Can we explain velocity without time?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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 : $$ \textbf v = \frac {d\textbf r}{dt} $$