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Generally, when an inelastic collision occurs, the nonrelativistic energy of two observable objects is not conserved because some energy has been transformed into other forms such as heat and sound. I understand the fact that the energy of the entire isolated system is conserved if the total mass of the system is a constant. However, why should not some momentum of the two objects transform to the air molecules to generate airflow and sound?

Please note that I am asking the reason why momentum is conserved while energy is not conserved in terms of the collision of two observable objects.

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    $\begingroup$ Possible duplicate of Why is momentum conserved in inelastic collisions? How is it related to momentum-impulse theory? $\endgroup$ – BioPhysicist Oct 29 '19 at 22:52
  • $\begingroup$ Thank you for answering but why is energy not conserved here? $\endgroup$ – Leo Liu Oct 29 '19 at 22:57
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    $\begingroup$ From what I can tell, OP isn't really asking the same question here. They are asking why momentum is conserved if the process generates sound/airflow, which carry momentum. $\endgroup$ – JMac Oct 29 '19 at 23:21
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    $\begingroup$ Momentum is directionally dependent, are you arguing that more airflow and sound is going in certain directions more than others that isn't radially balanced? $\endgroup$ – user234190 Oct 29 '19 at 23:21
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    $\begingroup$ Energy is conserved even in an inelastic collisions. The quatity that is not conserved is the macroscopic kinetic energy (and there is no general conservation rule for that quantity). In other words elastic collisions are a very special sub-set of interactions in which no energy is diverted from the macroscopic kinetic channel. $\endgroup$ – dmckee --- ex-moderator kitten Oct 29 '19 at 23:30
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There are two things you've mentioned, airflow and sound. They both apply differently.

Any net directional airflow caused by the objects will be accounted for by the drag force acting on the objects as the travel towards each other. This momentum is lost from the moving objects the entire time they move, and will depend on size, shape, and velocity. It does slow the objects down, changing their momentum.

The sound will carry momentum; but because of it's nature, it will extend radially from the colliding objects. The momentum isn't focused in any particular direction, so the net momentum change should be negligible for the final velocity of the balls.

That momentum is generated from the energy lost. I would say you could very loosely compare it to a rocket. In a rocket the momentum is conserved when you consider the rocket and the fuel; but the energy which causes the rocket and fuel to separate comes from the chemical energy of the fuel reaction. Energy changes form to create new particle movement (such as air or fuel movement), but the net momentum is still conserved when you consider everything affected and the directions.

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You are correct that momentum of the two objects will change if there are extra forces involved in the collision. For example:

  • If one of the colliding objects is a rubber bladder full of air, the collision will force the air out and will push the objects to one direction, gaining momentum.

  • If the colliding objects are elementary particles that emit a photon due to the collision, the photon will carry some momentum.

But in both of those cases, the whole system includes anything that gets emitted from the collision. After the collision, there aren't just the two objects, but also a bunch of air molecules leaving in various directions. The momentum over all the objects involved will be conserved.

Further:

  • If the air flow leaving the rubber bladder hits other air, it will slow down.

  • If the collision causes directional sound to be emitted, the sound waves can carry a small amount of momentum.

In these cases, there isn't a single collision, but multiple collisions. Sound is propagated by many elastic collisions between air molecules, which each conserve momentum and also energy. Thus the system that needs to be considered expands as the sound waves move outwards. Over the whole system, momentum is still conserved.

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You are correct in that sound waves would carry momentum. However, there are three reasons why it's not relevant for the results of the inelastic collision.

The first is that the inelastic collision is a theoretical model and therefore happens in the famous frictionless vacuum. In fact, the model doesn't include any consideration about where the energy goes which disappears during the collision. It's simply assumed to be lost. A friend of mine once did the force integration approach of modelling a collision and could show that an inelastic collision is equal to assuming that at the point where the two centers of mass are nearest the elastic force stops working.

So where does the energy go? This leads us to our second reason. Where exactly it goes is a complicated solid state physics rabbit hole, but basically, it at first plastically deforms the object (by overcoming the attraction between its constituent molecules) and from there decays into heat. You can have other channels, like the sound you mentioned, but all of these can be assumed for the purpose of the excercise to be isotropic, meaning they give off the same energy in all directions. This means they are momentum-free, because the momentum carried by the sound wave travelling in one direction is exactly cancelled by the momentum of the sound wave travelling in the opposite direction. In total, no momentum is carried, and thus we can assume the inelastic collision to preserve momentum on its macroscopic scale.

And thirdly, even if our sound is not isotropic, and therefore carries some momentum, as momentum is proportional to mass, and the mass of the air carrying the sound is consideraly smaller than that of the two colliding objects (and their speed not enough orders of magnitude larger to cancel that), the momentum loss would be below 1% of the total, which is negilible.

Additionally, both the elastic and inelastic collision are theoretical models meant to show students the possible spectrum of results of two objects hitting each other. In reality, no collision is purely elastic or inelastic, but always somewhere in between. They are taught not because they're perfect models of reality, but because their assumptions makes the math really easy. You can calculate more complicated collisions (even including parts flying off and carrying momentum), but that math gets horrible quickly.

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  • $\begingroup$ Thank you very much! $\endgroup$ – Leo Liu Oct 30 '19 at 16:31

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