Author Topic: A 'simple' Physics postulation...  (Read 6583 times)

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Online TimFox

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Re: A 'simple' Physics postulation...
« Reply #25 on: July 10, 2021, 03:56:53 pm »
Probably (1/137) furlongs
 

Offline vad

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Re: A 'simple' Physics postulation...
« Reply #26 on: July 10, 2021, 06:39:16 pm »
The answer is: it depends. Depends on the observer’s ability to take velocity measurements.

Imagine that the object (let say a 100kg steel sphere) was thrown up in the air with the initial speed of 1 m/s in complete darkness of the night, and all the observer has is a crappy civil radar with velocity resolution of 1 m/s. To the radar and hence to the observer the object would seem stationary for the most part of its trajectory.

Now imagine another observer, who has the finest laser interferometer and the fastest equipment to measure the smallest wobble of the 100kg sphere at the shortest period of time possible using the current era technology. What would the second observer see when the object reaches the apex? Most likely he/she will observer thermal fluctuations of the ball in the air. The object would never seem stationary to the second observer.

Now imagine another observer - technologically advanced extraterrestrial, who has equipment to take measurements with 10 to -53 power seconds resolution, and freeze the object to the temperatures low enough to rule out thermal jitter. Would the observer be able to dial in timescale resolution in its state-of-the-art measuring equipment and catch the moment when the object is stationary in the apex? The quantum uncertainty principle gives definite “not” as the answer.
« Last Edit: July 10, 2021, 06:41:06 pm by vad »
 

Offline dietert1

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Re: A 'simple' Physics postulation...
« Reply #27 on: July 10, 2021, 07:08:32 pm »
You should not write about extraterrestrials when considering a physics question. That's fantasy. As are all answers referring to quantum effects. No healthy scientist will ever apply quantum theory unless necessary. Not even an extraterrestrial physicist..

Regards, Dieter
 

Offline vad

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Re: A 'simple' Physics postulation...
« Reply #28 on: July 10, 2021, 07:41:04 pm »
You should not write about extraterrestrials when considering a physics question. That's fantasy. As are all answers referring to quantum effects. No healthy scientist will ever apply quantum theory unless necessary. Not even an extraterrestrial physicist..

Regards, Dieter
There is a Russian saying: “don’t tell me what to do, and I will not tell you where the <censored> you should go”.

Back to physics. The classical mechanics models discussed in this thread have their limits. One limit is thermal fluctuations. Another limit is drawn by quantum mechanics. There is no such thing as exactly zero velocity in the real world
« Last Edit: July 10, 2021, 07:43:03 pm by vad »
 

Online TimFox

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Re: A 'simple' Physics postulation...
« Reply #29 on: July 10, 2021, 07:46:47 pm »
In general, the only things that can be exactly zero are integer variables (e.g., the number of people in the next room or the number of direct flights between Des Moines and Sheboygan today). 
(No--there are no pregnant women in that room.)
However, in mathematics, a continuous variable going from a positive value to a negative value, must pass through zero, even if it does not dwell at zero for a finite time.
« Last Edit: July 10, 2021, 07:56:56 pm by TimFox »
 

Offline Ed.Kloonk

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Re: A 'simple' Physics postulation...
« Reply #30 on: July 11, 2021, 06:32:48 am »
iratus parum formica
 

Offline GlennSpriggTopic starter

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Re: A 'simple' Physics postulation...
« Reply #31 on: July 11, 2021, 11:29:57 am »
O.P. here... I was intrigued & overwhelmed by some of the mathematical/physics  comments here!   :-+
By far the largest 'problem' is defining what 'stop' means, but the following might 'negate' the concern for such thoughts??  :phew:
We generally agree, that to 'stop', should mean to be a rest for a known period of time...  However!!!... Consider ALL the other speeds
that the object was traversing through while initially decelerating and then re-accelerating through???  It was NEVER at ANY of those speeds
for ANY amount of time!!  So in the same manner, it was never at say 5 m/s or 10 m/s for any time greater than zero!! if you know what I mean!   8)

On a similar note, I had an argument with my maths teacher when I was young, about right-angled triangles....  2 sides, 1x1, hypotenuse = Root-2.
Now Root-2 goes on for ever, so 'technically' can not be drawn as a physical distance between to points, as you can keep adding digits to the end,
making it realistically ever so further, even though microscopic!!  However, I went on to say that if you DO draw such a triangle, with theoretically
zero line thickness, then there the hypotenuse lies...  exactly between those 2 actual physical points!!!  If you grasp what I mean!?   :phew:
Diagonal of 1x1 square = Root-2. Ok.
Diagonal of 1x1x1 cube = Root-3 !!!  Beautiful !!
 

Offline vad

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Re: A 'simple' Physics postulation...
« Reply #32 on: July 11, 2021, 01:02:44 pm »
Now Root-2 goes on for ever, so 'technically' can not be drawn as a physical distance between to points
This statement is false.

Square root of 2 is irrational number. The only thing that distinguishes irrational numbers, from other real numbers (from rational numbers), is that the irrational number cannot be expressed as a/b, where a and b are integers, while the rational number can be expressed that way. Nowhere it follows that if you cannot express distance between two points as a/b, such distance is somehow forbidden.

Contrary to that, the distance can be irrational number. You can prove this by contradiction:

Suppose distance between two points must always be a rational number.
Take a right triangle with both legs of size 1.
The distance between two vertices of hypotenuse of such triangle is square root is 2, according to Pythagorean theorem.
It can be proven elsewhere that the square root of 2 is not a rational number.
Therefore the initial assumption that the distance between two points must always be a rational number must be false.
:)
 

Offline Brumby

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Re: A 'simple' Physics postulation...
« Reply #33 on: July 11, 2021, 01:07:45 pm »
Now Root-2 goes on for ever, so 'technically' can not be drawn as a physical distance between to points
This statement is false.
Yes, it is absolutely false.

Just because you can't define a value to a known number of decimal places, does NOT mean you can't draw an object with that value.  I can offer several examples without even trying.

Quote
Square root of 2 is irrational number. The only thing that distinguishes irrational numbers, from other real numbers (from rational numbers), is that the irrational number cannot be expressed as a/b, where a and b are integers, while the rational number can be expressed that way. Nowhere it follows that if you cannot express distance between two points as a/b, such distance is somehow forbidden.

Contrary to that, the distance can be irrational number. You can prove this by contradiction:

Suppose distance between two points must always be a rational number.
Take a right triangle with both legs of size 1.
The distance between two vertices of hypotenuse of such triangle is square root is 2, according to Pythagorean theorem.
It can be proven elsewhere that the square root of 2 is not a rational number.
Therefore the initial assumption that the distance between two points must always be a rational number must be false.
:)

Absolutely correct!   :-+
« Last Edit: July 11, 2021, 01:10:40 pm by Brumby »
 

Online TimFox

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Re: A 'simple' Physics postulation...
« Reply #34 on: July 11, 2021, 01:52:46 pm »
Again, modern mathematics as taught in freshman analysis courses rigorously defines real numbers, including simple irrationals like root-2 and transcendental numbers like e and pi.
There is no need to re-fight 19th century battles.
 

Offline T3sl4co1l

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Re: A 'simple' Physics postulation...
« Reply #35 on: July 11, 2021, 05:04:54 pm »
Well, if you want debate the physicality of numbers, it's pretty clear rationals exist; that's just a matter of definition.  The rest, whatever you like.

Irrationals, maybe, but it's not like we can construct nor measure a perfect right triangle.  Even if one made a crystallographically perfect shape and counted the atoms along each side, one would find whole numbers, a mere approximation to the irrational ratio.  How should you even count the corners?  They aren't perfect right angles, there's at least one ionic radius around the edge.  Should you project the lines/planes towards their ideal intersections, or should you measure the real distances, taking account of the corner somehow?

One likely cannot even construct a 3-4-5 right triangle, as the hypotenuse is not on a crystal plane -- unless there happens to be a symmetry group with exactly such a plane, in which case merely pick any other of the infinite Pythagorean triples you can choose from and check again.


I'm fond of a little rant about "real" numbers, as there are so very few indeed that can be constructed or computed in any meaningful way -- pi and e happen to be two of the most powerful ones, showing up in a great many places.  The whole space of real numbers -- let alone just those that can merely be computed, through any arbitrary (and likely impossibly slow) process -- is truly, unsettlingly vast.  It's not just an uncountable infinity, but most -- indeed, "nearly all", well and truly defy any possible description, by any finite means, in infinite time, in the real universe-as-we-know-it.

For example, we can construct the number which is the decimal concatenation of the maximal number of steps of execution (with halting) of an N-state Turing machine.  That is, the "Busy Beaver" function.  The first few values of which are known, and which grow incredibly quickly.  The currently known bounds on BB(6) are unimaginably large, and no computer can exist in the known universe to compute it directly.  The problem in general, is a statement about the decidability problem, a problem which is provably unprovable.  Therefore, the exact value of this constant cannot be known.  And it doesn't have any handy identities to allow a shortcut; this isn't your average pi we're talking about.

See: https://math.stackexchange.com/a/462835
They give a slightly less "dense" form, using powers of two rather than concatenation.  Obviously, there are infinite ways we could encode this function/sequence as a real number, so there's already a small infinity of real numbers constructed with this uncomputable function, in part.

So, I would humbly submit that, you can give or take algebraic (irrational) or computable numbers, but "real" Real numbers, most definitely aren't!

Tim
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Online TimFox

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Re: A 'simple' Physics postulation...
« Reply #36 on: July 11, 2021, 05:45:06 pm »
Again, this was all worked out to generate rigorous definitions of "real", "algebraic irrational", and "transcendental" numbers by mathematicians years ago, especially after the rigorous development of limit theory.  Numbers are mathematical constructions that evolved from the "counting numbers" or "natural numbers" (positive integers, excluding zero) found in everyone's day-to-day life.  As Leopold Kronecker reportedly said, "God created the natural numbers; all the rest is the work of man."  (The work of man started with adding zero, then the negative integers, and the rest is history.)
An interesting detail, found by Georg Cantor, is that infinite sets such as the set of rational fractions and the set of algebraic irrational numbers can be put into a one-to-one correspondence with the natural numbers, and are therefore "countable infinite" sets.  However, the set of real numbers cannot be put into that one-to-one correspondence, and are therefore called "uncountable infinite" sets.  He assigned cardinal numbers to these infinities, but I don't know how to enter Hebrew letters into this post.
Mathematics is very useful to describe physical phenomena, but should not be confused with the phenomena themselves.  Pythagoras was correct about 3 by 4 by 5 triangles in geometry, but that does not imply that you can produce an exact 3 in by 4 in by 5 in triangle in the machine shop from tool steel made from atoms.  That triangle is still useful for human purposes:  supposedly, the ancient Egyptian tax authorities used it to produce right angles and to measure the area of peasants' fields (after each inundation) by dividing the irregular shapes into triangles for tax assessment, hence the term "geometry".
« Last Edit: July 11, 2021, 05:48:51 pm by TimFox »
 

Offline magic

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Re: A 'simple' Physics postulation...
« Reply #37 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
 

Offline vad

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Re: A 'simple' Physics postulation...
« Reply #38 on: July 11, 2021, 06:41:10 pm »
Natural numbers, integer numbers, rational numbers, real numbers, complex numbers - are all abstract concepts invented by humans for one reason or another. They are all useful for mathematical modeling of real world phenomena. The real world, on the other hand, appears to be more complex than any abstract model humans came up with so far.
 

Offline dietert1

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Re: A 'simple' Physics postulation...
« Reply #39 on: July 11, 2021, 07:01:23 pm »
Philosophers and psychologists say the exact opposite: It is absolutely fascinating and surprising to which extent the human brain can model and understand reality. The failure of a human brain to accomplish this in perfection or let's say even minor difficulties to do so are called mental illness, and it is more abundant than we imagine.

Regards, Dieter
 

Online TimFox

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Re: A 'simple' Physics postulation...
« Reply #40 on: July 11, 2021, 07:38:03 pm »
T3sl4co1l is right, real numbers are a kludge invented to justify calculus after it had been handwaved into existence :-DD

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".
 

Offline jeffjmr

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Re: A 'simple' Physics postulation...
« Reply #41 on: July 13, 2021, 01:17:56 am »
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

 
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Offline Rick Law

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Re: A 'simple' Physics postulation...
« Reply #42 on: July 13, 2021, 03:10:00 am »
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".
 

Online TimFox

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Re: A 'simple' Physics postulation...
« Reply #43 on: July 13, 2021, 03:19:49 am »
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.
« Last Edit: July 13, 2021, 03:37:26 am by TimFox »
 
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Offline LaserSteve

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Re: A 'simple' Physics postulation...
« Reply #44 on: July 13, 2021, 03:58:01 am »
This is the EEV blog thread of the year! I'm in awe.
"When in doubt, check the Byte order of the Communications Protocol, By Hand, On an Oscilloscope"

Quote from a co-inventor of the PLC, whom i had the honor of working with recently.
 
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Offline magic

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Re: A 'simple' Physics postulation...
« Reply #45 on: July 13, 2021, 04:57:34 am »
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".
No, that's backwards. The early work I called "handwavy" and the kludge is all the mental gymnastics that followed to make it "strict".

Such as:
Quote from: some mathematician somewhere
Let's consider all infinite sequences of rationals divided by the relation of Cauchy co-convergence, that's clearly the set of numbers that are real. I mean, because we need it that way to make our theory work.
Makes a lot of sense :D

Or, you said something about transfinite numbers, any progress on figuring out how many alephs there are before the continuum?
Quote from: yet another mathematician
So we have that problem that we don't really know what a "set" is, let's try some common sense axioms and everything will clear up.
:-DD

Yes, I'm kidding somewhat. But that's because the whole thing is funny :P
 

Offline Nominal Animal

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Re: A 'simple' Physics postulation...
« Reply #46 on: July 13, 2021, 05:08:25 am »
Thinking that there is no single moment when the object has zero velocity because moments have duration is mixing completely different scales together ignoring their associated assumptions/contexts and replacing them with another, and getting garbage as a result.

As stated, the only sensible answer – both in the human scale and in the physics sense – is that yes, the trajectory does include a point where the vertical velocity component is zero.  Anything else is randomly picking one set of details and completely ignoring others.  (When we do do that, we try to pick a consistent set.  That is the difference between an approximation and an informed guess.)

If we assume the object is made from ordinary matter, it is actually a writhing mass of atoms and molecules that never "stop" unless cooled to so close to absolute zero that you cannot measure the movement anymore.  How do you measure the location of such a blob?  Its center of mass?  In the human scale, objects pretty much perfectly follow Newtonian physics, because of the inherent assumptions about scale and precision.  Mixing in Planck timescales is utterly silly, because that object itself is just a vaguely defined fuzzball of atoms and molecules; probably losing and accruing atoms and molecules on its surface constantly.

If we assume it is an uniform solid, it is no longer a physics question; it is a speculative question about fictitious physics.  Sci-Fi.
« Last Edit: July 13, 2021, 05:11:36 am by Nominal Animal »
 
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Offline T3sl4co1l

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Re: A 'simple' Physics postulation...
« Reply #47 on: July 13, 2021, 01:01:25 pm »
Another way of putting it: where is a line?

Say you mark a measurement with a chalk line.  What position is that line actually representing?  The chalk mark is obviously fairly wide.

Do it again with a pencil, or a scribe.  It still has width, and subsequent uncertainty.  (The scribe at least might leave a v-shaped groove that we could find the root of.  But other than that, it still has width.)

What if, instead, we declare that the position shall be the edge between two regions.  We can use a permanent marker held tightly against a rule, which marks a sharp ink line in the same position as the ruler's edge.  The position is perfectly defined because the ink doesn't spread out (heh, well assuming a nonporous material and gel-like ink/paint, but you get the idea).  Of course the rule and pen tip and everything won't be perfectly smooth and repeatable and all that, and the materials are ultimately made of atoms, but the important difference is taking the edge of some region, rather than trying to eyeball the middle of it.

It's the same idea over time rather than space.  An instantaneous moment is a zero-time boundary between times before and after that moment.  A finite moment, is some period of time around a moment.

Tim
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Online TimFox

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Re: A 'simple' Physics postulation...
« Reply #48 on: July 13, 2021, 01:48:00 pm »
Some of these snide comments about technical language in mathematics make me think that 19th and 20th century mathematical progress is something that happened to other people.
 

Offline aneevuser

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Re: A 'simple' Physics postulation...
« Reply #49 on: July 13, 2021, 04:55:00 pm »
Some of these snide comments about technical language in mathematics make me think that 19th and 20th century mathematical progress is something that happened to other people.
Well, 19th and 20th century mathematical progress is something that happened to mathematicians, I think it's fair to say. You won't find many "users of mathematics" (physicists, engineers, etc) who have studied formal definitions of limits, or who have heard of the IVT, or of Dedekind cuts or whatever. But I don't think it matters too much, most of the time; you don't need to be able to make an epsilon-delta argument to learn enough calculus to get a physics or engineering degree.
 


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