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A 'simple' Physics postulation...

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dietert1:
It's long time ago, but i also think i learned that at school, during the last years - what anglo-americans call college.
An example i remember from mathematics lectures during the first years at university was about functions that are in general continuous except in an infinite number of points. Useless nonsense in retrospective, but somewhere i may still have notes..

Regards, Dieter

TimFox:
My memory from 50 years ago is hazy, but that sounds like the generalization of Riemann integral (the one we all use) to the Lebesgue integral, which I have not needed since.

Nominal Animal:

--- Quote from: TimFox on July 13, 2021, 06:38:42 pm ---My personal experience is that we used epsilon-delta arguments in freshman calculus class, probably in the first academic quarter, to prove limits as part of understanding the nature of the differential calculus, etc.
--- End quote ---
My own from 1994 in Finland was exactly the same.  Not sure since.


A key point here to remember is that math is the language used to describe physics.  We discover physics, don't invent it.  Whatever expressions we use to describe physics, the first thing we always must to do is to examine whether it makes sense or just happens to match the data in the limited region we used to construct the expression in.

Going below say nanosecond scale without at the same time considering the "object" a collection of atoms and the associated phenomena (including Brownian motion) is picking a subset of the physically applicable rules and knowingly rejecting the rest, and yields garbage physics, even if mathematically correct.

Here, to repeat myself, the only sensible approach (i.e. that makes physical sense given the context of the question) is to consider the trajectory (height component) a second-degree polynomial (given the limitations of the question), which is obviously continuous and differentiable, and therefore has an extremum where the derivative is zero.

Even if we define "stop" to mean "a period of time when the object does not move", the answer does not really change.  You see, near the extremum, velocities are low, so if we pick two points in time symmetrically before and after the extremum, the difference in height is minimal.  Mathematically, the trajectory of the object does change, but considering the kind of approximations we are already making and the associated limits to precision, we can tell those two points in time as separate before we can tell a change in height.

As an example, if you throw a baseball up just right, so its apex is at shoulder height, your friend can hit it with a bat at just the apex, and make it fly perfectly horizontally.  Well, okay, the ball-bat angle with respect to ground can compensate for small vertical velocities.. but you could make a machine that does the same with a flat vertical hitting surface (a pool cue with a plate at the tip).  So, the only sensible interpretation of the original question is that there indeed is such a moment, because we can act (or create a machine that acts) during that moment, poking the object with a bat or cue right then.  The precision at which we can do so, is "just" an engineering problem.

magic:
Here's an interesting idea:

If we consider a function whose value is 1 on [0,⅓]∪[⅔,1] and zero elsewhere, any Riemann integral thereof with a mesh finer than 0.1·⅓ will be within 20% of ⅔.

If we then set the function to zero on the middle third of each of those intervals, any Riemann integral with a mesh finer than 0.1·(⅓)² will be within 20% of (⅔)².

We clearly have a limit in zero here. Therefore function which is 1 on the Cantor set and 0 elsewhere is Riemann integrable and the integral is zero. The Cantor set being, of course, an infinite, uncountable and continuum-cardinality abonimation :scared:

This appears to complete the proof that you don't need to understand all that :bullshit: to be a physicist for your whole life :-DD

TimFox:
This is why the "other people" invented the Lebesgue integral for those who find it necessary.
By the way, if you sample a real-world waveform inappropriately, you may find these high-falutin' concepts useful in understanding the result.

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