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Just to recap, Rayquaza flew into space, smashed open a six-mile wide meteor, fought Deoxys so I could capture it, and flew back to Earth, all with a rider on its back who wasn't even injured when Rayquaza smashed through the meteor. I'm not much of a scientist, but this seems a little impossible.

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I've never thought about it this way but this makes so much sense ;P
I'm guessing Rayquaza puts a protective layer against the player or something.
Thanks for supporting my answer sumwun and GammaDragon425. These people are too afraid to face the reality.
Thanks for speling my name wright.
Your welcome.

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Ok, here's a serious answer.
First off, you are talking about a fictional game called Pokemon, where Pokemon can direct electricity, use telekinesis, teleport and produce matter out of nowhere. Pokemon breaks so many laws of physics in the real world it's ridiculous to even correlate the two worlds. However, I'm up to the challenge.

The first law Rayquaza breaks is Newton's Law of Gravitation. Rayquaza weighs a reasonably heavy 206.5 kg, and its basal surface area is too low to allow it to float in the air, since no upthrust is provided in the form of visible wings, propellers or such. Since Rayquaza is approximately a metre off the ground, it should take it a tenth of a second to fall down like the pathetic snake it is.

Next is in regards to its move Dragon Pulse, which violates the law of conservation of energy, as well as Einstein's General theory of Relativity. Dragon Pulse emanates energy from Rayquaza's gaping mouth, which should not be possible since energy can't be formed instantaneously without the help of a nuclear reaction, which obviously doesn't take place. The alternative is that Rayquaza draws energy of some form from its surroundings and directs it towards the opponent, which is again not possible.

Rayquaza's ability Air Lock violates the law of common sense, you just can't make weather disappear. Does Rayquaza absorb sunlight? Hold an umbrella? Wear shades to protect against sand? We would never know.

Next Rayquaza fits into a bloody Pokeball, which violates the Quantum mechanical model of the atom and the Valence Bond theory if you assume that Rayquaza shrinks in size to fit into the ball, since nuclear forces become unstable at such close proximity, along with e-e repulsion since atomic orbitals shall overlap. This problem could be solved if instead of being composed of hydrocarbons like regular living creatures, Rayquaza is made of inert material, in which case Rayquaza's regular form should be gaseous.
Or the above mentioned law of conservation of energy and Einstein's General theory of Relativity if it is converted into energy or data.

Mega evolution violates the law of conservation of matter, since Mega Rayquaza weighs almost double of what Rayquaza weighs. It violates a few laws of biology as well, but you asked only for physical laws.

Quite surprisingly, the scene of Rayquaza smashing into the asteroid is quite possible within the laws of physics. Since we don't know the hardness of Rayquaza's mouth blades, the only law broken here is the law of conservation of momentum because the fragments of rock upon it breaking move sideways rather than forward to maintain constant momentum. This is if it is an elastic collision which I assume it to be, since Rayquaza collided with it headfirst and contact was maintained for a very small duration of time and Rayquaza was moving pretty fast.
Yeah and since the player is human, air resistance and friction with the help of gravity should have been enough to tear open his suit and throw the idiot back to earth, what he deserves for riding the magic snake into space. This violates the concept of viscosity and friction, not exactly laws but more like definitions.

Rayquaza shouldn't be able to stop after hitting the meteor, since space has virtually no matter and any effort Rayquaza makes to stop should have no effect since internal forces have no effect on a system of objects. Again not a theory, but a Postulate based on common sense.

Deoxys violates a few laws as well, but the question does not ask that.

The answer in brief? Darn too many.

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I have taken like 1 or 2 years of physics, so I can attempt to out-nerd you when you talk about Newton mechanics but not things like thermodynamics or quantum physics.
There's no way you can assume anything here is elastic. Rayquaza smashed the meteor, which I'm pretty sure requires turning kinetic energy into heat. To decrease kinetic energy while conserving momentum, Rayquaza should slow down, but continue to move forward and accelerate some of the meteor pieces forward. The other pieces can move sideways or backward as long as they slow down a bit. Then again, decreasing kinetic energy isn't the best assumption either, because Rayquaza might have smashed the meteor using the power of MAGIC instead of its own kinetic energy.
Heat is only emitted in inelastic collisions. At high speeds the impulse is large thus collisions tend towards being more elastic.
Even assuming that the collision was inelastic, the net momentum of the bodies should be the same even after collision. Look at the animation, Rayquaza passed through the rock with nearly the same velocity with which it was heading towards it, while the rock stopped in place and all the fragments scattered sideways. Rayquaza should have had no realistic means to stop.
I get that the in-game animation is stupid. That's not what I'm arguing here. Do you agree that in order to conserve momentum, Rayquaza should slow down (but not stop), and some of the meteor pieces can fly sideways or other random directions?
Yes, I agree with you. In fact, it is possible for Rayquaza to completely stop too. But the animation defines the game, if all animation is dismissed as 'stupid', we can't really apply any laws of physics anymore.