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.
Jupiter pass in STIII, at "warp 0.5," i.e., impulse, shows 0.33-0.5c.