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A 'simple' Physics postulation...
TimFox:
--- Quote from: magic on July 11, 2021, 06:20:20 pm ---T3sl4co1l is right, real numbers are a kludge invented to justify calculus after it had been handwaved into existence :-DD
--- End quote ---
Like many things, calculus was invented to be useful, then mathematicians worked to define the concepts rigorously.
Due to lack of acceptance by his peers, Newton probably used his "fluxions" to derive his important results in Principia, but had to show proofs in very cumbersome geometrical arguments, since his peers accepted Euclidean geometry that was, by then, ancient.
Leibniz, and many others since then, worked to rigorously define concepts such as the continuum, real numbers, etc. and there is no longer a useful controversy in mathematics about these definitions. It is very easy (and smug) to dismiss early work on a concept as a "kluge" in hindsight, just like the 20th-century physics innovation of the "Dirac delta function", which was useful for decades before the mathematically-rigorous definition in terms of "distribution theory".
jeffjmr:
I am not a physics major, but a perpetual armchair student.
The OPs question immediately brought to mind the “fly stops a freight train” mental exercise; wherein a freight train is moving down the tracks at 60 mph, and a fly is traveling over the tracks in the opposite direction. When the fly is splattered on the front of the locomotive it is “stopped” for an instant before changing direction, therefore since at that instant the fly and train are one, the train is stopped as well.
To me, logic would dictate that at no time, not even for a Planck time, is the fly’s velocity 0. Neither is the OP’s object.
The rest of the math discussion reminds me that .999…. can be “proven” to equal 1.
.999… = x
10x= 9.999…
Subtract x from both sides
9x=9
X=1
Mental gymnastics. Everyone knows .999… does not equal 1. Math exercises don’t always comply with reality.
Cheers,
Jeff
Rick Law:
The OP's question is actually covered nicely by Heisenberg's uncertainty principle. The OP is asking to know about two complementary variables. Heisenberg's uncertainty principle say they cannot both be known to precision because they are complementary. The more you know about one, the less you can know about the other.
In the OP's case, the two complementary variables are the object's position (at the "top") where the object's speed=0 (momentum=0). If you know for sure it is at the top, then you can know nothing about it's momentum. If you know for sure it's momentum is zero, then you can't know it's position.
The trouble with analytical solution to a physics problem is that it gives us the illusion that we can have more resolution than reality offers.
How well can you for example halve the current follow? At some point, your current becomes merely thousands of electron, then hundreds, then tens, then what? Can we have half an electron flowing? You will run into the same issue say cutting a rod to 1/2 length. At some point, you need to halve a molecule, then halve an atom, then halve a proton?
This is less earth shaking as it seems. You can see the problem with electron/x-ray microscope if you look at the ends of a rod. There is no defined flat-surface at the end of the rod. The more you zoom in, the lumpier it gets. The little bumps are the molecules or the atoms making up the rod. So how long is the rod? Just average the lumps. We pretend the average of the bumps (atoms) is the flat surface at the end of the rod - but it can only be done in our imagination only. We should remember that there is no flat surface at the end of the rod if we zoom in really really really close. Nature only works when you "average it out". Otherwise, you don't know something as simple as "the exact length of a rod".
So, comforting as analytical solutions are, they are virtual. They are not reality. Accept that there are things in nature that you can't know. You can "average it out" and say: It "kind of stopped" at "kind of at the top".
TimFox:
Jeffjmr:
In mathematics, the notation 0.99999… does equal 1, and the calculation you cited is the proof thereof. There is no logical flaw in that calculation, and the method is the normal way to compute the rational fraction that equals an arbitrary repeating decimal fraction.
An exercise for the reader: write the repeating decimal fraction for 1/3 and apply the same method to recover the fraction from the decimal. Then, for a more complicated calculation, apply the method to the repeating decimal fraction for 1/7 = 0.142857142857142857... The mathematical notation for such a repeating fraction draws a line above the repeating bit (3 or 142857, respectively) and omits the ellipsis, which at the end of the expression means it repeats forever.
How can you say “everyone knows 0.9999… does not equal one” when anyone who got as far as limit theory in freshman mathematics knows otherwise? The standard mathematical notation with an ellipsis or overhead line explicitly means the limit of the expression. "0.999" without an ellipsis is close to 1, but no cigar.
LaserSteve:
This is the EEV blog thread of the year! I'm in awe.
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