I recently heard that motion, the observation that things move, or rather change, was considered a real philosophical problem. What is the status of that question? Can someone e point me to an essay that describes why exactly that was (perhaps still is) a problem?
I recommend to start with the arrow paradox by the Greek philosopher Zeno, see
Afterwards you could study how calculus formalizes the limit process, and physics defines velocity as the derivative of the position with respect to time.
Adding some calculus: To obtain the velocity v(t) at a time t one computes the quotients
covered distance/time needed
and takes the limit for the time interval going to zero. The limit of the quotients is the velocity at t. Note: The limit of quotients is not necessary the quotient of the limits. That explains the wrong conclusion in the arrow paradoxon.
In addition, the distance covered during a finite time period T is the definite integral with respect to t over the velocity v(t) in the bounds t=0 and t=T.
To enlarge upon Jo Wehler's answer, Zeno's Paradox was only a paradox for philosophers, not for Zeno's neighbors who sailed boats, walked down roads, built buildings, plowed the earth or carried rocks from one place to another. Those people did not simply lay down their loads, remove their shoes, and chill while waiting for the philosophy community to settle the issue.
Beside the usual Zeno Paradoxes and their treatments, there is a bit more subtler problem with motion, it not being an intrinsic property of objects. Standard, Russellian, account of motion is not able to capture the motion state in an instant of time. The state of motion of an object is derived from it's positions at two different time instants. For a property, or state, to be intrinsic, it needs to be invariable to any other properties external to object, including the object at different time points (I say this knowing the premise and the expected conclusion, it would be great if someone more educated on this topic than me could confirm or correct me there).
Another account of motion, Hegelian, seems to fare better with this, but might not solve the issue completely. Here is the first article I found discussing this https://www.researchgate.net/publication/334385562_Change_and_contradiction_a_criticism_of_the_Hegelian_account_of_motion
I believe that Nemanja's answer wraps up the current understanding nicely. I will instead try to explain the issue in simple terms and give a bit of a physicist's perspective.
The problem is not that things change - there is a little dispute that they do. The problem is with the time itself: our thoughts and perceptions seem instantaneous. We intrinsically have a notion of the "moment in time", where it is possible to capture the "snapshot" of the world around us. But there is no obvious mechanism for transferring from one such snapshot to another - in other words, the world around us changes, but we only can slice the time finer and finer, much like Zeno did, with still no mechanism for the transition. Many more centuries later, mathematics came up with the formalism to resolve this subdivision into infinitely small slices, but this still does not provide a transition from one to another.
What we could do, then, is to deny the notion of a "moment in time" - modern physics leans towards this somewhat, and it matches naïve perception in some other spots: there is a macroscopic uncertainty principle with human psychophysiology, the minimally perceivable difference (just noticeable difference, per some notations). Anyone watching a celestial object such as the Moon could attest that they are only able to "kind of" capture where the object is, and only after a certain time elapses it is in the "new" position. Between those two, it is mooooooooooving over that roof. This is largely a Russellian view, but more closely tied to the human observer. The modern twist comes with the microscopic uncertainty principle from quantum theory, which states that this agnosticism is in the very nature of things.
Stating that we only can know the nature to the extent it is revealed to us as observers, however, is a thorny stance and the one dissatisfactory for many thinkers. So yes, the nature of time is very much still a problem in natural philosophy, and the problem of motion and change equates to it.
Not a philosophical answer but a physical one: While the earth spins around it own axis, it spins around the sun. The sun itself spins around the center of our galaxy (most likely), and the galaxy... well I don't actually know.
But still while every thing is moving in an absolute sense, we normally do not notice such movements. People at the equator move with at least "1670 kilometers/hour (1037 miles/hr)." according to https://image.gsfc.nasa.gov/poetry/ask/a10840.html
I'm not much familiar with the philosophical problem of motion, but physically you should perhaps be thinking about fundamental units -- speed (aka motion) is just distance divided by time, so if you have a problem with motion, then you must be having a problem with space or with time (or with both). If you had problems with neither, then "motion" would just be their ratio, and you presumably don't have much problem with that idea.
Space/time can be taken more-or-less for granted as in classical Newtonian physics, or can have many physical subtleties. Even relativity takes their existence pretty much for granted, with non-Newtonian subtleties about their interrelated geometry. Existence questions typically involve the emergence of relationships we perceive (and can measure) as space/time from more a more primitive underlying reality. Causal set theory takes "events" as the fundamental elements of reality, with "causality" the poset (partially ordered set) structure relating them. And then space/time can be derived as emergent from that structure, e.g., http://philsci-archive.pitt.edu/18063/1/3EmergenceCausets_archive.pdf (The emergence of spacetime from causal sets).
So philosophically, I'd think you should maybe recast your problem of motion as more fundamental problems like those addressed by causal set theory. The math's maybe a bit more than you might have hoped for, but that's how it seems to be. And once that stuff is all settled, the observation/perception of motion is just an emergent phenomenon.
In addition to paradoxes related to locomotion already mentioned, other types of change, like change of parts or qualitative change, are also subject to well known puzzles. Although you can characterize these equally as puzzles of identity, persistence, etc.
The growing argument was noted by the Greeks: Assume that object A grows by acquiring in some sense a new part B. But is B part of A? It would seem not because more properly B is only appropriately conjoined with A. At best B is part of some arrangement A+B. But that arrangement didn't grow either because it didn't exist previously, or even if it did exist B was already part of it. So actually A has failed to acquire a new part. Since nothing specific about A and B was assumed the very idea of growing by gaining new parts seems impossible. A similar argument can be given to the effect that things can't shrink by losing parts.
Or consider qualitative change like some banana turning from yellow to black: At t1 some banana b is yellow, at time t2 b is not yellow but black. So at times t1 and t2 b has different properties. But things that have different properties can't be identical (by indiscernibility of identicals). So things can't persist through qualitative change, and qualitative change is thus impossible.
These can be described as paradoxes because the conclusions are fairly incredible but hard to resist. See
"The Problem of Change" Ryan Wasserman https://compass.onlinelibrary.wiley.com/doi/full/10.1111/j.1747-9991.2006.00012.x (online somewhere)
On the growing argument search "paradox of increase".
The problems of motion and change, come down to the problem of definition of identity, in a way.
We do not observe motion or change as such. We observe states that are different, or observables that we associate with motion because pragmatic experience says that's a useful thing to do. But motion and change themselves, as processes, inherently seem to be creatures of time and comparison of the world at different time, and observables we associate with change over time.
These all come down in the first case to change, because velocity is simply, changed location over time, rather than change of some other kind. So we can simplify it to examining change itself. We don't directly observe it. But we believe it exists and occurs.....
Which raises the Yellow banana/Black banana question posed by others, that a Thing such as a banana, is not the same now, yellow, as future, black.
So if we only observe two states of the world, and in one, appears to be a yellow banana, and the other a black banana, what entitles us to say it is the "same banana"? Does the phrase even mean anything?
Because before we can draw any conclusion that "the banana" has"changed" (in condition or location), when we do a comparison of the two states we observed, we must be able to state that they are in some sense "the same thing", which is being compared. If its not identifiably the "same banana" we cannot conclude "the" banana has changed.
Thus, whenever we describe change, we inherently seek some matter which exists in both of the compared states, and which we identify as the same matter in two states rather than two different matters. So as concrete examples we say "you've changed" of a person (same person, two states), "a pixel" has changed on a screen, "the sky" has changed, "a culture" has changed, or a persons "[set of] beliefs" has changed. But we always seek an underlying "one thing", that we claim exists in both of the two states, and whose attributes or situation we propose to compare in the two states.
So the problem of change comes down to one of identity. What is it that allows us to say this is the "same" banana? Or the same person, or same pile of sand, or same anything?
Before we can hope to understand and analyse change, we must be able to formalise identity, or the quality of "same-thing-ness" rather better than we usually do. Because change is built on that.
The locus classicus of this problem are Zeno's paradoxes of motion. Today, these are usually resolved by calculus with the suggestion that the ancient Greeks didcnot solve this problem because they didn't have calculus.
However, as Aristotle pointed out, the solution by calculus does not get to the real nub of the problem. Although Aristotle didn't have calculus he understood the notion of limit perfectly well and said that the explanation due to this was 'adequate'. But, he went on to say that it missed important considerations and hence was not a comprehensive solution.
He resolved the question of motion by refering to motion as the actuality of a potential movement of change. This is not a million miles away from how QM with the collapse postulate views change and hence motion. It's also worth noting that Hegel suggested that a solution may be possible by a change of logic because he said at the point of motion an object is both here and there. Again, this is reminiscent of the superposition principle in QM. It should be remarked in connection with this that in the view of Wallace, the most radical solution to the interpretative problems of QM may come from changing the underlying logic. This is useful even in such traditional subjects as calculus where we can synthetically and concretely posit the infinitesimal rather than get interminably bogged down in epsilon-delta arguments.
The problems with motion arise from the attempt to break up spacetime and even stop time.
For example, in Zeno's paradox, the moving arrow is pictured as stationary at different positions from where it then moves to a next position. Giving the impression that the arrow can't move at all as it has to travel infinite broken pieces of space, from which the false conclusion is drawn that the arrow is stuck forever.
The discretization of spacetime, apparently avoiding this problem, introduces another problem though. For example, what determines how long a basic unit of time takes? Is there a clock ticking behind the scenes, giving signals every Planck-time (10exp-43 seconds) to change the scene to a new configuration?