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The NX-01 Enterprise has left space dock and been gone for awhile. They had to travel 8.2 hours at maximum warp with the engines running at 105%.

Enterprise "Affliction":

T'POL: The freighter's warp trail has terminated.

REED: They may have dropped to impulse.

ARCHER: How long to intercept?

TRAVIS: At our current speed, eight point two hours.

ARCHER: Archer to Engineering. Commander Kelby, respond.

[Engineering]

KELBY: Kelby here, sir.

ARCHER [OC]: We could use a little more speed.

KELBY: The injectors are running at a hundred and five percent.

The Klingons board the ship and rewrite the warp subroutines when the Enterprise reaches the site 8.2 hours away at maximum warp. After this point Enterprise is forced to travel above warp five or else the ship will blow up.

Enterprise "Affliction":

TRAVIS [OC]: Bridge to Captain Archer. Pressure's increasing again.

ARCHER: Take us to warp five point two.

[Bridge]

TRAVIS: Sir, we can't hold that speed for long.

ARCHER [OC]: We're out of options, Travis. Five point two.

TRAVIS: Acknowledged.

Then conveniently the Columbia which is pretty far away at this point can catch up to the Enterprise.

Enterprise "Divergence" (Part 2):

ARCHER: I've just been talking with Trip. Columbia can rendezvous with us in thirty minutes. He says he can restore the subroutines.

REED: He can't do that from Columbia.

ARCHER: We're going to bring him onboard.

REED: The transporter won't operate at warp, and docking's out of the question.

ARCHER: Trip told me you have experience transferring personnel at warp speed.

REED: In training exercises, yes, and at warp speed one. Every move was rehearsed in simulators for weeks.

ARCHER: We don't have weeks.

REED: We'll have to merge our warp field with Columbia's.

ARCHER: T'Pol's doing the calculations now. At warp five, we'll have to move within fifty metres of each other.

Captain Archer states that the Columbia can catch up to the Enterprise when they have essentially the same specs, and the same maximum speed of 5.0 and are a considerable distance from each other. Since the Enterprise has been traveling at greater than warp 5 speeds, they have the same effective maximum speed, and the Enterprise is ahead of the Columbia how is it possible for the Columbia to catch the Enterprise? (And in less then a half hour no less)

Is it that they simply changed heading and turned around back to earth? Even if that decision was made it didn't happen for awhile, they were more than 8.2 hours away from earth. Also Archer doesn't give the command to alter heading and the Columbia did catch up to the Enterprise from behind.

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    Is there any indication that Enterprise has been traveling straight away from Earth, at warp most or all of the time? One might chalk it up to a meandering course at slower speeds on the way out, with the two ships now heading straight for each other at maximum warp on the way back. Plus, Trip Tucker engineering magic. – Politank-Z Jul 16 '15 at 19:46
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    Is there any reason to think they aren't simply heading on a converging course? That we don't see a course change isn't important. Episodes don't show captains giving every detailed order. – DJClayworth Jul 16 '15 at 19:47
  • I'll add more @Politank-Z I usually get criticized for long questions. – JMFB Jul 16 '15 at 19:50
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    yeah if there intersecting at an angle is not unrealistic that they can meet. Also without trip onboard the enterprise is going 5.2, so i imagine with trip the columbia probably can get 5.3 eheh – Himarm Jul 16 '15 at 19:53
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    Why would somebody downvote this question? lol – JMFB Jul 16 '15 at 19:59
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TL/DR : It was a writer's plot hole that they ignored so that they could have that suspenseful problem, and yet have someone come to the rescue in an interesting way.

If you wanted to try to use reason, you could assume that the Columbia, even though the same class type of the Enterprise, is of newer construction. In real life, newer builds of a class of ship usually benefit from technologies that were refined in the years or even months of separation between even one ship to the next. Even though the structural specs would be very similar, maybe "they researched a better warp field geometry" that would allow for the Columbia to have a higher warp threshold, or at least maintain maximum speed for a longer duration.

Now, considering that ST:ENT was trying to use the same warp scale as The Original Series, even then, increasing warp factors was more than a linear increase in actual speed. As quoted in Memory Alpha:

In ENT: "Regeneration", Trip states that warp 4.8 (approximately 111 times the speed of light in the TOS-scale) is double the speed of warp 3.9 (approximately 59)

source: http://memory-alpha.wikia.com/wiki/Warp_factor

Also considering that it doesn't really mention in the episodes which direction Enterprise is heading while taking their Speed-like trip, it makes sense that if they have to run, they would run towards help, which would be the Columbia racing to them from space-dock.

And that is the closest we can try to explain how the Columbia could possibly make it to the Enterprise in time.

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