Why does striking two balls of equal mass and equal magnitude of velocity turn the system to rest?

Question

In the answer by Ron Maimon in this stack post, he writes:

if you smack two identical clay balls of mass $m$ moving with velocity $v$ head-on into each other, both balls stop, by symmetry. The result is that each acts as a wall for the other, and you must get an amount of heating equal to $2m E(v)$.

What exactly is this symmetry that he speaks of?

If you do momentum conservation then you can get that the total momentum of system must be zero but I'm not sure how you could deduce that each clay ball must have velocity of zero after collision.

Answer 1

The answer you are referring to must be interpreted in the following way:

By saying 'clay ball' the author means, that the two objects combine into one single object after the collision (That means we have a completely inelastic process). So before the collision, you have two objects with mass $m$ and momenta $\vec p_1=-\vec p_2$ so before the collision, you have a total momentum of $$\vec p_{tot}=\vec p_1+\vec p_2=0$$ after the collision you have an object of mass $2m$ which has to have a momentum due to momentum conservation of $p_{tot}$ as $$p_{after}=p_{before}=p_{tot}$$

After the collision, the combined balls do not move anymore so the kinetic energy has to be converted into 'other forms of energy' (binding energy) of the particles as energy is conserved.

The 'symmetry' answer you are referring to is the fact that the center of mass momentum is zero therefore when combining everything to a single particle this particle can not have any momentum.

Answer 2

The symmetry argument is as follows. Suppose we invert the initial scenario about the midpoint between the two balls. If the combined ball after collision has a velocity vector $\vec v$ then in the inverted scenario it will have a velocity vector $- \vec v$. But because the clay balls are identical and have equal and opposite initial velocities, the original scenario and the inverted scenario are identical (this is the symmetry). Therefore $\vec v = - \vec v$ (because Newtonian mechanics is deterministic so the same scenario cannot have two different outcomes) which means that $\vec v = 0$.

Answer 3

If in an inelastic collision particle A has a velocity magnitude greater than that of particle B, then the velocity of the combined particle will be non zero and in the direction of A before the collision. If B has the greater velocity magnitude then the combined particle will a non zero velocity in the direction of B. In the example given where A and B have equal velocity magnitudes, the combined particle will have a velocity of zero as there is now no preferred direction. This is because of the symmetry of the velocities (equal magnitude) in this particular reference frame. This is the symmetry he speaks of. The reference frame that has symmetry has particles with equal momentum magnitudes and is called the centre of mass (COM) reference frame and if the particles have equal mass (as in this case) the COM reference frame is the one where both particles have equal velocity magnitude. In the COM frame the total momentum is always zero. For an inelastic collision the particles stick together and the final velocity is always zero in the COM frame. If the collision is elastic, the particles just change the sign of their velocity after the collision, but the total momentum is always zero in either case.