement itself could be noisy/side-effectful (note: I’m still referring to classic physics here).

Example of how not just speed as a property but also its value is preserved:

Let’s say you take some period of time (let’s say a second) and start dividing it by two. No matter how many times you repeat it - you still will be able to get some non-zero number and continue.

You will not reach zero no matter how many finite steps you take. You only reach zero at infinity, which I personally would say is physically impossible.

Moreover, if you take a finite interval between two positions (let’s say 1 meter) and divide it by 2 for every step (simultaneously with time) - you can notice that no matter how many steps of making them both smaller you take, the instantaneous speed would still be 1 m/s.

Note: the linear (inertial) example above is only an example of mathematical possibility: AFAIK physical methods of measuring speed do not allow such perfect circumstances (linear sync between position and time), so things can even get as complex as QM where speed has different semantics, but even classic theories rarely (or never?) talk about fully precise instant speed, as linear relationship between time and position is some artificial ideal we invented. Actual measurements mostly reject this assumption. IMO, except in computer programming where interpreting a program run as a physical experiment makes border between natural and artificial thin to the point where “physics = math” - but even digital computers are subject to noise so maybe not.

In any case, perfect linearity/inertia being an artificial construct is the practical reason why subdivisions of time periods are thought to improve precision (while natural noise actually interferes with this artificial tendency in reality) - so we could have idealized perfect inertial instant speed but only in a physically unapproachable limit, where things would look linear but really don’t :).

When it comes to other possibilities of “time erasure”, ”mathematical” time can disappear, but speed would disappear consequently too

Time does not have to be measured in seconds and position does not have to be measured in meters. Any relationship between two variables could be called speed in terms of “rate of change” like speed of pixel brightness on a still digital image along the eye tracking path.

Any order can be seen as “time” so time can disappear as order disappears (informally there is no time without “more than” comparison). But speed looses meaning without at least some notion of time, purely by definition (as “rate of change of ordered events” so on) . So the answer is still no, canonically there is no speed without time.

In other words, time can be weird and appear non-linear comparing to intuitive notion of time (think of “Pulp Fiction”) but speed cannot exist without it.

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