Calculating Ship Time w/ Varying Acceleration in Relativity

[stefan10]

My only experience with special relativity has been in an introductory Modern Physics course. In that course I think we mostly used constant velocities or if we did do anything with accelerating objects the acceleration was constant or changed very few times (you can just add the few changes.) I currently want to write a short story about a generation ship that varies between non-relativistic velocities and various relativistic velocities, so that it can slow down/speed up to collect resources/prevent ship damage in more dense interstellar mediums. I'm assuming that such a ship would have a non-constant acceleration depending on the circumstances of what it needs to do. Eventually I want the ship to return to our solar system, but for story purposes I have to know the estimated time it took the round-trip for an observer (or set of observers) on the ship versus earth.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

^ From that website, I found a set of equations for a ship moving with a constant acceleration. Do I have a lot of leniency here where I can just say the average velocity they were going was like .5*c so therefore the ship time interval (proper time right?) is gamma( with beta = .5) *earth-time interval, or should I consider an average acceleration and use that as a constant in the linked set of equations? Is there another way to do this (I've been reading about a line integral method, I think?) Which would you say is more accurate for my potential scenario? I'm thinking I will just make up a scenario in which the accelerations change 10-15 times based on circumstances and then just calculate the time dilation at each change and add them together, but I don't know if that is the right way to go about it. For example, maybe something like the ship is accelerating at 1g until it reaches .5*c, and then slows down by 1.5g until it reaches 0 m/s, and then speeds up again by 1g until it reaches .5*c, etc, etc.

Thank you very much and sorry for any misconceptions I might have.

 

[Mentz114]

My only experience with special relativity has been in an introductory Modern Physics course. In that course I think we mostly used constant velocities or if we did do anything with accelerating objects the acceleration was constant or changed very few times (you can just add the few changes.) I currently want to write a short story about a generation ship that varies between non-relativistic velocities and various relativistic velocities, so that it can slow down/speed up to collect resources/prevent ship damage in more dense interstellar mediums. I'm assuming that such a ship would have a non-constant acceleration depending on the circumstances of what it needs to do. Eventually I want the ship to return to our solar system, but for story purposes I have to know the estimated time it took the round-trip for an observer (or set of observers) on the ship versus earth.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

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Thank you very much and sorry for any misconceptions I might have.

To get an exact answer you would need the 4-velocity that describes the journey, from which you could calculate the time on the ships clock. There is a formula see here

http://en.wikipedia.org/wiki/Proper_time

You'd also need the disposotion of matter nearby. The best thing would be to break it up into segments and add heir proper times.