straight line because that's how physics works
Yes, according to physics teachers. However, there are many many clues for why this isn't so for real. Can you spot them?
Clue #1: anybody who has been to bowling may answer the question correctly, unless they're a physics teacher.
nope straight line
1) We have a ball and a tube, therefore we can assume that things like Van der Waals forces are in effect. Otherwise our ball and tube would vanish into a mist of atoms.
2) We are assuming the ball is moving due to the force of gravity and affected by the tube. In other words we're assuming Newtonian physics are in effect. We have inertia, centrifugal force, etc.
3) From 1,2 we can deduce that the ball must roll down the tube with friction, and its axis of spin must align with the rotational axis of the curved tube which is slightly tilted from the vertical. Therefore when it hits the table, it will instantly take on a curved trajectory like a bowling ball when given a spin.
Even if the table was oiled perfectly frictionless to prevent this, the Magnus effect would still make the ball curve as it travels through air. If you take out the air, you wouldn't be there to set the ball into motion in the first place, and the previously mentioned oil would boil into the vacuum. There's no reasonable way to make the ball not curve even slightly.
This trick question was actually used to show that peoples intuitions about physics fail to reflect reality. In reality it's the physics teachers' abstraction of the case that fails to account for the real phenomena that produce the intuitions that people have. People answer the question "wrong" because they observe baseballs, soccer balls, basket balls, bowling balls… etc. do curved paths in similar situations. People who have been taught physics answer the question "right" by the teacher's insistence that the ball does not spin and there is no friction.
And this is why engineers hate physicists.https://en.wikipedia.org/wiki/Spherical_cow>Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer, "I have the solution, but it works only in the case of spherical cows in a vacuum".
>>3575327>This trick question was actually used to show that peoples intuitions about physics fail to reflect reality.
except the riddle's "answer" is wrong, the ball would continue in a straight path. I knew exactly what the "trick" to this riddle was when I read it.
The ball, would in fact, go in a straight path and I don't have the patience to explain why - but i'll give you a hint, it has to due with a property of Pi
Nope. Geometry hasn't got anything to do with it. You're simply wrong.
Your answer shows that you didn't understand the question or the answer. The tube doesn't need to be exactly a half-circle. The crucial part is whether the ball spins or not when it rolls across the table, because that produces the curving path.
You could try to explain why "a property of Pi" would make a ball not spin when rolled down a curved tube, but you won't because you're simply trolling.
>>3575333>The tube doesn't need to be exactly a half-circle.
That's the exact thing that results in the straight path, actually. Any more obtuse an angle and it would curve.
Keep thinking about it and do NOT reply to me until you've burned at least 15 calories mulling it over.
>>3575334>Keep thinking about it
You're wasting time to troll. You know full well that any curve would make the ball spin.
>>3575335>You know full well that any curve would make the ball spin.>any
Unless a unique property of the half-circle caused this curve to act.. differently?
This is your last hint.
I'm obviously not talking about completely different curves or something esoteric. Roughly circular curves.
>Unless a unique property of the half-circle caused this curve to act.. differently?
It doesn't. When the ball moves inside the curved tube, it rolls due to friction. The curvature of the tube produces a centrifugal action which shifts the ball up the wall of the tube. Rolling up the wall forces the ball to roll around a tilted axis relative to the table. When the tube ends, the ball continues to spin with an off-horizontal axis and this makes it curve over the table.
There is nothing different about a half-circle in this respect.
>>3575342>I'd really just be solving it for you
You're the one presenting the wrong answer. You're supposed to prove yourself right, not me, burden of proof and all.
And as I already mentioned in brief: gyroscopic precession is also turning the axis of rotation of the ball to point up. Took me a while to remember where I saw the video though. https://youtu.be/tLMpdBjA2SU?t=154
When the ball enters the tube, it's rolling in with a horizontal axis. When it goes around the curve, it must change direction. This causes a precession torque that tries to lift the axis of rotation up - to align with the axis of the curve. Again, the exact shape of the curve doesn't matter - just the fact that you're going around at an accelerating rate, which is true since you're rolling down a slope. Both the centrifugal force moving the contact patch up the wall AND the gyroscopic precession conspire in the same way to put the axis of rotation of the ball pointing somewhere up as it exits the tube. It cannot come out rolling horizontally. If there is any reason related to "a property of the Pi" to explain why this isn't so, I'd be really interest to hear.
I didn't know what Lesswrong was. Idk what that last thing you said means either. You got the right answer, even though you didn't show why the 3 possibilities have equal probability.https://en.wikipedia.org/wiki/Sleeping_Beauty_problem>>3575317>O has information which is physically unattainable without breaking the speed of light
Wrong. If you know how far you are from a light source when you detect its emitted light pulse, it's trivial to work out when the pulse was emitted.>>3575318>The question cannot be solved.
Wrong again. It's a math problem; just treat them like points.
>The turn-around time is specified as zero.
Yes, this is how we know it's a math problem. Zero turn-around time implies infinite acceleration, which is physically impossible for massive objects.
>>3575816>If you know how far you are from a light source when you detect its emitted light pulse, it's trivial to work out when the pulse was emitted.
That information is still unknowable and undefined in the question. When the two lights flash, O' can measure a 17 ns difference in timing at any point where the observer is 17 light-nanoseconds further away from one source than the other. You get some kind of curve of possible locations that extends to plus and minus infinity away from the X axis. Relativistic corrections will twist the curve into different shapes, but it doesn't change the point that if O' can be any distance away from the X axis, they can be moving at any speed relative to O.
You didn't even define what units X is? Is it yards, feet, inches?
>>3575816>Yes, this is how we know it's a math problem.
The solution you're begging doesn't match the situation you're setting up. It's like saying "If you have three apples and I take away four, how many apples do you have?".
You cannot have less than no apples, nor can I take away more than there is, so no answer can satisfy the case. Either you end up with an impossible number of apples, or I fail to act according to the description of the situation. The very problem is not coherent and the sentence lacks meaning; it's not even a question.
This is why everyone hates mathematicians.
How to make a 5-sided square:https://youtu.be/n7GYYerlQWs>>3575831>When the two lights flash, O' can measure a 17 ns difference in timing at any point where the observer is 17 light-nanoseconds further away from one source than the other.
Now we're just going around in circles. I already explained here >>3572010
the meaning of the 17 ns, and yet you still think it's the time difference between when the pulses are detected. It's not.
>You didn't even define what units X is? Is it yards, feet, inches?>two sources of light at x = 0 and x = 10 m
See that small "m"? That means meters. Welcome to SI units. Welcome to science.>>3575833>This is why everyone hates mathematicians.
I thought that's why you hate physicists. I guess you hate all experts.
Pi is like a circle defining its surface with a line but no "line" is the curvature of a circle. All lines are squared to fit points while a circumference is algorithmic to its origination and orientation. While a line is flat it can be matched to an approximate deviate curve in length of point to point, while a circumference is contiguous of its value. Some people try to square the circle using an approximate math and it has always seemed to not be correct, there is a margin where consistencies fall off accuracies and is alot like how a drop-point inevitable reaches absolute value, the margin is nearly the same proportionately of a line to curve / curve to line, comparison. But that is like some quasi-math that doesn't exist, so quantum physics happens, things like thermodynamic and surface tension get involved, its very stupid.
There is a deritive inclusive .01% to any whole or something where pi just is what it is. But it takes all of those things for people to accept a number they picked out because of a flawed understanding, the understanding of pi is flawed if correalating to C=pi(2r), the radius is the all determining factor, and eventually the radius can be used to determine quadrants with the imaginary number bullshit. "Its such bullshit". Whether or not its squarable, imaginary, or some arbitrary number string, Pi is the reason why civilization has been continually occupied with itself for the last few milleniums. But infact its the same that it is outside as it is inside, a reciprocal of even more bullshit. Not the answer of everything but at tool or footnote for every aspiring Socratic Astronaut. When people start using moonstones on it and making it into something else you start opening up to Masonic Austism.
One of the most clarifying explanations ever!
Why Math Might Be Complete BShttps://www.youtube.com/watch?v=d9wfw_98NJ4
>>3575842>the meaning of the 17 ns, and yet you still think it's the time difference between when the pulses are detected. It's not.
Now you're just talking nonsense. The only thing the observer can do is observe when the light pulses arrive to where they are - otherwise you're talking about measuring something at a distance without a delay - magic.
>See that small "m"?
I see it now. What you need to see is that your question still makes no sense.
> I guess you hate all experts.
Sophism isn't expertise.
Math is fundamentally the examination of rules. You start by defining something, like the concept of a number, and then you go on to figure out what you can do with such things according to the definition.
The reason why mathematics seems to "work" for everything is because you can literally come up with any rule - by defining different things like the concept of an imaginary number, and finding that its behavior as defined matches some real phenomenon.
Or you pretend it does because it's a neat simplification, like the Fibonacci sequence, or the golden spiral, which is seen everywhere, even though it rarely actually matches the things people claim it does.http://web.sonoma.edu/Math/faculty/falbo/cmj123-134
For example, if you apply the Fibonacci series to rabbits, like in the video, what actually happens is that one is followed by zero, and two is followed by anything between 0 - 22 in the common case.
I'm gonna start explaining these problems one at a time, starting with >>3571574
The key to solving this is knowing what it means for two events to be statistically independent. Two events are independent if and only if their joint probability equals the product of their probabilities. I'm going to label the entries for convenience:
The probabilities are the variables divided by either their row totals or column totals. The joint probabilities are the variables divided by the grand total (240). Take entry a for example, its value determines the probability of making the first shot (a/180), the probability of making the second shot (a/192), and the joint probability of making the first and second shots (a/240). So by the independence condition, we get:
(a/192)(a/180) = a/240
solving for a:
a = (192*180)/240
This means that each entry can be found by simply multiplying its corresponding row total and column total and dividing by the grand total. This gives you:
No need to solve a system of equations.>>3575964
It's a special relativity problem, more specifically a relativity of simultaneity problem.https://en.wikipedia.org/wiki/Relativity_of_simultaneity
All the relevant information for solving the problem is on that page, but I'll fill in the details tomorrow. Maybe.
At first I 'tried' to solve this as a system of linear equations but ran into a problem. I think some equations were linear combinations of others and you can't get unique solutions when that happens.
If O sits at x=5m, immobile relative to the light sources, that means in his and the lights' frame of reference the lights blinked simultaneously. There's no other way or reason this would be the outcome of the observation.
The light takes 17ns to cover 5 meters, so O' sitting immobile at x=2.5m will observe the difference of 17ns. That's one possible solution, dubious considering "O' is moving". so let's find a working one that requires O' moving.
Now speed of light remains the same in all frames of reference but distance between the lights changes with speed of O'.
There's infinity of solutions for various speed and x position of O' in between x=[0,10] as combination of Lorenz contraction and difference in distance between the two lights, but let's find one outside [0,10]. O' speed must be such that distance between the lights contracts to 5m. They still blink simultaneously but light from the nearer will arrive 17ns before light from the farther one. This comes to 259,628km/s, where Lorentz contraction is 50%.
Let me add this is, along with the text, "according to second observer". From O frame of reference they will reach O' a long time apart, and for O to observe O' being hit by the lights 17ns apart the pretty obvious solution is O' traveling towards the lights at 0.5c.
The problem says O' is moving parallel to the X axis, but it doesn't specify at what distance Y to the axis - only that O' is traveling in the same direction - so the number of solutions is actually infinite and the problem doesn't have an answer.
For every possible speed in the direction of X, you can find a location on X-Y where the two lights appear to be 17 light-nanoseconds apart.
Ah. *cringe*. True, depending on position and distance… Poorly stated problem, if it was 'along the X axis' it would be less ambiguous.
The key to solving this problem is a solid grasp of special relativity. First, let me address an issue that seemed to come up repeatedly. Regarding this line:
>According to a second observer O', moving at a constant speed parallel to the x-axis, one source of light flashes 17 ns before the other.
A core feature of special relativity is that there are no preferred reference frames. O' is just as valid as O. O' is smart. He can tell he's moving with respect to the two light sources. Or he could say the light sources are moving with respect to him. Anyway, the point is he knows that the time between when he actually measures the two light pulses isn't necessarily the time interval between when they were emitted. There's a very simple way for O' to determine when exactly a light pulse was emitted: take the time he measured it and subtract (his distance from the light source)/c. That is,
t_e' = t_m' - d'/c
where t_e' is the time the pulse was emitted, t_m' is the time the pulse was measured, d' the the distance from where the measurement took place to where the pulse was emitted all in the O' frame of reference. c is the speed of light. Using the above equation, he finds t_e' for both sources, notices that they are different, and subtracts them to get 17 ns. This 17 ns is the time between when the pulses were actually emitted
in the O' frame of reference.
This is odd, isn't it? Events that were simultaneous according to O were not according to O'. But this is how reality works, known as relativity of simultaneity
Simultaneity is not an absolute.
So with that out of the way, how do
you solve the problem? Well, we need a way of relating events that happen in one reference frame to those in another. A coordinate transformation. Special relativity provides this with what's called the Lorentz transformation
, which has it's own section in the previous linked article. The Lorentz transformation is four equations, but we only need one. Which one?
Well, what the knowns and unknowns? Let's set the light pulse that happened first according to O' as the origin of both reference frames. That way I don't have to type out a bunch of deltas. With that in mind, the original problem gives us for the second event: x = 10 m, t = 0, t' = 17 ns. We are told to solve for v. The Lorentz transformation for time contains all these variables, so all we need to do is plug in the known variables and solve for v. I'm actually going to solve for v/c, which is the fraction of the speed of light O' is going relative to O. See attached image.
Dropping the negative since we're concerned with speed and plugging in the remaining two variables yields:
v/c = 1/sqrt(1+(10 m/(3*10^8 m/s * 17*10^-9 s))^2) = 0.454
O' is traveling at a speed of 0.454 c relative to O. That's pretty damn fast.
the knowns and unknowns
So simultaneity is something that counts as both a thing, so that's like saying this over here and this over there is the same. That's using subsets dood.
Known's and Unknown's in simultaneity is like a giving the technical yes-no on-off status of same shit-different story complex.
So a lorentz transformation is basically schroedinger's cousin. That somehow you have something you aren't sure of and you actually would know what it is. Almost like you are proving casualty. Since one formula says you don't know and this one says you do know….hmmm.
Hypocritical Math. Oh yes, this will upset those number buffs.
Edit for Proof:
0/0=0*0 (no imaginary required)
You don't have something just because you say it is a different value.
I.E. in the case of special relativity and light or time travel. Reference to Laws of conversation and how energy cannot be created or destroyed. Unless in the case its applied, for instance by a constant if you are in quantum physics. Or maybe some portal or wormhole since it would be tasked with time.
Basically Having any Polynomial, calculate the difference quotient.
Just look up how to do this here. But is given by:
Example : f(x)=x2+3x+5.
To find f(x+h), plug x+h instead of x:
So that f(x)=x2+3x+5 looks like f(x+h)=(h+x)2+3(h+x)+5
then go back to the that with
f(x+h)−f(x)h f(x+h)=(h+x)2+3(h+x)+5 - x2+3x+5
———— = ———————————-
then simplify. f(x)=x2+3x+5 is h+2x+3.
This is in algebra somewhere.
Shortly after that is linear algebra and in linear albegra that uses linear equalities. Basically where you have to know this to know. (There is alot of falloff here.)
But that is not my point. In Linear Equalities the Solving difference quotient is "helpful" in simplifying solutions, Solutions that are later substituted as secondhand Quantum-Equations. (You would already know this if you studied rocket science.)
This basically breaks down to a standard quantum formula(look it up), but can be used to define that the original polynomial (going back a couple of diagrams or so, can be rewritten again.
As a Set of
Polynomial=Undefined (without casualty)
Polynomial = Nullset(1) (with casualty)
(i know i need citation here i dont have it)
The problem is using polynomials it is implied the solution is real. When infact, it is not KNOWN to be real or imaginary. All polynomials are in "in good faith". Don't forget that. So to express the polynomial as real or imaginary there with respect to time we can use the diagram depicted. That it is either real or imaginary, or rational, or irrational. Express through i,-i,1,-1.
(Don't ask me for the answer I dropped out around that point. But right now I have a pretty good proud going)
So that you "might" know if Casualty is Inclusive or In Exclusion to its current Quantum State. In theory anyway. I.E. Don't ask your Prof if its a bad day, not worth the risk.
My opinion is that quantum comptuers can just do this up for us cuz my phone's on loan.
This would also look good in logic based programming where
If True, Then Object Is.
If False, then Object Is.
If Object is False then Imposter is true.
If Object is True then Imposter is false.
so that Polynomial of
f(x)=x2+3x+5 is true. Then f(x)=h+2x+3 is true. If Object is True.
Is generally accepted. But.
f(x)=x2+3x+5 is true. Then f(x)=h+2x+3 is false. If Object is False.
f(x)=x2+3x+5 Is Imposter. (with casualty)
f(x)=x2+3x+5 Is Object (without casualty)
So then Nullset(1) and Nullset(2) are quantum entangled and can be written as a single quantum formula(look it up) which can prove quantum logic. (Because the nullset configured for the quotient or its counterpart)
Using true and false turns of the same sets would then its define unknowns. Have that with all of the references, you could rewrite a quadratic formula with respect to time.
Would be the idea of Lorentz Model.
And once you have one quadratic formula from having a Lorentz Model you can in theory measure two different Quantum particles with proof of Casualty. It would plot any knowns or unknowns as being entangled for most of it and look cool.
Have all integers plotted of the set, forms the bulk of the absolute value and any result proves its determinant. So the absolute value is measurable and apart from the original value. Fits accordingly in scale, that a measurable data holds retention in respect to time, or proves retrocausality. Or "futurepast" or the world we see is the past or an approximation.
Because f(x)=x2+3x+5 is f(x)=h+2x+3 then any linear change occurs as quantum waveform because of the isolated absolute values of them. That means an account for a .01 virtual percentile or something like the Planck Constant.
Except the origin would be misplaced, something about singularity. You could also work the quotient as a source anchor. A 'fringe' anchor.
I botched the explanation in my previous reply, so I just deleted it. Here's take two.
For those who missed the last reply, I admitted I hadn't tried to solve the system of equations. But Sorrowful is right that it's impossible, because the equations are linear combinations of each other. I tried to explain how this came about from the independence constraint, but upon further reflection I realize that's not the case.
The real reason it's impossible to find a unique solution to the system of equations is because of how the equations are obtained. If we frame the problem in the language and notation of linear algebra, we get the matrix equation in pic related. Matrix M is always going to be the same no matter the vector on the right hand side of the equation. Because this matrix cannot be inverted (the matrix is singular), a unique solution cannot be found no matter what row totals or column totals are given in the problem or whether or not independent events are being described. This is because of the invertible matrix theorem:https://en.wikipedia.org/wiki/Invertible_matrix#The_invertible_matrix_theorem
This means you cannot use a system of linear equations to find entries in a table by using row and column totals. Well, except for the trivial case where your table is just one entry.
If they instead flipped the problem and asked if the events are independent given the table entries, then you can just turn the table into a matrix and take the determinate (that's what det means in the image). If the determinate is zero, you can conclude the events are independent. Otherwise, the events are dependent. From the solution to the problem, the determinant of the 2x2 matrix is ad - bc = (144)(12)-(36)(48) = 0. So this means the events are independent, which we already knew.
In this latter case of the 2x2 matrix, the determinant equaling zero for independent events is equivalent to what's known as the odds ratio equaling one. Can you see why?https://en.wikipedia.org/wiki/Contingency_table#Odds_ratiohttps://en.wikipedia.org/wiki/Determinant
Where have I seen this before?
Just consider my "experiement" autistic tripe for developmental use, if anything. The way I look at bootes void is as some universal pressure, I do not think it is any different than visible dark matter, dark matter is supposed to be visible under extreme conditions, sort of like overcast weather.
It is because of the minutely scoped measurements (which are considered extreme by human observation and comparison to other cosmic anamoly) that makes it seems unreal. But it is actually very standard in newtonian laws, dark matter is like a sublime property of the universe not an alternate dimensional type thing.) Sort of like the primer for other particles to situate themselves on. The expanse of that area is like having nothing but that, things can be there just at a different condition due to a gravity blockade or something space-agey.
Humoring the idea of traversing a space-time continuum, such as the bootes void, I think that a long range wormhole is not feasible without the construction of multiple gates, this would require obviously alot of resources, and drones to do such with since we would also have to program them to function blindly in an unmonitorable enviroment. At best.
There would be the chance it would require not only that operation but also a feeding rail to do so, where resources are just dumped per location within the void. Not feasible. But having lock points within the field on a targeting system can in theory generate a path through it, the path however may be broke or in fluctuation. You would have to counteract the need of redirecting routes.
In space, velocity is a real problem because it happens on very many levels to an object relative to its trajectory, it is not a single direction when particles are operating in a quantum motion. You would require a quantum rangefinder plus a targeting system for your waypoint, which sounds easy enough. If you had it. Basically a quantum Comptroller.
Then you would have to also be able to manage your traversal under critical mass through super massive speeds which would turn you into a black hole. You would need some sort of free-energy machine to also stabilize your propulsion as the rest is being used for "onboard rangefinding" and waypoint targeting system which cannot fail. If any go out your velocity would go all over the place and you would basically departiculize your transport into nothing or be lost or unable to move. You would need all three on all the time due to unmonitorable hyperspace, plus you would be a miniature black hole at the same time.
So it would be possible except it would also be broken with many detours and very turbulent especially if you lose your velocity tracking. You would be stranded in it. The only way to do this is through a proposed spiral propulsion using some sort of toroidal coil. Which they have already, because only a spiral can feed energy seamlessly into itself without disrupting itself linearly. Course its just all conjecture.
The problem is how to sustain the powerdraw. In a continuum, power is disrupted by the travelling, we have to ensure that a generator can adapt to hyperspace, such as a dark matter generator or a higgs-hadron collider. Either would generate a quantum pulse of energy (either by the allowance of dark matter or the breaking down of matter into dark matter) Ideally, a quantum compatible source of energy will carry over in hyperspace instead of having power loss through unprocessable travel.
Warp-travel is movement that is based on a curve in space time to simulate kinetic inertia, the curve is made through sustain quantum energy, but is "routed" all over the place during quantum tunnelling, it cannot be adequately traced. When it is directed, a "wormhole" can then shift in space-time, then act as a flexible sieve through an open plane in outer space, though it is dangerous to pass through other objects for instance if the tunnel were cut short you would "phase" into obstacles like a particle meatball or just get smacked, not sure which. While warp gates can do this, having a warp drive to shut off by input or engine braking like a switch is important. (which is like what tesla coils would do I guess.) Though your ship would need a pilot or a program or an ai to do that correctly, anyway.
The warp curve is like simulating a ball rolling down a hill, that accelerates very fast depending on the power used, but dark matter must sustain itself through any fallout of the curve.
Touch Up to Newtonian Physics:
1.Objects at rest remain at a rest (indicative of inertia, so while in the space of the warp field it will act upon the inertia of any object)
2 The orientation of the warp field might help with the push, depending on how or when it is used in leiu of its course. The orientation might be more important than usual. (Angular diffusion that would have to be adjusted for like a paper airplane is designed for forward flight)
3.Required sustained energy or power draw hopefully from a quantum engine source.
4.Warp drive makes a flexible sieve through a plane where reorientation of quantum travel might be neccessary (or normalization or dewarping) through power mitigation. (Basically imagine refolding yourself once you are a crumpled piece of paper)
I really dont have anything else to add, I just didn't want to leave out any holes in the idea since the chance someone will do it because they read it online.
In maethmatics the property of math is usually defined as interchangeable but the translation of that result may or may not be the same value as the actual value is being represented. This is one of the first philosophical problems that comes in math. So to quantum mechanics being physically approximated of the original math; this develops into further complications where congruents are required to the original values of the integer.
When all due singularity(or balance of any displacement), these approximations come together to form an otherwise expressed sum of the original parts. This forms a group of the entire state of physics, which is important in mechanics to know due to simulteneity. The group is usually based on meta, congruents, and other variants. Or basically a "slieghted" control panel to the system, shows both quantum and non quantum sides as a virtual simultaneity of the group, and rounds the edges of entire 'state' by approximating alot of it "together" in a represented matrice or formula. This helps with sorting invalid and valid results (or some Euclidean context).
Because of approximation the less accurate it is and because of stuff like probability and casualty it is very "exclusive" to the original expression. However the intent is to summarize any loophole or oversight it may have, and therefore give a better description of the math. Though the simplicity always is boiled down to whether or not 2+2 apples = 4 apples or some only 1 apple exists at a time fallacy. It is still important in rocket science especially when the notation of a half life is prefered, but impossible to correlate except with quotients or some lineation of quotients.
In actual particle physics, usually any proofing involves a previously rendered state and then charting it on a "curve" or some math jargon to dissect it all for an "increment" within or of the state. Such as ex.:(if center mass is shifted by x-amount then state of LineAb changes to some proximity angle or coordinate or stressed or invalid function or temporarily gains casualty.) Basically this helps to seal the deal between quantum and non-quantum states especially for Smart stuff like Morse Theory or Golden Ratios Or Point Curves. So glad for summer.
So in an isolated system such as described, you would have a state of integers described, barring any modulated, baselining, or gentrifying of the state it can be compared to another state. This comparison is simplified down into an balance where any definition can is indistinguisable so that you could have, 2x=1x+1 suddenly make sense or something completely inane. Like quantum function(where the function has two sides or determinates and has to satisfy it to chart anything, which is what all of this is geared to do). And curves literally be silly straws.
The theory would suggest anynway, that an isolated product is term. Terms are reverse engineered too based on their work(so writing it out is important but you know, fat chance). When two are availiable they can be compared in approximation, these simplify too and are useful for imaginary half lifes and imaginary particles like nuetrinos. In math there is a chain blocking or something where terms occur (usually they just farm the second terms from the first but a related equation helps so much to have independent comparisons). If the approximations can be considered true, when no other answer is availiable it suggests an isolated instance. I don't know I think its useful in predicting thermal dynamics with particle diffraction or something. But its really stupid and no one cares and something like "flipping the diode is involve" which is code for jerking around.