Momentum Definition: Inertia, Classical & Quantum

[syang9]

i have been struggling to come up with a qualitative definition of momentum. for example, i can say that an object's inertia is a measure of it's resistivity to changes in velocity. are inertia and momentum (in the classical sense) the same thing? what about in quantum mechanics?

 

[D H]

"Inertia" per se is not a measurable quantity. This http://en.wikipedia.org/wiki/Inertia" nails it just about right:

Inertia is a non-quantifiable property of matter by which it remains at rest or in uniform motion in the same straight line unless acted upon by some external force.

Newton, by the way, did not use the word "inertia" in his writings. The word is too colloquial.

When you say

i can say that an object's inertia is a measure of it's resistivity to changes in velocity.

what you are really talking about is mass, not momentum. Momentum is the product of mass and velocity. Think about it this way: Which would you rather stop with a baseball glove -- a baseball going 60 miles per hour, or a Mac truck going 60 miles per hour?

 

[Jax]

To add to what DH said, you can think of an object's momentum as "how difficult it is to stop."

 

[syang9]

so then momentum is kind of like.. a more comprehensive measure of an object's resistance to changes in velocity (or acceleration), right? since two objects can have the same mass but different momenta.

 

[D H]

That's correct. In fact, that is precisely what Newton's second law of motion says. An external force is needed to change the momentum, and the rate at which the momentum changes is proportional to the force. This means there is some constant of proportionality that relates forces and changes in momentum. In mathematical form,

$$\vec F = k \frac {d\vec p}{dt}$$

where $\vec F$ and $\vec p$ are the force and momentum vectors respectively and $k$ is some constant of proportionality. In the metric system of units, the units of mass, length, and time are intentionally defined to make that constant of proportionality equal to one.

In the case that an object's mass is constant, change in momentum is proportional to change in velocity, and the constant of proportionality is the object's mass. Thus for a object with constant mass, Newton's second law can be expressed as

$$\vec F = m\frac {d\vec v}{dt} = m\vec a$$

This is the form of Newton second law with which you have probably familiar.

Note that I dropped the constant of proportionality here -- I am assuming a system of units such as the metric system where the constant of proportionality is one. The more general form is

$$\vec F = k m\vec a$$

and this is the form you need to use if you measure force in pounds-force, mass in pounds-mass, and acceleration in feet/second/second (in which case $k=1/32.1740486\,\text{lbf}\;\text{s}^2/\text{ft}/\text{lbm}$).