On what ground do you multiply $m$ with $v$ in the momentum equation $p=mv$?

Question

I've read several other posts that says the momentum equation is the definition of momentum, and it has no proof. However, I would like to know what is the experimental observation where the multiplication of m = mass and v=velocity is justified.

In other words, what is the meaning of this multiplication of 2 physical quantities of mass and velocity?

There is no need to justify a definition. It's just made for convenience --- as are all definitions. We know that $F=ma$ (which is the definition of $F$, and $a=dv/dt$, so $F= dp/dt$. — mike stone, Apr 28 '22 at 21:03
Related question: Why is momentum defined as mass times velocity? — Thomas Fritsch, Apr 28 '22 at 21:05
@mikestone so the multiplication here is arbitrary? and I can replace multiplication with division? — John Sall, Apr 28 '22 at 21:06
If you want to define some quantity other than momentum you can certainly form $z=m/v$. It may b useful for something..... All defintions are made as a convenient packaging of something. For example the cosine is defined as the adjacent over the hypotenuse. — mike stone, Apr 28 '22 at 21:09
@mikestone if it's a mere definition then isn't all of classical physics that is based on the momentum equation become a tautology? — John Sall, Apr 28 '22 at 21:10
It's simply a word . You can always start from $F=ma$ and never mention momentum. Not having the $p$ word defined in your vocabulary means only that it will require longer describe things like collisions. As for a tautology, all mathematics is at heart tautology. — mike stone, Apr 28 '22 at 21:15
Does this answer your question? Why is momentum defined as mass times velocity? — BowlOfRed, Apr 28 '22 at 21:22
@BowlOfRed no, it wasn't useful. There was no accepted answer even in that post. — John Sall, Apr 28 '22 at 21:24
In the Principia, the first three definitions are of the mass, the momentum, and the inertial mass; the “three laws” are presented in the next chapter as axioms. — rob, Apr 28 '22 at 21:39

Dale · Answer 1 · 2022-04-28T21:18:05.297

1

Strictly speaking, a definition doesn’t need justification. However, some definitions are useful and important, and it is relatively easy to show why they are useful. In the case of momentum, basically all of the experimental observations that confirm Newton’s third law justify using $\vec p=m\vec v$ as an important quantity.

If Newton’s 3rd law holds then for any isolated mechanical system the quantity $m \vec v$ is constant, or conserved, regardless of the details. This is a very useful property and so it makes sense to give it a specific name and traditional symbol.

Observations supporting Newton’s 3rd law include (in no particular order):

Recoil of a gun

Rocket thrust

Newton’s cradle

Collisions

Etc.

edited Apr 28 '22 at 21:18

answered Apr 28 '22 at 21:12

Dale

99,825

Would you mind to explain simply how momentum is related to Newton's third law? – John Sall Apr 28 '22 at 21:25
Need the second law, too, since the third doesn't mention mass. But second and third taken together demand momentum conservation. – John Doty Apr 28 '22 at 21:41

On what ground do you multiply $m$ with $v$ in the momentum equation $p=mv$?

1 Answers1