I am just an absolute beginner to physics. I've seen a proof of the formula for momentum using Newton's second law of motion, but to prove Newton's second law of motion you have to use the formula for momentum. You see ? There is a circular reasoning here and I am so confused.
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2$\begingroup$ That is not proved. Prior to Newtonian mechanics, the concept of momentum is non-existant. We are defining what momentum is. If you read Newton's Principia, you will see it prominently written as a definition before N2L is even written down. $\endgroup$– naturallyInconsistentCommented Dec 13, 2023 at 6:38
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$\begingroup$ What do you mean by "prove the formula for momentum" ? To prove that $p=mv$ ? If so, it's not provable, some things in Physics are just definitions, like axioms in math which also is not provable. It's like if I would ask you "prove" our star name "Sun". Or "prove" that acceleration is ${dv}/{dt}$. Such questions makes no sense. $\endgroup$– Agnius VasiliauskasCommented Dec 13, 2023 at 7:38
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3 Answers
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Physics does not prove its formulas.
Basically, they are mathematical models that fit our experiments. I.e. we do some experiments and collect numbers, and then we see some regularity in the numbers and come up with a formula that matches the numbers.
Of course, our experiments aren't independent, so any new formula must not contradict the results from other ones we already had (if this happens, we have to think again much harder and eventually come up with a completely new model, e.g. Einstein's relativity).
So, in a physics context, "to prove" a formula can only mean to find out that this very formula can be derived as logical conclusion from other ones. So, it's not a proof of the formula itself, but a proof that it must be true if some other given ones are true.
The "holy grail" of physics is to find a basic, minimum set of formulas that "completely" explains the physical world, a set that allows us to "prove" all other formulas from this minimum set.
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$\begingroup$ While in general your post is correct,- there are cases in Physics when mathematical proof IS needed, for example proving that some wave solution comes from generic wave equation, or that some state equation comes from momentum/energy conservation law and etc. So in general, if you want to derive something from basic premises,- you need a strict mathematical steps, which basically is a proof that solution follows from assumptions. I.e. mathematical part also exist in Physics, while deriving theories or solving Physics problems. $\endgroup$ Commented Dec 13, 2023 at 15:50
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Conservation of momentum( linear and angular) is taken as granted. Law of conservation of linear momentum and angular momentum doesn't need proofs.( At least in classical physics)
This conversation cannot be violated.
We get it by observing universe. Physical laws don't change at any time or place.
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I assume that you are talking about the formula,
$\vec{F}= \frac{d\vec{p}}{dt}$,
where symbols have their usual meaning.
This formula is just the Newton's second law written in mathematical form. Force, acceleration and momentum is defined in such a way that it satisfies this equation and therefore, the law must hold true.