A 'simple' Physics postulation... - Page 1
Forum section: electronics / general

author: GlennSprigg
author_rank: Super Contributor
timestamp: July 09, 2021, 10:58:32 am
content: In reference to JUST the local surface of the Earth where you stand, (NOT the solar system, galaxy etc!!)...If you were able to throw an object 'exactly' vertical, such that it came back to you due to gravity, did theobject ever 'Stop' at it's apex?? Obviously, it can't have, even for 1/1000000000-th of a second, as that woulddefy the laws of physics. So it is only ever going up, or coming down. Without stopping!!?

author: Brumby
author_rank: Unknown
timestamp: July 09, 2021, 11:09:34 am
content: Define "Stop".

author: retiredfeline
author_rank: Frequent Contributor
timestamp: July 09, 2021, 11:42:56 am
content: Maybe you've discovered oneZ's Arrow paradox.

author: Ian.M
author_rank: Super Contributor
timestamp: July 09, 2021, 01:50:57 pm
content: You'd need to go to the Earth's North or South pole to conduct your experiment, otherwise, as you throw it upwards it will drift off to the West, due to the Earth's rotation. (As the tangential velocity required to remain 'stationary' with respect to a rotating frame of reference is proportional to the rate of rotation and the radius from the axis of rotation, it doesn't have enough tangential velocity to maintain its angular position at a greater radius from the center of the Earth.) Therefore if you throw an object so that it lands exactly where thrown from anywhere other than the poles on the Earth's surface, to do so you *CAN'T* thow it vertically, you have to tilt your throw ever so slightly East to give it the extra angular velocity required. Its trajectory will therefore be a tall extremely skinny*loop, west-bound at the top, and, in flight, it will *NEVER* have zero velocity with respect to the point it was thrown from, and lands at.At the poles, for an infinitesimal instant at the apex of its vertical trajectory, it has zero velocity. Its speed with respect to the surface is a continuous function so how else is it going to get from positive (distance increasing = ascending) to negative (distance decreasing = descending) without going through zero speed?* The tangential velocity difference over 100m height difference at the equator is (if I've done the maths right) only 45um/sec, so is normally absolutely imperceptible.Quote from: Brumby on July 09, 2021, 11:09:34 am*EXACTLY* *THIS*Velocity and acceleration both zero for a non-zero time interval (i.e. Physics: 'At rest') would probably be a reasonable definition.

author: Brumby
author_rank: Unknown
timestamp: July 09, 2021, 02:00:57 pm
content: Aside from the polar conversation (which is valid) it is this which contains the key elements of the answer:Quote from: Ian.M on July 09, 2021, 01:50:57 pmThe word "instant" is key here. The OP is considering aperiodof time - which cannot deliver a zero value....Unless you take that period to be equally before and after reaching the apex in which time the net velocity will be zero. (But, I don't think the OP will be happy with that.)However - this statement not only allows, butrequiresa zero velocity at a point in time:Quote

author: TimFox
author_rank: Super Contributor
timestamp: July 09, 2021, 02:22:28 pm
content: This problem is straightforward in calculus, since velocity is the time derivative of position. For the parabolic curve of position vs. time, the derivative is zero at the extremum (maximum) position, but for an infinitesimal time.I usually think that nothing continuous can go to zero unless it can go negative: in this case, the if the initial (up) velocity is positive, it reverses at the maximum and changes to a negative (down) velocity.

author: Ed.Kloonk
author_rank: Super Contributor
timestamp: July 09, 2021, 02:26:29 pm
content: Quote from: GlennSprigg on July 09, 2021, 10:58:32 amWhen ever I pondered such things as a young man, my father would find and confiscate my stash.

author: esepecesito
author_rank: Regular Contributor
timestamp: July 09, 2021, 02:30:19 pm
content: Quote from: GlennSprigg on July 09, 2021, 10:58:32 amI cannot understand why you say stopping would violate a physics law... it doesnt. It does stop.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: July 09, 2021, 03:47:23 pm
content: Yes, without a fully defined scenario, there's no reason, value or meaning in saying something might "stop".If you mean in terms of a nonrotating, pointlike, classical object, in pure vertical motion with no lateral velocity, following a Newtonian trajectory, without reference to a rotating or accelerating (other than gravity) frame, and no other outside forces (like wind resistance) -- and please take note of how many conditions one must apply to be rigorous here -- then one can say it "stops", as in velocity = 0, at one point during its path: the apex of the parabolic trajectory. This does not occur for a defined duration of time, but is an instantaneous event only. We use calculus to describe such conditions. If you prefer nonzero bounds, then we can define the limit as there being a duration of time dt such that the velocity is v +/- dv, for some dv and dt that exist. In the limit, we take both dt and dv to zero, but any will do.Note also, there is nothing special about the point where velocity goes to zero. We can make just as remarkable a claim about the two points adjacent to zero, or where it's going 1 m/s, or 10, or whatever; any velocity that happens to appear during the trajectory. (Or including velocities that don't, if you allow extrapolating to earlier and later times while ignoring possible collision, or allowing that the number of points may be zero, one or two, or many, for any velocity we choose to ask about.)Tim

author: TimFox
author_rank: Super Contributor
timestamp: July 09, 2021, 03:51:24 pm
content: Your discussion of nonzero bounds is handled in mathematics as an "epsilon-delta" proof of a limit. Stolen from the web:"The epsilon-delta definition of limits says that the limit of f(x) at x=c is L if for any ε>0 there's a δ>0 such that if the distance of x from c is less than δ, then the distance of f(x) from L is less than ε. This is a formulation of the intuitive notion that we can get as close as we want to L."This is the modern explanation of Zeno's paradox.

author: SiliconWizard
author_rank: Super Contributor
timestamp: July 09, 2021, 05:08:10 pm
content: Well, I guess it all comes down to the thought that velocity is mathematically zero when the object reaches its maximum altitude. Then the OP assumes "velocity = 0" means "stop".Of course, as others have pointed out, first thing would be to define what "stopping" means. Even if it's a term we use in everyday's language without giving it much thought, I'll just dare a simple definition that looks good enough: an object stops iif its velocity reaches 0 *and* the velocity function at this point is non-derivable.In the OP's example of parabolic motion, velocity indeed reaches 0 at the apex, but is still derivable. So, the object never "stops". Just my 2 cents.

author: TimFox
author_rank: Super Contributor
timestamp: July 09, 2021, 05:26:30 pm
content: It stops, then reverses, with the velocity variable being continuous in time.

author: Zero999
author_rank: Super Contributor
timestamp: July 09, 2021, 07:46:49 pm
content: It sounds similar to a child on a swing, who stops, before swinging back the other way, or the voltage and current in the resonant tank of an LC oscillator.

author: TimFox
author_rank: Super Contributor
timestamp: July 09, 2021, 08:33:47 pm
content: Exactly. All of these time-dependent variables (as functions of time) have extrema (maxima and minima) where the time derivative goes to zero. (When I learned math, we said that the derivative "vanishes", which usage bothered me as a smart-ass student.)

author: dietert1
author_rank: Super Contributor
timestamp: July 09, 2021, 09:49:54 pm
content: This reminds me of a professor at the university, when i was student of physics. I remember the name, but i won't tell. One day in a lecture he started to explain: Since velocity is a vector, it can never become negative. The vector will just change its direction. Isn't that crazy?The question in this thread was posed such as to focus on the vertical component of velocity. And that becomes zero before getting negative. Voting about physics questions, that's modern times!Regards, Dieter

author: TimFox
author_rank: Super Contributor
timestamp: July 09, 2021, 09:59:53 pm
content: One can express the vector velocityvin terms of its three scalar componentsvxx+vyy+vzzwherevis the velocity, andx,y, andzare "unit vectors" in the three Euclidean directions.The componentsvx, etc., can be positive, negative, or zero, but the magnitude of the vectorvcannot be negative. If not moving, the magnitude ofvin that reference frame can be zero.

author: TerraHertz
author_rank: Super Contributor
timestamp: July 09, 2021, 11:30:19 pm
content: Quote from: Ian.M on July 09, 2021, 01:50:57 pmYou're only considering the Earth's rotation. But that is a meaningless exclusion. There are other imposed motions: The Earth's orbit around the Sun (which is a complex spiral due to motion of the solar system) and influences of other objects such as the Moon and other planets.Glen's question only applies to a hypothetical system of two masses in an infinite void. Initially set in motion moving apart, but not fast enough for gravitational escape. Yes, they both stop and reverse. But then you get into questions of quantum physics, and what is the smallest units of time, space and velocity you can theoretically measure. Even for atomic particles there are limits, and for macroscopic objects with non-zero thermal motion of the component parts, good luck.

author: Alex Eisenhut
author_rank: Super Contributor
timestamp: July 09, 2021, 11:45:35 pm
content: When Captain Picard orders a "full stop" of the Enterprise 1701-D, does it stop?

author: Rick Law
author_rank: Super Contributor
timestamp: July 10, 2021, 12:00:46 am
content: Quote from: GlennSprigg on July 09, 2021, 10:58:32 amThink of this problem as looking at a video of the event. You can measure where the object is on each frame, but in between frames, you don't know for sure. That is how nature works. In the case of a video, you can increase the frame rate and get more resolution. In the case of Nature, there is a point where you cannot speed up the frame rate anymore. There is a limit to the resolution one can get out of Nature.When one uses Newtonian mechanics to describe an object's travel, onecannottake that solution to apply it to the very fast (near speed of light), very small (quantum scale, or the even smaller Planck length), or very quick (Planck time).What is stop?One cannot define position smaller than one Planck length(about 1.6x10-35m), nor can one definetime duration smaller than one Planck time(time it takes a photon to travel one Planck length). So is a particle moving when it is wiggling within a cube where each side of the cube is one Planck length? There is no answer to that question. Can something happen quicker than one Planck time? There is no answer to that question because you cannot measure time less than a Planck time.That is the resolution Nature is willing to give.So, at the best nature can offer is: the object wentUPto within one Plank-length to the top, for a period of one Planck time, we don't know exactly what happened. By next Planck time, the object is movingdown. That is the theoretical best one can do, if one can indeed get that accurate.

author: Brumby
author_rank: Unknown
timestamp: July 10, 2021, 11:34:05 am
content: Quote from: TerraHertz on July 09, 2021, 11:30:19 pmI did think that as well, but considered the polar perspective was far enough.Quote from: Rick Law on July 10, 2021, 12:00:46 amAlso crossed my mind, but I constrained my journey down that track with:Define "stop".

author: Ian.M
author_rank: Super Contributor
timestamp: July 10, 2021, 12:11:23 pm
content: Over the at most couple of minutes*that one *COULD* put an object into a free fall trajectory in vacuo at or near sea level, treating the Earth's orbital motion (including its 'wobble' round the Earth - Moon barycenter) as constant velocity motion in a straight line is a reasonable simplification.* Deepest mineshafts are around 4Km which gives a free fall time from the top of slightly under half a minute. One could conceivably wait for such a mine to play out, lease the shaft, vacuum seal it and pump it down and conduct the experiment from the shaft bottom. You'd need a rifle to launch the projectile, and some mighty fancy doppler laser measuring gear at the top of the shaft!For convenience:https://www.omnicalculator.com/physics/free-fall

author: TimFox
author_rank: Super Contributor
timestamp: July 10, 2021, 02:20:38 pm
content: The late Prof U Fano showed me (and the other guys) how to do quantum-mechanical calculations of macroscopic problems. To be relevant to a large-scale question, one tracks the "expectation value" of the variable in question, which is well-defined in quantum mechanics.

author: TimFox
author_rank: Super Contributor
timestamp: July 10, 2021, 03:04:53 pm
content: Going back to the original simple question about the velocity at the top of the trajectory.Stripping away the practical complications, such as air resistance and the Earth's rotation, the question goes back to the nature of "infinitesimals", which Leibnitz and others discussed ca. 1700, as calculus was being invented.In modern mathematics, the question was answered by a rigorous theory of limits, applied to the definition of derivative in the calculus.A different example of limits: consider the functionf(x) = sin(x)/x, used in the reconstruction of a continuous analog waveform from discrete sampled data. (xin radians, of course)Whenx= 0, the function would be indeterminate (0/0 is not uniquely defined).However, applying proper limit theory (left as an exercise for the reader), the limit offasxgoes to zero (from either direction) is unity.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: July 10, 2021, 03:05:58 pm
content: One that's always stuck with me is calculating the quantum number of the Earth-Sun system (applying the parameters to the hydrogenic atom). N of course is uselessly large, the neat part is calculating the wavelength of the particle (i.e., graviton) emitted in a transition from N to N-1. Can you guess the wavelength?Tim

author: themadhippy
author_rank: Super Contributor
timestamp: July 10, 2021, 03:17:08 pm
content: Is the answer 42?

author: TimFox
author_rank: Super Contributor
timestamp: July 10, 2021, 03:56:53 pm
content: Probably (1/137) furlongs

author: vad
author_rank: Frequent Contributor
timestamp: July 10, 2021, 06:39:16 pm
content: 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.

author: dietert1
author_rank: Super Contributor
timestamp: July 10, 2021, 07:08:32 pm
content: 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

author: vad
author_rank: Frequent Contributor
timestamp: July 10, 2021, 07:41:04 pm
content: Quote from: dietert1 on July 10, 2021, 07:08:32 pmThere 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

author: TimFox
author_rank: Super Contributor
timestamp: July 10, 2021, 07:46:47 pm
content: 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, acontinuousvariable going from a positive value to a negative value, mustpass throughzero, even if it does not dwell at zero for a finite time.

author: GlennSprigg
author_rank: Super Contributor
timestamp: July 11, 2021, 11:29:57 am
content: 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??We generally agree, that to 'stop',shouldmean to be a rest for a known period of time... However!!!... Consider ALL the other speedsthat the object wastraversing throughwhile initially decelerating and then re-accelerating through??? It was NEVER at ANY of those speedsfor 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!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 aphysicaldistance between to points, as you can keep adding digits to the end,making it realistically ever so further, even thoughmicroscopic!! However, I went on to say that if youDOdraw such a triangle, with theoreticallyzero line thickness, then there the hypotenuse lies... exactly between those 2 actual physical points!!! If you grasp what I mean!?

author: vad
author_rank: Frequent Contributor
timestamp: July 11, 2021, 01:02:44 pm
content: Quote from: GlennSprigg on July 11, 2021, 11:29:57 amThis 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.

author: Brumby
author_rank: Unknown
timestamp: July 11, 2021, 01:07:45 pm
content: Quote from: vad on July 11, 2021, 01:02:44 pmYes, 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.QuoteAbsolutely correct!

author: TimFox
author_rank: Super Contributor
timestamp: July 11, 2021, 01:52:46 pm
content: Again, modern mathematics as taught in freshman analysis courses rigorously defines real numbers, including simple irrationals like root-2 and transcendental numbers likeeand pi.There is no need to re-fight 19th century battles.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: July 11, 2021, 05:04:54 pm
content: 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 defyany possibledescription, 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/462835They 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

author: TimFox
author_rank: Super Contributor
timestamp: July 11, 2021, 05:45:06 pm
content: 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".

author: magic
author_rank: Super Contributor
timestamp: July 11, 2021, 06:20:20 pm
content: T3sl4co1l is right, real numbers are a kludge invented to justify calculus after it had been handwaved into existence

author: vad
author_rank: Frequent Contributor
timestamp: July 11, 2021, 06:41:10 pm
content: 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.

author: dietert1
author_rank: Super Contributor
timestamp: July 11, 2021, 07:01:23 pm
content: 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

author: TimFox
author_rank: Super Contributor
timestamp: July 11, 2021, 07:38:03 pm
content: Quote from: magic on July 11, 2021, 06:20:20 pmLike 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 inPrincipia, 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".

author: jeffjmr
author_rank: Regular Contributor
timestamp: July 13, 2021, 01:17:56 am
content: 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… = x10x= 9.999…Subtract x from both sides9x=9X=1Mental gymnastics. Everyone knows .999… does not equal 1. Math exercises don’t always comply with reality.Cheers,Jeff

author: Rick Law
author_rank: Super Contributor
timestamp: July 13, 2021, 03:10:00 am
content: The OP's question is actually covered nicely byHeisenberg'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 theobject's position(at the "top") where theobject'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".

author: TimFox
author_rank: Super Contributor
timestamp: July 13, 2021, 03:19:49 am
content: 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.

author: LaserSteve
author_rank: Super Contributor
timestamp: July 13, 2021, 03:58:01 am
content: This is the EEV blog thread of the year! I'm in awe.

author: magic
author_rank: Super Contributor
timestamp: July 13, 2021, 04:57:34 am
content: Quote from: TimFox on July 11, 2021, 07:38:03 pmNo, 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 somewhereMakes a lot of senseOr, you said something about transfinite numbers, any progress on figuring out how many alephs there are before the continuum?Quote from: yet another mathematicianYes, I'm kidding somewhat. But that's because the whole thingisfunny

author: Nominal Animal
author_rank: Super Contributor
timestamp: July 13, 2021, 05:08:25 am
content: Thinking that there is no single moment when the object has zero velocity because moments havedurationis mixing completely different scales together ignoring their associated assumptions/contexts and replacing them with another, and getting garbage as a result.As stated, the onlysensibleanswer – 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 aconsistentset. That is the difference between anapproximationand 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 aboutscaleandprecision. 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.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: July 13, 2021, 01:01:25 pm
content: 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

author: TimFox
author_rank: Super Contributor
timestamp: July 13, 2021, 01:48:00 pm
content: 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.

author: aneevuser
author_rank: Frequent Contributor
timestamp: July 13, 2021, 04:55:00 pm
content: Quote from: TimFox on July 13, 2021, 01:48:00 pmWell, 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.

author: TimFox
author_rank: Super Contributor
timestamp: July 13, 2021, 05:17:25 pm
content: My undergraduate education required several specific mathematics courses to qualify for a physics major. The epsilon-delta method for defining and proving limits was an important part of that syllabus.I do not think "it's fair to say" that there are not many physicists who have studied formal definitions of limits or epsilon-delta arguments, even if our expertise and study is limited compared with that of professional mathematicians.

author: dietert1
author_rank: Super Contributor
timestamp: July 13, 2021, 05:49:01 pm
content: Absolutely. In fact, as you explained before, often enough physicists invented new calculus to model reality and only after that mathematicians got interested. Sometimes mathematicians invent entities that no physicist will ever need. But you can't know in advance. Sometimes instances of those constructs were found in nature later on. One example i remember are certain types of mathematical groups useful to discover families of fundamental particles. Hope that's correct.Regards, Dieter

author: TimFox
author_rank: Super Contributor
timestamp: July 13, 2021, 05:56:22 pm
content: This reminds me of my first year in graduate school, before I married, when I lived in a dorm with other graduate students in many different departments.One mathematics student, who was otherwise a nice guy, did not like that the Mathematics department was in the Division of Physical Sciences (along with Physics, Chemistry, etc.), since he considered Mathematics to be one of the humanities (that happened to be sometimes useful to us scientists). He was proud (and smug) that his two personal interests were of no use to science: differential geometry and topology. I had to disillusion him, since differential geometry is at the heart of Einstein's formulation of General Relativity, and topology was used in Feynman diagrams.I did not pursue particle physics, but group theory is very important to that discipline.(When Werner Heisenberg said to me, and the other thousand guys in the auditorium, that he saw no reason why particle physics should be less complicated than quantum chemistry, I lost interest in that field.)

author: aneevuser
author_rank: Frequent Contributor
timestamp: July 13, 2021, 06:19:21 pm
content: Quote from: TimFox on July 13, 2021, 05:17:25 pmIt'll vary from country to country, of course, and from university to university, and from precise course to course (and probably from decade to decade), but certainly in the UK, there are plenty of universities where a physicist won't formally study limits (and definitely won't study construction of the reals, or anything like that, which is certainly 19th C mathematics), and I suspect very few where the typical engineering student would formally study limits - they may have a brief intuitive overview of the idea of a limit, but most won't need anything more. Plenty will need to learn some "find the limit" techniques, of course.In terms of numbers having studied such stuff, physicists are something of an intermediate case though, between mathematicians and engineers. If I had to summarise, I'd say I wouldn't be surprised if a physicist told me that they had studied epsilon-delta proofs, and I would be surprised if an engineer had.

author: TimFox
author_rank: Super Contributor
timestamp: July 13, 2021, 06:38:42 pm
content: Your last sentence is a more reasonable statement than your earlier statement "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, ... 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"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. (Perhaps times have changed since 1967.) We did have one math-department course on "applied mathematics for scientists", but the other required courses were taken by science and math majors alike. My undergraduate college did not have an engineering school, but some physics students went on to engineering school after graduation. The epsilon-delta argument is not very difficult.

author: dietert1
author_rank: Super Contributor
timestamp: July 13, 2021, 07:01:59 pm
content: 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

author: TimFox
author_rank: Super Contributor
timestamp: July 13, 2021, 07:29:26 pm
content: 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.

author: Nominal Animal
author_rank: Super Contributor
timestamp: July 14, 2021, 03:38:13 am
content: Quote from: TimFox on July 13, 2021, 06:38:42 pmMy 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 onlysensibleapproach (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 sensibleinterpretationof 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. Theprecisionat which we can do so, is "just" an engineering problem.

author: magic
author_rank: Super Contributor
timestamp: July 14, 2021, 01:44:37 pm
content: 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 abonimationThis appears to complete the proof that you don't need to understand all thatto be a physicist for your whole life

author: TimFox
author_rank: Super Contributor
timestamp: July 14, 2021, 01:48:55 pm
content: 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.

author: magic
author_rank: Super Contributor
timestamp: July 14, 2021, 02:06:42 pm
content: I don't even remember what Lebesgue integrals were useful for that Riemann couldn't solve. Supposedly, a function is Riemann integrable iff it is bounded and discontinuous at most on a set of Lebesgue measure zero; that seems to cover all reasonable functions and many unreasonable ones as well.Now, the Lebesgue measure itself I remember as being an interesting and elegant concept, but the details of it are quite gone from my memory.

author: TimFox
author_rank: Super Contributor
timestamp: July 14, 2021, 03:38:19 pm
content: Yeah, that is how my education worked as well. All I remember about Lebesgue integrals is that one could reduce the Lebesgue integral to a Riemann integral that can be solved normally.Here's how I (now retired) look back on my formal education. (I had excellent teachers and professors, two of whom later won Nobel prizes due to no fault of mine.)Over the years, I encountered several things that I thought were "cool stuff", worth remembering. Here are a few examples:1. Formal theorem proofs, from definitions and axioms and other theorems, in modern treatments of Euclidean geometry.2. The epsilon-delta method for proving a limit (which I didn't realize would kick up such a ruckus in this thread).3. The method in Purcell's electricity and magnetism textbookhttps://archive.org/details/ElectricityAndMagnetismPurcell3rdEditiondemonstrating that the magnetic field from a current results from special-relativity effects on the charge carriers in motion.4. Variational principles, such as Fermat's principle of least time in optics and Lagrangian mechanics for systems with interesting degrees of freedom.I do not claim expertise in any of these areas, nor can I remember every jot and tittle of the proofs, but memory allows me to know what to look up in textbooks or online, should I encounter a relevant problem.

author: TimFox
author_rank: Super Contributor
timestamp: July 14, 2021, 05:33:24 pm
content: Let us go back to the original legitimate question, from July 9, that invoked the ancient problem of the “continuum”:“If you were able to throw an object 'exactly' vertical, such that it came back to you due to gravity, did the object ever 'Stop' at its apex?”,where Glenn Sprigg is concerned about the change from positive to negative velocity at the apex, and explicitly is not interested in astronomical scales or non-terrestrial situations.Here, I try to answer that question in three sections: Physics, Engineering, and Quibbling. I start with the velocity directly, since that is more direct than starting with the altitude. Applying this question to pop-up foul balls in baseball could lead us to a “Unified Infield Theory”. (For those who may not be familiar with baseball, that was a joke. Everything else below is serious.)Physics discussion:Consider the (idealized) one-dimensional vertical motion of an object in an inertial reference frame with a uniform gravitation field, where the velocity’s vertical componentv=v0> 0 att= 0, and we define the vertical positiony(altitude) att= 0 to bey= 0 . We do not specify how the object attained that velocity at that position and time, only that, fort> 0, there is no force on the object except gravity. The object goes up (v> 0) to an apex, then falls down (v< 0). The gravitational acceleration (downwards) is -g< 0, where the constantgis positive.Newtonian physics tells us that, in this model, gravity constantly decelerates the velocity:v(t) = dy/dt=v0–g twherev= 0 at a timet=tA=v0/g, and goes negative thereafter.Simple calculus tells us thaty(t) =v0t– (gt2/2) , where the constant of integration givesy(t) = 0 att= 0 ,and that, at timet=tA=v0/g, the apex altitude isyA=y(v0/g) = (1/2) (v02/g) ,which is a well-known result, but not needed to discuss the velocity. The altitudeyis a parabola (quadratic function of time).We need discuss only the velocity. In the first equation, the velocity is continuous and monotonic, and a simple linear function of time. Looking at the equation for velocity vs. time, we see that it goes continuously through zero.One could object that the variable does not remain at any value, including zero, for a finite amount of time. However, we define the average velocity at any timet0, over a finite symmetrical time interval 2s, as the mean value fromt= (t0–s) to (t0+s).Since the velocity is a linear function of time, the average velocity is independent of the interval length 2s, so long as the interval stays within the time range before the object hits the ground (proof left to the reader). Therefore, the limit of the average velocity assgoes to zero is a trivial computation (since the average velocity is independent ofs), equal tov(t) as in the equation above, including the value 0 at the apex (t=tA), where the velocity is positive before the apex and negative thereafter, with an average value of 0 (independent of the interval length).Engineering discussion:When higher accuracy is required, engineers modify the above ideal case by adding other details to the calculation, including other factors for the specific problem. Among others, these include:--Aerodynamics and air resistance. Boeing and General Motors have good software packages for computing these effects. Baseballs have interesting aerodynamic features due to their spin and external stitching. Wrigley Field is famous for winds off Lake Michigan.--Non-inertial reference frame (earth’s rotation underfoot). Baseball is often played at roughly 40 deg N latitude, and this effect can be added, along with deviations from a true vertical initial velocity.Quibbles:Here, I group phenomena that are physically real, but whose effects on the basic question are unmeasureable, especially with respect to uncertainties in the engineering corrections above. Examples include:--Planck distance. Trivial compared with the 74 mm diameter of a baseball: any effect is not measurable in this problem.--Non-uniform gravitational field. If the pop-up attains 50 m altitude, the variation ing, with the Earth’s radius of roughly 6400 km, is approximately 16 ppm.--Special relativity. This is trivial compared with the typical baseball velocity of 45 m/s.--Heisenberg’s Uncertainty Principle: left as an exercise for the reader.

author: CatalinaWOW
author_rank: Super Contributor
timestamp: July 15, 2021, 04:36:06 am
content: I think the way to put this question in perspective is to build on T3sl4co1l's observation that the velocity is never constant. Velocity can only be defined as a derivative or as an average over an interval.So in the sense of the OP's thought that the ball never stops, it never has any velocity at all. The most elusive thing there ever was, moving without velocity.If you look at things the wrong way you get very weird answers.

author: vad
author_rank: Frequent Contributor
timestamp: July 15, 2021, 10:32:31 am
content: Quote from: CatalinaWOW on July 15, 2021, 04:36:06 amWhat if we postulate the velocity, and define distance as integral of v dt? E.g. like SI defines meter.

author: CatalinaWOW
author_rank: Super Contributor
timestamp: July 15, 2021, 11:45:00 am
content: Quote from: vad on July 15, 2021, 10:32:31 amSure. An integral is an average. This can also be misinterpreted if you think of it incorrectly. (There exist velocity histories for which the integral is zero over some interval. The object is in the same place at the ends of the interval, hence there was no motion. This is analogous to the original posters "conundrum".)

author: TimFox
author_rank: Super Contributor
timestamp: July 15, 2021, 01:39:28 pm
content: In my long post above, I pointed out that for this particular case, where the velocity is a linear function of time, if you calculate the average velocity over a symmetrical interval around any time, the average is independent of the length of the interval.

author: GlennSprigg
author_rank: Super Contributor
timestamp: July 16, 2021, 11:23:40 am
content: O.P. here.... Oops!!, I think I opened Pandora's Box !!!