As faras I rmember, the only specific speed of V'ger mentioned in the movie is warp factor seven.
And much later V'ger slows down when approaching Earth.
[Enterprise bridge]
UHURA: A faint signal from Starfleet, sir. Intruder Cloud has been located on their outer monitors for past twenty-seven minutes. ...Cloud dissipating rapidly as it approaches.
SULU: Starfleet reports forward velocity has slowed to sub-warp speed. We are three minutes from Earth's orbit.
According to the official TOS war scale, which is not exactly canon, but close to canon, warp speeds equal the speed of light muliplied by the warp factor cubed. And of course there are questions about ohow such speeds fit the episodes.
In any case, warp facotor seven equals the sped of light times seven cubed, or 343 times the speed of light.
If V'ger traveled 343 times the speed of light, lt would travel 343 light years in one year, 3,430 light years in one decade, and 34,300 light years in one century.
DECKER: NASA. National Aeronautics and Space Administration. Jim, this was launched more than three hundred years ago.
KIRK: Voyager series, designed to collect data and transmit it back to Earth.
So if "more than three hundred years" is between 300 and 400 years, and V'ger can travel at about 343 light years per year, the maximum distance V'ger could have traveled since being launched would be 102,900 to 137,200 light years.
So the galacitic barrier around the Milky Way Galaxy probably hugs the surfaces of the galactic disc of the Milky Way Galaxy. It is possible that V'ger could have possibly have entered the disc of the Milky Way Galaxy thorugh the galactic barrier. But there is no evidence that it did so. Thus it s possible that V'Ger has been heading to Earth from some point within the galactic disc of the for 300 to 400 years - or perhaps only for a proportion of that time span,. Voyager 6 might have spent years, decades, or even over a century wandering space before finding the Machine Planet, and then being rebuilt from Voyager 6 into v'ger. at the Machine Planet.
So is it possible for someplace within the galactic disc of the Milky Way Galaxy to be 102,900 to 137,200 light years from Earth - or maybe a much smaller distance?.
According to Wikipedia:
The
Sun is near the inner rim of the
Orion Arm, within the
Local Fluff of the
Local Bubble, and in the
Gould Belt. Based upon studies of stellar orbits around Sgr A* by Gillessen
et al. (2016), the Sun lies at an estimated distance of 27.14 ± 0.46 kly (8.32 ± 0.14 kpc)
[2] from the Galactic Center. Boehle
et al. (2016) found a smaller value of 25.64 ± 0.46 kly (7.86 ± 0.14 kpc), also using a star orbit analysis.
[1] The Sun is currently 5–30 parsecs (16–98 ly) above, or north of, the central plane of the Galactic disk.
[154] The distance between the local arm and the next arm out, the
Perseus Arm, is about 2,000 parsecs (6,500 ly).
[155] The Sun, and thus the Solar System, is located in the Milky Way's
galactic habitable zone.
https://en.wikipedia.org/wiki/Milky_Way#Sun's_location_and_neighborhood
Since the diameter of the galactic disc of the Milky Way Galaxy has often been given as about 100,00 light years, the farthest that any place within the galactic disc can be would seem to be about 72,860 light years, a distance which V'ger could travel in 212.419 years..
But:
The Milky Way is a
barred spiral galaxy with an estimated visible diameter of 150-200,000
light-years,
[9][24][25] an increase from traditional estimates of 100,000 light-years. Recent simulations suggest that a
dark matter disk, also containing some visible stars, may extend up to a diameter of almost 2 million light-years.
[11][12] The Milky Way has several
satellite galaxies and is part of the
Local Group of galaxies, which form part of the
Virgo Supercluster, which is itself a component of the
Laniakea Supercluster.
[26][27]
https://en.wikipedia.org/wiki/Milky_Way
https://www.nbcnews.com/mach/science/milky-way-galaxy-may-be-much-bigger-we-thought-ncna876966
https://web.archive.org/web/2015031...y-way-may-be-much-larger-previously-estimated
https://www.space.com/29270-milky-way-size-larger-than-thought.html
So if the Machine Planet was directly across the galactic center from Earth, and was at the outer edge of the galactic disc, The galactic disc would have a radius of about 75,760 to 110,060 light years, and thus a diameter of about 151,520 to 220,120 light years for the Machine Planet to be 102,900 to 137,200 light yearsfrom Earth.
And if V'ger took many years, or decades to wander to the Machine pPlanet and be rebuilt, it might have traveled a much smaller distance at warp factor seven than 102,900 to137,200 light years.
DECKER: Voyager VI ...disappeared into what they used to call a black hole.
KIRK: It must have emerged sometime on the far side of the Galaxy and fell into the machine's planet's gravitational field.
Why would Kirk think that a black hole womrhole which could lead to anywhere would lead to "the far side of the galaxy" instead of to any other place in the universe?
Possibly because the speed at whcih V'ger was observed to travel, warp factor seven, was a speed which the
Enterprise could travel at, and so Kirk must have sometimes idly calculated how far the
Enterprise could travel in a decade, a century, or a millennium, at warp factor seven, and how long it would take the
Enterprise to reach the Magellanic Clouds, or the Andromeda Galaxy, at warp factor seven.
Thus when Kirk learned that V'ger was a very enhanced Voyager 6 which had been launched "over three hundred years ago" Kirk could immediately get a rough idea of how far V'ger could have traveled in that time.
And Kirk certainly knew what direction V'ger was reported to be coming from. If V'ger was coming from a dirction which was within the galactic disc and thus a direction very close the plane of the galactic disc, Kirk would naturally assume that V'ger was coming from some palce within the galactic disc.. And in my opinion V'ger probably was also reported to be coming from a direction very cloe to the direction to Saggittarus A West, the supergiant black hole at the center of the Milky Way Galaxy.
So my opinion is that V'ger was reported to be coming from the direction of the galactic center. And considering how long V'ger had been gone, and how fast it was travelling, V'ger probably came from beyond the galactic center, from somewhere that can be considered part of "the far side of the galaxy".
So it is my opinion that V'ger was coming from the Machine Planet, straight to Earth, and that the Machine Planet was on the far side of the galaxy, beyond the galactic center, and almost directly across the galactic center from Earth.