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A.T.

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With zero duration for each.they are falling through an infinite amount of equipotential surfaces, all with values equal to 9.8.

But the changes in speed are infinitesimally small.And an infinite amount of non-infinitesimally small numbers is infinity.

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The changes in speed can be measured, ...right?

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You could say that for any process that has a definite, non-infinitesimal, magnitude, that the changes are infinitesimally smallWith zero duration for each.

But the changes in speed are infinitesimally small.

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You could use this to refute the basic postulate of quantum theory, that all changes take place over infinitesimally small amounts of time.With zero duration for each.

But the changes in speed are infinitesimally small.

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Sounds to me like you need to check out Zeno's Paradox

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Even if the changes are infinitesimally small, my argument still makes sense, because if things were continuous, then acceleration would increase exponentially, because the faster you fall, the faster you gain speed, and the faster you fall. so acceleration would increase and velocity would increase exponentially. The only way around this is through quantization.You could use this to refute the basic postulate of quantum theory, that all changes take place over infinitesimally small amounts of time.

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That all changes take place over an infinitesimally small amount of time is just an idea that quantum mechanics just negates, without explanation. They take the opposite to be true, as an axiom.You could use this to refute the basic postulate of quantum theory, that all changes take place over infinitesimally small amounts of time.

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This just follows human logic, but galilean acceleration is an axiom, not based on logic. If it were then the ancients would have discovered that all objects gain speed uniformly as they fall to earth.Even if the changes are infinitesimally small, my argument still makes sense, because if things were continuous, then acceleration would increase exponentially, because the faster you fall, the faster you gain speed, and the faster you fall. so acceleration would increase and velocity would increase exponentially. The only way around this is through quantization.

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If they all have the same value then it will gain no speed. You are confusing the potential with the acceleration of gravity. The gravitational potential decreases as you approach the Earth. Your confusion seem to be on several levels though.

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Dale

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Almost nothing about this argument is correct. First, the value of equipotential surfaces would be measured in units of energy or energy/mass, not in units of acceleration. Second, the equipotential surfaces do not have equal values. Third, if the equipotential surfaces did have equal values then there would be no acceleration since the change in KE is proportional to the change in the potential. Fourth, when falling through unequal equipotential surfaces the potential change between neighboring surfaces is infinitesimal, not non-infinitesimal.

No, the argument is pure nonsense, it is already wrong on many levels. Infinitesimally small changes would indeed debunk your argument if other aspects did not already invalidate it. Further, you make yet another wrong argument here: not all infinitesimal changes lead to exponential functions when integrated.Even if the changes are infinitesimally small, my argument still makes sense, because if things were continuous, then acceleration would increase exponentially, because the faster you fall, the faster you gain speed, and the faster you fall. so acceleration would increase and velocity would increase exponentially.

Wow! Yet another wrong statement. Quantization is not an axiom of QM, it is a derived result of the axioms. In QM you can easily derive systems that are not quantized, or systems that are quantized over certain ranges and not over others, like a particle in a finite potential well.That all changes take place over an infinitesimally small amount of time is just an idea that quantum mechanics just negates, without explanation. They take the opposite to be true, as an axiom.

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This is equivalent to the archer's paradox. Paradoxes are such because they point to an error in reasoning. You led with yours, though actually yours was not directly relevant to the paradox you then stated (answer is dx -> 0). The answer to the question in your title is: Because they are. We know this because of 'the ultraviolet catastrophe'.

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Nugatory

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That is not a postulate of quantum theory.You could use this to refute the basic postulate of quantum theory, that all changes take place over infinitesimally small amounts of time.

As others have pointed out, quantum mechanics is not needed to understand continuous motion; Zeno's paradox was resolved long befopre the modern development of quantum mechanics. Indeed, the normal progression of learning quantum mechanics is to to start with Schrodinger's equation in the position basis, in which position is a continuous variable.

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