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Warp 9.975 = 13486 times the speed of light, Voyager started out 75,000 light years from Earth. Traveling at maximum warp, Voyager could cover 37 light years a day. Equating to 13486 light years in a year. Times that by 75 and you have 1011450 light years, which is how far voyager would travel in 75 years, which exceeds the 75,000 light years that they are from home, so it would actually take voyager 5.56 years to get home.
9.975 is voyagers MAXIMUM warp. Even trek newbies know that maintaining maximum warp for more than a short time will cause serious damage to the nacelle coils, the intermix chamber and pretty much anything that deals with warp plasma! That said, 75 years is still probably a heavy estimate, since voyager was the fastest ship in the fleet. Many ships in star trek have put out 9+ speeds, but picard's enterprised never seemed to use anything above warp 6 except in emergencies, and even in emergencies they never went 8+ unless they were being actively pursued by an advanced aggressor (borg usually), so it stands to reason that voyager's main speed improvements lie in its ability to maintain a higher CRUISING speed, eg. the speed that the ship can comfortably maintain, without taking any significant damage beyond normal operational wear. Voyager seemed to have an average maximum cruising speed of warp 8, so that's what I would peg them at.
Further on the first answer which I too find to be a bit inaccurate. Indeed Voyager maximum warp speed in 9.975 however in order for them to make the "5-6" year long trip to Earth they'd need travel at that speed constantly for all 75,000LY. They could not slow so they'd be moving at warp 24/7 and as the second makes clear that is not sustainable the nacelles and other parts of the warp drive will breakdown. The nacelles have to maintainenced the main reactor has to be regularly re-supply with antimatter fuel and the other components must be aligned/cleaned/etc.
Now that's just the hardware aspect but there is problem of the actual course the ship is going to travel. As we all know "faster than light no left or right" this means a ship must make course corrections by dropping out of warp speed. The first post assumes the ship travels in a perfectly straight which would be impossible without flying through stars, planets and all number of other things. ("Other things" includes the supermassive black hole which astronomers beleive is at the centre of our galaxy. Quornhog 23:19, December 22, 2011 (UTC))
Lastly there's the inhabitants many of whom were hostile. So the ship would have to deal with those. All these things add to their total potential travel time.Darkseid253 04:08, December 22, 2011 (UTC)
Don't forget that they also had to make pit stops. Dilithium crystals work for quite a while, but they need to add fuel, either through installing new dilithium crystals (which of course has to come from somewhere), or by traveling at impulse speed for the bussard collectors to work. Warp speed creates a warp bubble that bends space, and thus material, around it.
Add to that the ship can't maintain it's maximum warp for more than 12 hours. if we assume a safe cruising speed of warp 8, it would take aproximately 68 years to travel the 70,000 lightyears home. if we were to factor in all the variables, 75 years is reasonable with an average speed of about warp 7.7.-Cpthunt 21:25, December 22, 2011 (UTC)
A couple of things.
Faster than light no left or right does not apply to warp drive. Ships have been shown altering course at warp speed many times. As a matter of fact they've changed course in Transwarp and Quantum Slipstream too.
Second the TNG tech manual clearly states that the warp field and the navigational deflector can be made porous to allow for the collection of hydrogen at warp speeds, which makes sense as you could suck in a heck of a lot more hydrogen at superluminal velocities than at impulse.
Another thing, Dilithium is less of an issue for Voyager than say the NCC-1701. Theta Matrix recompositing courtesy of Scotty in Star Trek IV allow Dilithium crystals to be refreshed while still in the engine, so unless something actively damages the crystals which can happen, it isn't a pressing concern.
Now that all that's said, most of what's here is generally correct. No starship except Borg ships can maintain their maximum speed indefinitely. The Enterprise-D had a maximum cruise of Warp 9.2, over that it got dicey. And it could only hold warp 9.6 maximum warp for ten minutes at first. It was extended to ten or twelve hours later on.
c=speed of light
As for Voyager the speeds on Memory Alpha are as follows...warp 9.975 1,554c - 1,721c or 132 light years 1 month. The journey of 75 years is 900 months is = 118,800 light years, or greater than the breadth of the Galaxy. As pointed out before clearly overshoots the mark. Furthermore unlikely as the ship isn't going to maintain maximum warp the entire time. A more likely speed of warp 9 on average by the Memory Alpha scale gives a speed of 834 times the speed of light and you get a travel time of 83years 341 days.
If you go by DITL.org, a respected website whose warp scale is based on not only canon but also backstage sources, in this case the ST TNG Technical Manual, then Warp 9.975=5559.1c with a travel time of 12 years 222 days. Clearly not what Captain Janeway had in mind, but goes to the fact that they can't maintain maximum warp. At warp 9 that's 1516.9 times the speed of light and 46 years 59 days. But Warp 9 is still pretty fast even for a state of the art spanking new ship like Voyager, warp nine for four decades with NO support services or Starfleet equipment is definitely pushing it. So if we bring it down to Warp 8, which is perfectly reasonable for either Enterprise-D or Voyager, then we have a speed of 1024c for 68 years 131 days. Which I think fits in perfectly with the highest regularly manageable speed with both fuels (as anti-matter will be a much bigger challenge than Deuterium), engine maintenance, wear and tear, and poor B'Elanna's frayed nerves.
One more thing. This is directly from Mem-Alpha- Star Trek: Voyager Edit
In the pilot episode of the series, VOY: "Caretaker", it is established that "at maximum speeds" it would take 75 years for Voyager to reach Earth, that was at that time approximated to be 70,000 light years away. In Star Trek: Voyager Technical Manual it is stated that this calculation is based on a non-stop direct journey at the speed of warp 9.6. This indicates warp 9.6 equals to approximately 933 times the speed of light.
For more information check out Mem Alpha's Warp Factor page. Also look at this and this article on DITL.org. For the record on DITL writing in white is stuff the man doing the writing came up with to fill in blanks, green is backstage stuff and things from tech manuals over the years, and yellow is canon stuff from on screen. But most of what he writes fits well and makes sense. Full Disclosure. Another useful link is to STO Geekipedia. This is the map from the Star Trek Star Charts book, that gives the general canon map of Voyager's route. Delta Quadrant Rayfire 01:37, December 23, 2011 (UTC)
Tom Paris stated himself in episode 2x01, "The 37s" that Warp 9.9 was about 4 billion miles per second. Light travels at 186,000 miles per second. Doesn't this mean that Warp 9.9 is about 21,500 times the speed of light? I'm not sure where 1,721 times the speed of light came from.
I wouldn't take any notice of the warp factor page on the memory alpha site. The factors are all over the place as you read down them, just arbitary figures plucked or randomly calculated from iffy statements in episodes. The rules that were established by the show although not necessarily used correctly during technobabble on screen were that in Kirks day the rule of warp factor cubed gave speed in c and in the next gen era the warp speeds were changed to a logarithmic scale to put warp 10 at infinity. This is maths and basing it on that rule means that every warp factor can be accurately calculated without doubt. Warp 9.975 is indeed exactly as the OP suggested: 13486 x c. Although ships cannot maintain their maximum rated speed for very long it was stated in the opening episode by Lt Stadi that "Voyager had a maximum cruise velocity of warp factor 9.975". If this was true then the OP has a point Voyager should be able to sustain that speed, however later on in the series the ship quite ofter topped out at warp 9.6 and ended up with an Enterprise style time limit on it too. I suspect that what really happened is that when the makers of the show realised that warp 9.975 was too fast they sneaked it back a touch in order to stop Voyager from simply flying home in 5 seasons.