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>where they almost use the momentum of the rocket to "fling"

this is compulsory anyway due to physics.

Not really, the rocket and the payload would have the same momentum unless an additional force is applied between them.

Why? You could also just use a spring to push it out.

And what happens to the rocket from the reaction force from the spring?

Nothing, because you used a spring to launch the other satellite on the other side of the payload at the same time.

Or, you just use thrusters, like the ones that launched from the Shuttle did.