Conservation of Momentum = Conservation of Energy?

[Islam Hassan]

Energy and momentum only differ by a factor of V/2. So in essence, at a fundamental level, does this mean that the conservation of energy is the conservation of momentum in disguise?

IH

 

[atyy]

Momentum is a vector, while energy is a scalar. There are elastic collisions where both are conserved, while only momentum is conserved in inelastic collisions. In inelastic collisions, energy and momentum are both carried away by particles we don't explicitly track. However, momentum is carried away equally in all directions, so that what is lost in one direction is "added back" by the loss in the opposite direction.

 

[Bill_K]

However, momentum is carried away equally in all directions, so that what is lost in one direction is "added back" by the loss in the opposite direction.

Please explain what you mean by this.

 

[Vanadium 50]

It is also not true that "Energy and momentum only differ by a factor of V/2" - that's kinetic energy, not total energy. Kinetic energy is not necessarily conserved.

 

[atyy]

Please explain what you mean by this.

I meant that in an inelastic collision, energy is lost as heat, say as radiation. This removes energy, but does not remove net momentum, even though radiation carries momentum, because the heat loss is non-directional.

 

[Islam Hassan]

Momentum is a vector, while energy is a scalar. There are elastic collisions where both are conserved, while only momentum is conserved in inelastic collisions. In inelastic collisions, energy and momentum are both carried away by particles we don't explicitly track. However, momentum is carried away equally in all directions, so that what is lost in one direction is "added back" by the loss in the opposite direction.

What about at atomic/particle physics level where *everything* is kinetic: heat, sound, etc. At this level, do the two conservation laws of energy and momentum essentially repeat the same message?

IH

 

[vanhees71]

No they don't. While momentum conservation follows, by definition, from spatial translation invariance since momentum is the conserved quantity of this symmetry, energy conservation is due to temporal translation invariance since energy is the corresponding conserved quantity of this symmetry.

A system might not be space-translation invariant (e.g., a particle in an external potential) but time-translation invariant (if the potential is time independent). Then energy is conserved but momentum isn't.

 

[atyy]

What about at atomic/particle physics level where *everything* is kinetic: heat, sound, etc. At this level, do the two conservation laws of energy and momentum essentially repeat the same message?

IH

Not everything is kinetic energy. Fundamentally, momentum is always conserved, but only total energy, not kinetic energy is always conserved. But to see where your intuition holds partially, let's restrict ourselves to collisions that conserve momentum and kinetic energy. Let's say we don't observe all the outgoing particles, and we find that we are apparently missing some momentum. This does mean that we will also be apparently missing some kinetic energy. However, we cannot infer the apparent kinetic energy loss purely from the apparent momentum loss and conservation of momentum, without using conservation of kinetic energy. For example, if there are 3 missing particles, two of them may be identical particles moving in opposite directions, carrying away kinetic energy without removing net momentum, while the third carries away both kinetic energy and momentum.

 

[BruceW]

vanhees71 has got the right answer.
For example, a classical object in a constant (external) gravitational field. In this case, the z component of momentum is not conserved, but the x,y components of momentum are conserved. And since the potential doesn't explicitly depend on time, energy is also conserved.

 

[matonski]

What about at atomic/particle physics level where *everything* is kinetic: heat, sound, etc. At this level, do the two conservation laws of energy and momentum essentially repeat the same message?

IH

At the atomic/particle physics level, there is still potential energy.