Question: Let's consider that we have a Newton's cradle in vacuum:

[![enter image description here](https://i.stack.imgur.com/NPea6.jpg)](https://i.stack.imgur.com/NPea6.jpg)

Considering that each ball has a mass of 100g or 0.1 kg we release the ball and at the time of contact, the ball has a final velocity of 1 m/s. So the momentum will be:

$$p = mv = 1\*0.1 = 0.1 kg m/s$$

If the momentum is conserved, the ball at the other side should also come out with a speed of **1 m/s**. Then it will come back with an equal magnitude of momentum and the first ball should again move back at 1 m/s. Note that there is another effect at play here, which is the 'pendulum effect' which reverses the direction of the momentum but perfectly conserves its magnitude. And this process should keep going. But since the collision is inelastic, the kinetic energy will not be conserved and even in a vacuum some energy will be lost in the form of heat, but not as sound because it is in a vacuum. But according to the law of conservation of momentum, the momentum should still be conserved even if the kinetic energy is not. But the problem here is that since some kinetic energy is lost, the speed of the balls should gradually decrease. At one point the balls should stop moving, and so they will have 0 velocity. And if that happens, the momentum will become 0, even though we started out with 0.1 kg m/s. Doesn't this seem to violate the law of conservation of momentum? I can't seem to make sense of it

Answer: 
I think that to understand your misunderstanding we have to consider the conditions for conservation of momentum: momentum is conserved so long as there is no net external force acting on the system in question. Considering the simple case in a vacuum, even friction between balls or between the strings and the post is acting. There may even be other forces that aren't listed here that you could think of. But in essence, this is why momentum is not conserved in this case: because it is not ideal. Hence, energy is being dissipated away as heat or sound (if we insert a medium), causing the balls to lose energy and therefore velocity/linear momentum.
You seem to dismiss the fact that conservation of momentum means conservation of momentum **vector**. If you consider the initial state right before the first collision, on the left hand side, the initial momentum is 0.1 kg m/s in the direction horizonatl to the right. Once the last ball on the right is moving up the momentum is already changed. It will change both in magnitude, decreasing to zero and in direction, gaining a vertical component. Then there is a point where its momentum is zero and then the motion is reversed. So you have a continous change in momentum way before you see the effects of friction or other kind of dissipative effects. Why the momentum is not conserved? The condition for conservation is to have no external forces and this is not the case here. You can only expect conservation if you take the states just before collision and just after collision between two balls. Or even before the first collision and right before the last ball starts to rise. But after that the tension in the string and gravity act to change the momentum.