3 ms·

Did you see my response to the parent, where I give a sketch of how conservation of momentum follows directly from conservation of energy in all frames of refer

• cynicalkane 10y ago
  Remove either conservation law and work out the equations for an elastic collision, and you'll see a new degree of freedom pop up. In classical mechanics they are logically independent, and since the space is affine, frames of reference don't help you. Your sketch takes as a premise, something that is actually a consequence of conservation laws: the way quantities change under frame-of-reference changes. You can get both by looking at the least action principle under frame of reference changes and using Noether's theorem, but that's not what your sketch is about.
  • tprice7 10y ago
    Is there some objective sense in which conservation of momentum is more fundamental than conservation of energy in a different reference frame? Since either can be derived from the other, how do you determine which is "actually a consequence" of which? (I ask out of genuine curiosity)
    • cynicalkane 10y ago
      Well, what is fundamental isn't the point. If you imagine a world where different quantities are conserved, there would no doubt be different change-of-frame equations describing how they are. The point is you can imagine a world where energy is conserved but momentum is not. Actually it's possible to imagine physical systems, with affine frames of reference, where nothing is conserved at all. Fundementality is a different thing, but it's more natural to assume conservation laws and take frame-of-reference equations are a consequence. You can do it the other way (I'm pretty sure, but not 100% sure because my physics math is rusty) but you're still making assumptions, and you still don't have one conservation law following from the other.
  • akiselev 10y ago
    You've got it backwards: in an elastic collision, kinetic energy is conserved by definition. Removing one conservation law makes the whole exercise physically meaningless because elastic collisions require as a given that kinetic energy will not be converted to other forms of energy. This premise guarantees that the momentum is treated as a quantity separate from the total energy of the system (which, as a reply to your first post pointed out, when expressed in relativistic units, includes not just the energy-mass equivalency but energy-momentum as well). In reality, momentum in almost every collision will be converted into other forms of energy such as when intramolecular bonds are broken and materials are deformed in a car accident or when momentum is converted into mass in particle colliders. The elastic collision equations are only useful as a rough approximation precisely because they go out of their way to simplify the energy-momentum equivalence to conservation of kinetic energy just like Newtonian mechanics is a simpler but useful rough approximation of relativity at low energies. You can't use elastic collisions to reason about energy-momentum equivalence because their most basic assumption is that there is no energy-momentum equivalence.