The problem of instant velocity

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."

( https://en.wikipedia.org/wiki/George_Berkeley#Philosophy_of_physics )

@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.

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.

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.