Well, the ship has to achieve a certain velocity in real space before engaging warp engines, at least that is my understanding.
Umm, no. Visually, we do see ships jump to warp from a standstill, or at most accelerate to very modest speeds before the warp flash takes place. They also do this at the command of "Ensign, warp seven, engage!", not at the command of "Ensign, warp-enabling speed and then warp seven!", FWIW.
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Not sufficient proof. First, considering what the ship is doing, I imagine any acceleration to warp would look like "standing still", no matter how fast the ship was going beforehand.
And in TOS "Obsession", Scotty had the impulse engines offline for repairs yet Kirk still went to warp on the heels of the dikironium cloud creature; apparently, the acceleration to warp happened by using the warp engines alone. "Engaging warp" would thus have to be possible from a standstill, even if the initial speed was just warp 0.01 rather than warp 1.
The ship does have maneuvering thrusters. And besides, having the impulse engines off doesn't mean the ship is at a dead stop. The impulse engines are sublight and provide Newtonian maneuvering. When one cuts off the impulse engines, it doesn't stand to reason the ship comes to a stop.
Okay, so perhaps we can argue that a ship going from zero to warp 1 does so by going through all the speeds in between, even if this takes just a picosecond. And perhaps it takes longer with primitive engines, which is why Cochrane in ST:FC accelerated his test rig using some alternate engine before engaging the warp engines. But there isn't a technological limitation that would force modern starships to do a separate acceleration maneuver before engaging their warp engines: it's all well integrated, and no "preacceleration" is evident in the drama even if it theoretically lingers in the tech details.
Timo Saloniemi
Considering "preacceleration" is utterly routine, to even note it would be pointless under normal circumstances. Cochrane would note it, because "routine" warp acceleration was nonexistent to him.
I stand by my suppositions.