# Why Is Momentum Calculated by Multiplying Mass Times Velocity?

1. Oct 1, 2012

### vanmaiden

Good Afternoon,

How is it that we are able to get the value of momentum from multiplying mass times the velocity of a moving body? A lot of people settle for knowing how to get momentum, but I want to know why this equation works. I understand as much calculus as calc II permits me with a bit of vector calculus and linear algebra as a fair warning on my level of the comprehension of your explanations. I thank you in advance for helping me.

Vanmaiden

2. Oct 1, 2012

is momentum not simply defined as mass * velocity?

3. Oct 1, 2012

### vanmaiden

Possibly. Is there not a reason that we know of that can explain why we can combine mass and velocity in a multiplicative fashion to yield momentum?

4. Oct 1, 2012

### Muphrid

People identified that mass x velocity was a useful quantity. They gave this quantity the name momentum.

Actually, it took quite a while for people to realize that "mass x velocity" was the quantity they should be looking at or to call this momentum instead of a host of other names, but at core, this is a definition. It is the name we give to the thing.

Perhaps you thought there was some deeper definition of momentum?

5. Oct 1, 2012

### vanmaiden

Yeah, I thought there was. I should have asked it a different way actually. How does the mass times the velocity of an object yield the characteristic we've named momentum?

6. Oct 1, 2012

### soothsayer

But if momentum has no meaning outside of it being mass x velocity, how is it that we have determined different forms of momentum for different branches of physics? p = mγv for SR, and momentum is an operator stated by -ih∇ in quantum mechanics. Momentum may be defined as mv in classical mechanics, but is it not more general than that?

$d\mathbf{p} = \int_{t_1}^{t_2}\mathbf{F}\cdot dt$

almost seems a little more general. In which case, vanmaiden, it's easy to derive p=mv from that as well, if you feel more satisfied with that explanation. Of course, force is not a super well-defined concept in some branches of physics either...

7. Oct 1, 2012

### Ben Niehoff

Momentum is important because it is a quantity that is conserved in the absence of external forces.

Consider a 1-dimensional collision between unequal-mass objects. Looking at the velocities of each object both before and after collision, you can deduce experimentally that there is a conserved quantity which is proportional to mass x velocity. That quantity ended up with the name "momentum" (although historically it had many other names).

Conserved quantities are important to solving the equations of motion, to figure out what happens in a given system. As you know from 1-dimensional collisions, you start by writing down the two conservation laws: momentum, and energy. These two equations give you exactly enough information to solve for the two final velocities.

8. Oct 1, 2012

### Muphrid

Sure. Part of what I'm trying to get at is whether vanmaiden has some more fundamental notion of momentum in mind and is wondering how that can be shown to reduce to $mv$ in classical physics or some such. But without an idea of what that more fundamental notion is that vanmaiden may have in mind, it's difficult for me to simplify it down to $mv$.

One thing that is unifying regardless of exact discipline is the notion of an action principle. There, one gets the general idea of position and mass-velocity being conjugate variables, and this is one way to identify mass-velocity as a useful quantity and to decide to give it a name (momentum). From an action principle, one can derive all the properties of momentum that are familiar, even in quantum mechanics or relativity.

9. Oct 1, 2012

### Staff: Mentor

Also, the fact that this quantity is conserved can be derived from Newton's Third Law of Motion.

Including the name that Newton gave it: "quantity of motion." He considered it to be a fundamental quantity, which he defined before stating his laws of motion.

10. Oct 1, 2012

### vanmaiden

Is it fair to say the equation was obtained through experimentation and not so much mathematical manipulation?

11. Oct 1, 2012

### Woopydalan

Keep in mind at one point Isaac Newton thought mass times velocity was kinetic energy.

12. Oct 2, 2012

### MrRobotoToo

Because mass times velocity is a conserved quantity resulting from the space translation invariance of a system's Lagrangian.