Why does velocity add up to all objects inside a moving container? Imagine  I would be on a mobile belt-conveyor and throw a ball in the air, then the ball would not fall in my hand, because I move to the right but the ball does not, just like this:

However, if I would be in a wood-box and throw a ball up, it would go up and down in a straight line and land in my hand, while the box moves the the right:

Why is it like that? I would expect that the ball would not land in my hand. How come the ball also moves to the right in the box?
 A: Ideally, in both cases the ball will land in your hand. In both cases the thrown ball will move along with you.

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*The first case is non-ideal, though, since you might have sideways air resistance slowing down the ball. You yourself are not being slowed down since the conveyor belt forces you forwards and overcomes air resistance. So, although the ball will move forwards as well, it will not move as fast forwards as you and will be lacking slightly behind.

*In the second case, all air inside the box already moves along with you at the same pace. So there will be no sideways air resistance when you throw the ball.

So these examples are not really about velocity vectors "adding up" (or syncronising) inside a box, but rather about velocity not being affected when there are no forces to affect it. If you inside the box blew with compressed air sideways on the ball, the ball would naturally also be pushed away and will not land in your hand inside the box either.
A: Air resistance matters. If you performed the experiment in a vacuum, both the first and second cases would end with the ball landing in your hand. But in Earth's atmosphere the moving belt conveyor creates a wind (from your perspective) which pushes the ball away from its natural trajectory. The effect of the wind depends on the relative velocity of the conveyor through the air -- at low speeds you probably wouldn't notice it (the ball would fall in your hand). In the box case the box would shield you from the wind.
A: To answer your question directly; all vectors "add up", meaning we can decompose all dynamics into a sum of dynamics in different directions; the most basic of which being 2D motion decomposed into x-direction motion plus y-direction motion.
Here in the y-direction our equation of motion would be some initial velocity upwards, with constant acceleration down (gravity); you can use this to calculate the y-direction kinematics of the object.
Now in the x-direction, our equation of motion would be the initial velocity which we had on the conveyor belt (which the ball also had), and there is no net force in the x direction, so the acceleration is 0. Therefore it maintains the conveyer belt velocity in the air.
So the height of the object is changing with constant acceleration, but while this acceleration is happening its x-velocity, is the same as ours on the conveyer belt, as a result it will alway have the same x-coordinate as us moving on the conveyor belt, and inevitably when it falls back to our y-level, it will still have the same x-coordinate as us, and land in our hand.
