Momentum, of course!
They are fucking off away from the ground to launch vehicles into space. That’s a lot of momentum upward. And they are gaining velocity still while they are firing. So when the boosters shut off and disconnect, they still have a ton of momentum overcoming gravity. It takes a while for gravity to slow the boosters, stall them, and then start to pull them back down to Earth.
Just like this. Take a ball, throw it upward with your arm. Sure, the ball “disconnects” and has no more “thrust production” once you let it leave your hand, but it still goes up up up for a while, until gravity overcomes it and brings it back down.
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