Understanding Momentum: Definition, Intuition, and Properties

[Dadface]

This just goes to show that E and px,py,px are not independent quantities but components of the same thing.

In some contexts "not independant but components of the same thing" perhaps but they are not the same.Momentum is momentum and energy is energy.
Please read and take into account sophiecentaurs post 47 above.

 

[_PJ_]

Momentum is the massive component of the kinetic energy of a body when relative motion is considered.

 

[Studiot]

Momentum is the massive component of the kinetic energy of a body when relative motion is considered

Would you like to elaborate on this?
I sort of guess that you are identifying the kinetic energy as the dot product of two vectors viz the momentum vector and the velocity vector. But where does the factor of one half come in?

 

[_PJ_]

The 'other half' (I'm, guessing you mean as in KE=0.5*m*v*v) is for the remaining velocity contribution. It's from the integration of v

 

[Studiot]

Perhaps a derivation?

 

[Delta Kilo]

In some contexts "not independant but components of the same thing" perhaps but they are not the same.Momentum is momentum and energy is energy.

I was answering specific question. And I clarified it as "same but in different dimensions". And I did add Relativistic Caveat as requested, didn't I?

Please read and take into account sophiecentaurs post 47 above.

I understand the need to keep things simple, but I'm against dumbing them down. For, example this "down-to-earth" explanation looks sort-of-alright. But it only works in a very specific case where the force is constant and is known in advance. This case is in fact very rare as we don't usually know the forces and they tend not to stay constant. The example with vehicle is in fact incorrect because braking force depends on the weight of the car, so if you try measuring momentum of the car with the same initial speed but different loads you will not get expected results.

And I can't think of any typical school or college-type problem where this kind of "intuition" would be helpful. In fact, most of the time when you deal with momentum, the forces are unknown, like the proverbial billiard balls or rocket equation, and you just use conservation law directly.

On the other hand, it would be very nice to grasp the connection between momentum conservation and Newton's laws of motion as soon as possible. It doesn't require anything beyond school programme to see the that p=mv and F=ma are basically telling us the same thing, but it makes everything so much neater.

It is also worthwhile to mention that momentum conservation comes directly from translational symmetry, i.e. the fact that laws of physics do not depend on position in space (without going into details of Noether theorem of course). It just wraps it up nicely.

 

[sophiecentaur]

I agree with most of what you've written except the phrase "dumbing them down". It isn't dumbing down to establish a good understanding of things at a very worldly level. Believe it or not, there are many people for whom SR is and will always be MAGIC. They may be able to reiterate the consequences of traveling at near c but the associated sums and beyond are just too much. Nontheless, many of these same people can do calculations involving colliding cars and snooker balls.
The A level Mechanics level is a kind of 'base camp' from which a few hardy souls can scale the mountain.
Talking of dumbing down- I used to kick my Sons' arses when they would wind me up with "Momentum is speed times weight". They still do, at the age of 40.

 

[James_Harford]

I understand the need to keep things simple, but I'm against dumbing them down. For, example this "down-to-earth" explanation looks sort-of-alright. But it only works in a very specific case where the force is constant and is known in advance. This case is in fact very rare...

It is not really "dumbed down" if it is correct in its domain. Nor is it just a matter of keeping things simple, but of starting out simple if it can be properly related to everyday experience, and then elaborating step by step upon that insight.

For example, since the intuitive description happens to occur along a line, it can be formulated as (1) Δp=FΔt and (2) ΔE = FΔx, where F is constant. Since Δt is any nonzero value. This includes as a special case,

(1) dp = Fdt
(2) dE = Fdx.

which answers your first objection, and so on...