0

Today, I was watching a video, and the man on the video said that we only "accept" $F=ma$. Then said that no one can prove $F=ma$. But I know we can derive $F=ma$ by momentum with calculus. I am confused, so do we only accept $F=ma$, or do we know the proof?

Qmechanic
  • 201,751
Bora
  • 65
  • Try to link the video if you are referencing to it. –  May 11 '20 at 14:30
  • Very related: https://physics.stackexchange.com/questions/70186/are-newtons-laws-of-motion-laws-or-definitions-of-force-and-mass –  May 11 '20 at 14:32
  • 1
    What are your starting assumptions? You can't talk about proofs without stating your starting point. – BioPhysicist May 11 '20 at 14:54
  • 2
    Without an explanation of what assumptions you expect people to prove $F=ma$ from, it's not clear what sort of answer you want for this question. – ACuriousMind May 11 '20 at 14:57
  • I see guys, I understand. I apologize for my amateurship in physics even I haven't yet understand what you mean by assumptions. Thank you. @ACuriousMind – Bora May 11 '20 at 17:43
  • @Bora I would say it is more of a math idea than a physics idea. All proofs have assumptions behind them. Sometimes Newton's laws are taken as assumptions, so there is no proof. However, there are other formulations of mechanics (e.g. Lagrangian mechanics) where you start with a different set of assumptions, and then you can prove that Newton's second law arises from (can be proved by) these assumptions. Therefore, without telling us where you are starting from in your assumptions, there is no answer to if there is a proof or not. It all depends on what you start with as "true" or "given" – BioPhysicist May 11 '20 at 17:59
  • @AaronStevens I would say Newton's laws are taken as postulates in Newtonian mechanics, not assumptions. Also, Newton's laws are such succinct beasts that it is hard to be comfortable saying that they are taken as postulates even in Newtonian mechanics. They are more complicated, in the sense that they represent a jumble of postulates, assumptions, and lemmas within them: https://physics.stackexchange.com/a/340890/20427 –  May 11 '20 at 18:13
  • 1
    @DvijD.C. I am speaking a little lose here. By assumption I mean postulate. I think the rest of your comment is accurate, and shows why more work needs to be done on this question. – BioPhysicist May 11 '20 at 18:18

1 Answers1

2

Classical mechanics is a whole theory, it's not defined by an arbitrary equation you write on a piece of paper that means nothing at all.

By theory, I mean a whole "worldview", containing primitive concepts such as the concept of space, time, mass and force, and principles/relationship about those (say, the absoluteness of space and time, galilean invariance, etc). The equation of motion F = ma is a principle too, hence why it's called a "law".

Now, in mathematics the concept of proof is something that is more or less well defined: given some "worldview" (the latter necessarily comes with an existential axiom, as in, there exists something and this something is satisfying, by decree, the following properties), and given some logic/inference rules (telling you how truth is actually preserved), you can prove if some claim about these objects are true or false. Obviously, the hard part is to chose a worldview that is consistent and useful, as you have to be very careful with this choice if you want to produce something of value.

The same is true in physics. You cannot prove a theory wrong, and even less right, because that's not what they are for: they are here to make sense of some phenomenon, like the concept of number or velocity. Is velocity "true" or "false"? It doesn't make any sense to ask this question, and the same is true for classcial mechanics, or F = ma (which can, by the way, be justified extensively).

Note that even in relativity, the "moral" of F = ma, which can be understood as "absolute motion or change is equivalent to acceleration" is entirely preserved, even if the details of the equation changes a little bit, the philosophy of it never did for the past 400 years.

sure
  • 1,014