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Currently writing a scifi novel and am trying to do the math to establish time frame of the lore. The main points are that in 2068 CE Earth sent a generation ship to colonize Proxima Centauri. They were never heard from again because Earth was destroyed a 25 years later during WW3 and the survivors around the solar system have no means of contact. The main story partly evolves around first contact with what seems to be aliens, but is actually the return of the generation ship's descendants. I am trying to establish a solid timeline for when the generation ship reached Proxima Centauri.

So to the main question: If the ship leaves Sol accelerating at 1 g (9.8 m/s^2), how long would it take them to reach 4.5% the speed of light (13,490,661 m/s)?

I would like an actual equation, however, because I expect i might need to tweek the numbers for future situations in the story. Rounding to 10 m/s^2 and 13.5M m/s is fine, wouldn't mind knowing the difference though.

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ At these speeds, the relativistic correction is far smaller than the 2 % rounding error you are okay with. $\endgroup$ – Jasper Aug 14 at 18:51
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    $\begingroup$ This question has been already asked, just with different numbers in it... $\endgroup$ – L.Dutch - Reinstate Monica Aug 14 at 18:54
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    $\begingroup$ @L Dutch And it would be nice to include a link to that question and answer. $\endgroup$ – M. A. Golding Aug 14 at 18:58
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    $\begingroup$ Downvoted because asking about basic physics indicates lack of research and effort. $\endgroup$ – Zeiss Ikon Aug 14 at 19:03
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    $\begingroup$ Simple rule of thumb: one year at one G is lightspeed, discounting relativity. You want a little less than 1/20 that speed, so you get less than 1/20 of that time. $\endgroup$ – Zeiss Ikon Aug 14 at 19:23
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If you're dealing with constant acceleration, and you aren't interested in orbital mechanics, the basic equation relating how much Time "T" takes to accelerate from Starting Speed "S1" to Target Speed "S2" at Acceleration "A" is:

enter image description here

So you can solve for T, like this:

enter image description here

and then just plug in your numbers. So if you started at 0 m/s:

enter image description here

and the result is close to 1,376,598 s, which converts to just 16 earth days.

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    $\begingroup$ So I actually did the math before and got a fairly close number, but I didn't trust my math because I thought that number seemed too small. I guess it was right though. Thanks. $\endgroup$ – TitaniumTurtle Aug 14 at 19:18
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    $\begingroup$ This is only true for non-relativistic cases, though. . . If the final speed is 4.5% the speed of light, you need to consider this relativistically to get a good answer. $\endgroup$ – HDE 226868 Aug 15 at 0:36
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    $\begingroup$ @HDE226868 Which them means... "How long" from the perspective of the passengers, or the planet from which they departed? $\endgroup$ – Chronocidal Aug 15 at 9:06

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