newtonian mechanics - Equal and opposite reaction - forces coming in pairs - or just common sense?

I've never understood why the conservation of momentum law is taught in schools as "every force has an equal opposite reaction".

To me a gun's recoil is self explanatory; the explosion sits in-between the bullet and the back of the barrel and the gun and bullet fly in opposite directions (as would be expected). The bullet has a much higher velocity than the gun because of its lower comparative mass. Am I wrong here?

Same with stepping off a boat. You push against the boat and the mass differential between your body and the boat as well as the friction in the water enables you to jump off the boat onto the ground without falling on your face. The boat comparatively moves in the opposite direction minimally, but not because of some weird law, simply because you pushed it back!

To tell kids in school "every force has an equal and opposite force" is misleading I think. It makes it seem as if an equal mysterious force manifests out of thin air to keep the universe happy.

Just thinking - "direction of application" is only relevant to perspective, there really is no direction of application since the force and its counterpart occur simultaneously… so more accurately:

• The total force can never be propagated along a single vector?

Answer

First of all, mathematical definitions of force and momentum aren't really very intuitive or common-sensical. Just ask Aristotle for his common sense laws of forces! The fact that momentum is conserved in closed systems is a highly non-trivial fact, as is the Third Law. The reason that these laws exist at all is because you can't really 'see' or' feel' exactly what forces and momentum are referring to: they are ONLY mathematical constructions to make sense of the world. So no, this isn't common sense. Honestly, even if it were, 'common sense' cannot account for quantitative aspects of physics at all, so formal mathematical constructions are still essential.

Now, "any force applied in one direction is split in the opposite direction" would be far more misleading. This is because forces are not 'split'. Instead, they are exactly what the Third law says they are: TWO different forces, acting on TWO different objects. Otherwise, I promise you, your math is wrong.

If you still have any questions, talk back and let me know.