Yesterday, while studying a simple question, a rather strange doubt hit my mind:
Consider a ball of mass $m$ moving with velocity $v$ that collides with a wall in a perfectly elastic collision and rebounds back with the same velocity. Taking the direction of initial motion of the ball as positive, the change in momentum of the ball is $-2mv$ and the wall is clearly at rest.
So, in order to obey the law of conservation of momentum, the wall must gain a momentum of $2mv$.
Now, this itself confused me a lot. I then applied the law of conservation of kinetic energy to this situation, and clearly the magnitudes of the initial and final velocities of the ball are equal, so kinetic energy of the ball is conserved and hence the final kinetic energy of the wall must be zero in order to conserve energy.
Now, this contradicts with the law of conservation of momentum, which says that the wall must gain momentum of $2mv$.
One more point: If we are talking about a wall then it is an important fact, which must be considered, that the wall is fixed on earth and so the mass of the wall is equivalent to that of earth, say $M$. If the law of conservation of momentum is applied now then we get velocity of the wall, say $V=2mv/M$ which is no doubt an extremely small value. But, the law of conservation of energy strictly says that the velocity of the wall must be exactly zero, not even the smallest value is allowed (theoretically speaking).
Please clarify this and correct me anywhere if I'm wrong. I have tried my best to make it clear what I want to ask :).
Best Answer
Actually, it doesn't. Energy is conserved, yes, but nothing has to be zero for it to be conserved.
The mistake is that you assume the wall to be stationary also after collision. This is your own assumption, and as you clearly show with momentum conservation, that cannot be true.
You have already realized that if the whole Earth is included in the picture, the momentum conservation law makes sense in the way that the Earth is given a tiny, tiny, tiny speed after collision. Now redo the energy conservation considerations with this in mind - in other words, redo the energy calculations without assuming that the wall/Earth is stationary.
In fact, the acquired Earth speed is so tiny that it is negligible - that's why it is usually just assumed zero.