Really basic mechanics and Galilean Relativity question

Question

Consider two solid objects: A and B.

System 1:

A        <----- B
          10m/s

System 2:

A ----->        B
  10m/s

They are identical except for the reference frame.

Common relativity tells us they are equivalent and will evolve in the same way, right?

Now add masses: A weights 1kg and B weighs 10kg.

In system 1, B will hit A with 500J of kinetic energy. In system 2, A will hit B with 50J of kinetic energy.

So... the energy involved in the impact (and effects such as how far debris is flung) depends on what reference frame I'm in? This doesn't seem right.

How do we resolve this?

Answer 1

I think that your intuition is leading you astray here; KE and momentum are frame dependent quantities. To get a handle on, let's say, the violence of a collision, use the center of momentum (COM) frame of reference.

Let's use an absurd example to see how your intuition might mislead you. Consider the collision of some low-mass particle with an arbitrarily massive one. In the massive particle's frame, the low-mass particle approaches at a very slow speed and thus, has little KE and momentum. The collision is barely a kiss.

However, in the low-mass particle's frame, the KE and momentum of the massive particle is arbitrarily large!

The resolution to this "paradox" is, as mentioned before, to go to the COM frame. In this frame, the particles have equal and opposite momentum. In the example above, the COM frame effectively is the massive particle's frame.

In the COM frame of your example, particles A & B have KE of 41.3J and 4.13J respectively. Note that the total energy is less than the 50J in reference frame A. In fact, it is the case that the total energy of the system is minimum in the COM frame.

Answer 2

Yes the energy, and indeed the momentum is frame dependent (though conservation of energy and momentum applies whatever the frame i.e. the energy/momentum before the collision is the same as after it).

It would seem that when B hits A the collision must involve a lot more energy because B initially has a higher kinetic energy. This is true, but in the frame where B hits A, A ends up moving to the left with a high kinetic energy, so the "extra" energy has gone into making A move. In the frame where A hits B the total kinetic energy is intially lower but because B ends up moving more slowly after the collision, a greater proportion of the initial kinetic energy can be absorbed in the collision. The amount of energy dissipated in the collision will be the same in both frames.