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STI: The Enterprise passes Jupiter 1.8 hours after leaving Earth orbit. This suggests an acceleration of 28-46 km/s2.

STVI: There is a difference between maneuvering thrusters and impulse power - critically, as Lt. Saavik reminds Admiral Kirk, with regard to regulations.

STVIII: It turns out that warp power can be used for sublight speeds as well; also, note temporal shielding - not present on the original Pheonix - prevents modern starships from experiencing relativistic effects at high sublight speeds.
The Furry ConflictTFC:
Jupiter pass in STIII, at "warp 0.5," i.e., impulse, shows 0.33-0.5c.