wonderings

here's something that's been bothering me for a long time:

suppose you have two balls a and b. ball a weighs one kilogram and is moving two meters per second. ball b weighs two kilos and is stationary. if ball a strikes ball b and stops completely, how fast does ball b move?

easy, right? it's just conservation of momentum. momentum a = momentum b, momentum = mass * velocity, 1 * 2 = 2 * b velocity, so ball b moves at 1 meter per second.

but isn't energy supposed to be conserved as well? and since energy = mass * velocity squared, 1 * 2^2 = 2 * v^2, ball b velocity should be root(2), which is about 1.4. a lot faster than conservation of momentum predicts. what happened to all that energy? if conservation of momentum is strictly correct and ball b will move at 1 m/s, then we're losing exactly 2 joules, half of ball a's energy, to the aether.

"oh, that's because the ball loses a bit of energy in the collision to the air"

but what if it's in a vacuum?

"then it just sends out a few photons"

what if the the collision happened in reverse, with the heavier ball hitting the lighter one? then the system would appear to gain energy, with conservation of momentum predicting higher speeds than conservation of energy.

and most importantly, if energy isn't really conserved, at least in a meaningful, measurable way, why should momentum be conserved? is momentum somehow more important than energy? what's stopping the ball from losing momentum to the air or to friction or radiation? if we have to introduce external, unrelated factors like air resistance or heat radiation to our model, if it only works if you sort of fudge it around until everything sort of aligns, it seems like something is very wrong with our model -- or at the very least, its predictions shouldn't ever be taken fully seriously.