TNG "Booby Trap", using the gravitational field of an object (asteriod) to increase the speed of a spacecraft (Ent-D).
Though the principle was correct, there is absolutely no chance the asteroids the size shown in the episode (several hundred meters in diameter, a couple of kilometers at most) would've had enough gravity to be a significant velocity boost to the Enterprise. Even asteroids hundreds of miles in diameter like Ceres have neglible gravity. Most of the asteroids in "Booby Trap" look about the size of Deimos, the smaller moon of Mars. On Deimos, I've read the gravity is so low that a person can reach escape velocity by jumping.
I think sojourner is quoting Q from Deja Q. If I remember correctly, this is a possible solution that Q1 posed to Chief Engineer LaForge for stopping a moon from impacting a planet.
With those asteroids, if they are not moving with respect to the CM of the cluster, you could only change the direction of the Enterprise's velocity. If the asteroid was moving then you could gain some energy with the apppropriate maneuver. What actually happened in the episode?
Well, in TMP the Enterprise uses maneuvering thrusters to leave the drydock, which means that the ship is changing its angular momentum in relation to the dock, so the ship either leaves the dock behind or slows down its own orbit enough that the dock pulls away from it. The Enterprise also zings by Jupiter close enough that one can imagine its a slingshot maneuver. Voyager 1 passed only 348890 km from the center of Jupiter (277,540 km from it's "surface"), which is considerably closer than the Moon from Earth, and from that distance Jupiter would look about like it did in TMP size-wise. I dunno if it's realistic for such a slingshot to help at "warp point five" impulse power, but clearly someone did their homework. Likewise, someone did the math to figure out how long it would take the Enterprise to get to Jupiter, because the time from launch right after the flyby is given as 1.8 hours. At its farthest from Earth, Jupiter is about 3,220 light seconds away, or about 54 light minutes distance. At half light speed, it's 108 minutes away. 108 minutes = 1.8 hours. Ergo, if Jupiter were its maximum distance from Earth when the Enterprise launched, at half light speed the ship would get there in exactly the 1.8 hours Kirk cites, which is too on-the-nose to be an accident.
^what would be more impressive is if that is accurate as to where Jupiter will be in that year relative to Earth in that year.
Actually, simple relative motion applies on a scale this small; Enterprise could maneuver relative to spacedock as if both were motionless in space. For those kinds of small maneuvers, for anything above about 1000km orbit it'll take several hours for the differences in their orbits to become apparent.
Where was Jupiter in the movie? I mean, apart from "inside the solar system" which is about as specific as it gets, otherwise its position was never really defined. Also, I clearly recall that the estimates for Enterprise' speed were based on the closest and farthest positions of jupiter from Earth, so depending on the position Enterprise was either moving at .25C (the purported "full impulse power" of starships) or .5C (as in "warp point five").
Thruster are Newtonian, but the inertial damping fields are not. Maklng full impulse using the same IDF puts it in the same catagory. Sometime during a year, Earth and Jupiter should be opposite each other.
The .25c thing was a TNG tech manual thing. In TMP it's pretty clear warp point five means half light speed, because—as I noted previously—Jupiter at its farthest from Earth would take 1.8 hours to reach at half lightspeed.
However, if Spacedock is rotating, all bets are off. If the Enterprise is 'free floating' inside of the spacedock 'mushroom top', it would need constant thruster action to maintain relative position inside the enclosed space. One assumes there is some sort of inertia-imparting field that fixes this, but that's certainly not real physics!