In relation to universal space, yes, the rocket would be traveling as fast as the galaxy is moving, as everything in the solar system is already traveling that fast. Think of two pickup trucks driving next to each other, both traveling down a highway at 100 MPH. A guy in the back of Pickup A tosses a basketball to a guy in the back of Pickup B. The guy isn’t throwing the basketball at 100 MPH, but the basketball is still traveling 100 MPH, because the pickup trucks are driving 100 MPH.
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