Why is momentum equal to mass times velocity?

[logicgate]

Homework Statement

Why is momentum equal mass times velocity and nothing else such as mass plus velocity or mass times velocity squared or cubed ?

Relevant Equations

p = mv

I tried searching on the internet for hours to find an answer, but I didn't find any.

 

[DaveC426913]

For starters, science does not attempt to answer 'why' questions; it simply attempts to build models that are predictive.

p=mv because that is what we observe.

There's another way of looking at the question. It plays with what you might observe if p were actually equal to m+v, and what weirdness you'd observe, but we can go into that in a bit.

 

[Steve4Physics]

Homework Statement: Why is momentum equal mass times velocity and nothing else such as mass plus velocity

Note that you can't add values for different types of things. You can add 4kg to 5kg to get 9kg. But you can't add 4kg to 5m/s to make something - it doesn't make sense.

(But you can multiply and divide different things sometimes. E.g. 10 metres divided by 2 seconds gives you a speed of 5m/s.)

or mass times velocity squared or cubed ?
Relevant Equations: p = mv

I tried searching on the internet for hours to find an answer, but I didn't find any.

The rules here (see https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/) require that you provide some evidence of your own effort before we offer guidance. We won't simply answer your h/w for you.

Be prepared to use a bit of ingenuity when you search. I searched for 'why is momentum useful?' and found some good links. Try it for yourself. You'll need to read through a few links and think about the ideas. Then you should be able to put the information together to answer the question.

If you are still struggling, tell us what you have found and any ideas you have.

 

[kuruman]

Homework Statement: Why is momentum equal mass times velocity and nothing else such as mass plus velocity or mass times velocity squared or cubed ?
Relevant Equations: p = mv

I tried searching on the internet for hours to find an answer, but I didn't find any.

Quantity ##mv## is a definition of a product of two quantities which is useful for describing nature mathematically. It is given the name momentum. Quantity ##mv^2## is another useful quantity which is twice the kinetic energy. You are free to define gronk as ##\frac{1}{\pi}mv^3## but that would not be very useful if you want to describe nature mathematically.

 

[DaveC426913]

Note that you can't add values for different types of things. You can add 4kg to 5kg to get 9kg. But you can't add 4kg to 5m/s to make something - it doesn't make sense.

Way better answer than mine.

 

[jack action]

As @DaveC426913 mentioned, it is what was observed.

The first mathematical concept was done by Buridan, after studying work done a few hundred years before him, who referred to impetus as being proportional to the weight times the speed. (Not mass and velocity.)

Then, a few centuries later, Descartes claimed that the total quantity of motion in the universe is conserved. Quantity of motion was understood as the product of size and speed. (Again, not mass and velocity.)

A few decades later, Newton finally zeroed in on defining quantity of motion as "arising from the velocity and quantity of matter conjointly" specifying:

The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. ... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight.

Which is the modern definition of momentum, i.e. mass times velocity.

I found this info on Wikipedia on pages about Momentum and Mass.

So the keyword here is really observation.

 

[haruspex]

As others have noted, that momentum is mass x velocity is a matter of definition. The question you should ask is why it is interesting. The clearest answer is that is that it is governed by a conservation law. This is what drove the various attempts to define it listed in post #6.

 

[PeroK]

As others have noted, that momentum is mass x velocity is a matter of definition. The question you should ask is why it is interesting. The clearest answer is that is that it is governed by a conservation law.

@logicgate The conservation of momentum follows from Newton's second and third laws. If two bodies of mass ##m_1## and ##m_2## interact, then the third law says that the force that ##m_2## exerts on ##m_1## (let's call this ##F##) is equal and opposite of the force that ##m_1## exerts on ##m_2## (this would be ##-F##). The second law says that the force equals mass time acceleration. Therefore:$$m_1a_1 = F = -m_2a_2$$Now, acceleration is the time derivative of velocity, and as mass is conserved, we have:
$$\frac{d}{dt}(m_1v_1) = m_1a_1 = -m_2a_2 = -\frac{d}{dt}(m_2v_2)$$Rearranging this equation we see that:
$$\frac{d}{dt}(m_1v_1 + m_2v_2) = 0$$Now ##m_1v_1 + m_2v_2## is precisely the momentum of the two-body system. And, we see that this quantity does not change over time, no matter how the two masses interact.

As a corollary, this conservation applies to any closed system of ##n## particles:
$$\frac{d}{dt}(m_1v_1 + m_2v_2 + \dots + m_nv_n) = 0$$And that's what we mean by conservation of momentum.

Note that the interaction could be gravitational, electromagnetic or a direct collision. In all cases, Newton's laws imply that the total momentum of the closed system is conserved.