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I'm trying to wrap my head around inertia/velocity, or just change in general, I guess. I'm unsure exactly sure were my gap is, or what question to ask, so I'll explain with a scenario.

Lets say I'm an insubstantial observer inside a flawless virtual simulation of our universe, watching two identical metal spheres floating in empty space. From my perspective, Sphere $A$ is moving at $1$ unit of velocity, directly away from my position. Sphere $B$ completely stationary, relative to my position.

I pause the simulation multiple times over a minute, record the positions of both, and from those measurements, I can say that the velocity of $A$ is $1$, and the velocity of $B$ is $0$.

Questions:
1. What if I only paused once though, hadn't recorded anything before, and wanted to know where both spheres would be in the next frame (or shortest possible period of time, whatever that would be)? Is there an observable/measurable property of either sphere, or the space around it, other than a previous position, that would tell me that?
Edit: The consensus on #1 so far seems to be "no", so #2 is the only remaining unanswered question. Leaving this here for context.

2. If the answer is no, then why does sphere $A$ move at all? (To say the spheres simultaneously exist across all points in their eventual path is fine, but that's really just a different perspective of the same thing, so then, why are [$x,y,z$] for $t$ not equal to [$x,y,z$] for $t+1$, without looking at any previous point of $t$.)

Edit: This question was closed because the question needs clarity, and the answers so far seem to be answering a different question than the one I'm asking for #2, so I'll attempt to clarify "Why does sphere $A$ move at all?"

Lets say the invisible immortal Joe, every morning, without fail, moves a stone directly westward by $N$ inches, where $N$ is the number most recently spoken in proximity to the stone. Because Joe is flawlessly predictable, we can record the stone's positions for a couple days in a row to determine $N$, then know where that stone will be on any day until the next time a number is spoken.

  • I know how we determine $N$.
  • I understand that $N$ only changes when someone speaks the number in proximity to the stone.
  • I understand that there are many ways of expressing the path of the stone and its current/future positions, both individually, and as an infinite set.
  • What I want to know is why the hell the stone is mysteriously moving every morning. I don't know about Joe, or understand why he would want to push this rock around.

"We have no idea" and/or "we don't care" is a perfectly acceptable answer, if there's a reason not to care. I'm just assuming that this isn't the case, and the gap I'm perceiving is because something in this scenario IS actually the cause of the movement, and I'm missing it.

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  • $\begingroup$ You can't read velocity out of an instantaneous snapshot. If you think this through as an algorithm, you are asking your program to return all the position data. You won't get velocity data because you didn't ask for it. So with just an image, you also just have positions. What other ways of measuring velocity were you envisioning? $\endgroup$ – TBissinger Jan 21 at 14:03
  • $\begingroup$ I wasn't talking about just positional properties, but to your question: None really. Measuring velocities this way makes sense. I just assumed it was a simplified abstraction of some underlying property that results a force that generates an instantaneous change in each moment, but in equilibrium... like space gets compressed moving out of the way of the mass, and the compressed space draws the mass forward (not specifically this, just an example of what I imagined). Seems super weird that this is missing, which the answers so far seem to indicate. Doesn't it? Am I crazy? $\endgroup$ – Brandon D Jan 21 at 14:46
  • $\begingroup$ If you're not just talking about position, you need to give a more precise meaning to what you are then talking about. In classical mechanics and general relativity, the full information of a state is stored in position and velocity. If you read out this state without velocity, you're losing information and defniteness... If you were talking about a charged particle, you might read out momentum information from its retarded potential, but that is caused by motion and does not effect motion. $\endgroup$ – TBissinger Jan 21 at 15:10
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    $\begingroup$ ... To make them independent, we bake the derivative relationship into the functions we operate on in Lagrangian mechanics, rather than baking them into the coordinate systems themselves. $\endgroup$ – Cort Ammon Jan 22 at 0:24
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    $\begingroup$ Please make your post one cohesive question. There is no need to specify your edits; there is an edit history for those who are interested. I initially worked on edited the question down, but then realized it invalidated current answers, so I rolled it back. In any case, you can look at that edit to get a better idea on how to clean up your post. $\endgroup$ – BioPhysicist Jan 22 at 14:50
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  1. No
  2. The dynamical state of a system at any moment in time is not simply given by the positions of all of the constituents. We need more information - namely the velocities (or momenta) - in order to fully determine how the system will evolve in time.
  3. I'm not sure what you mean by this. Clocks tick and objects move; the former does not cause the latter. Indeed, the former is really a particular case of the latter. We simply quantify motion by e.g. specifying the distance traveled by an object in some number of clock ticks.
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  • $\begingroup$ 3. I mean that using time to measure/describe velocity and predict position is obviously expected, but it doesn't explain why sphere A changes its position between two sampled times. If you say it changed position because it had X velocity, you're kinda saying "time did it", since time and displacement are really the only factors there. I know nobody's really asserting that time moves things, but at the same time, if no other property of the sphere or surrounding space exerted force to cause a change in position, why did it change? $\endgroup$ – Brandon D Jan 21 at 15:35
  • $\begingroup$ @BrandonD You are asserting that position is a thing which is naturally fixed, and if that position changes then there must be some influence making it so. This is wrong. Position naturally changes in accordance with the object’s momentum. It is the momentum which needs an external influence in order to change. You can get position from a photograph, but for momentum you need at least two. $\endgroup$ – J. Murray Jan 21 at 17:59
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Analyzing motion comes in the form of differential equations; there is a bunch of ways of expressing these differential equations and you can look up reasons why but basically the crux is :

$$F=ma$$

This is a 2nd order ODE, and because of that to solve it for position we end up with a function in time that requires you to know both the Position and Velocity at some time $t_0$. Think about if you were to drop an apple from a tower, dropping it on at any floor would change the path, also the velocity you throw it at, even though the have the same $F=ma$. Now in the case you've outlined, $F=0$; thats fine...but we still need Position and Velocity at some time $t_0$, and a "snapshot" is not enough to obtain this information, particularly velocity.

From here we're done as far as physics is concerned, and if you are still wondering why we need time in our expressions, at its core it comes up because there are certain constants of motion (momentum), and to change them you need something to effect the system over time (a force), and from there we use the change in momentum to see how the position is affected.

From a "philosophical" point of view, we can discuss if it even makes sense to observe the velocity of an object, what does that mean? Can we really observe an infinitesimal change in a variable? Do variables in the real world even change smoothly, or are we just drawing constant secant lines and we've never had a "true" tangent line...this is all fun for conversation and interesting to think about, but it doesn't really matter in Physics; the fact of the matter is that we use Symmetries and Constants of Motion, and use our best observational tools possible to make predictions about motion, and in so far as our predictions work, Physics is happy.

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No, you can't define velocity without time. Because velocity is rate of change of position with respect to time. You can write velocity in terms of position but that position is also changing with time. So ultimately you will end up with time.

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Can we explain velocity without time?

Sort of. In terms of momentum :

$$ v = \frac pm $$

In terms of kinetic energy :

$$ v = \sqrt { 2 \frac {E_k^~}{m} } $$

But I think in one or another point - you'll need a time, cause position change over time is most natural : $$ \textbf v = \frac {d\textbf r}{dt} $$

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  • $\begingroup$ I don't think you can measure momentum or KE from a snapshot in time, so there's no way to derive velocity without time, regardless of how natural it is. $\endgroup$ – Nuclear Hoagie Jan 22 at 14:58
  • $\begingroup$ As I have understood OP statement - it is asking if velocity can be defined, not using time terms directly, so the answer to that question is - YES, as I've showed with formula. How momentum or kinetic energy is measured is completely a different question and not related to the first one, imho. $\endgroup$ – Agnius Vasiliauskas Jan 22 at 15:46

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