Travel time for a space ship at relativistic speeds

[lordl]

Hi

I'm writing a science fiction novel, and I'm trying to figure out how to make an approximation on the travel time of a hypothetical spaceship traveling at relativistic speeds. Given the following:

The spaceship is capable of accelerating at a constant 1G, until it reaches a top speed of 0.8c, after which it will travel at constant speed until beginning deceleration. Deceleration will also be at a constant 1G. For this instance, the travel distance is 100 light years, and the ship will start from a standstill, and should come to a complete stop at it's destination.

What I want to find out, is how long the journey will take from the viewpoint of the passengers, and how long it will take from the viewpoint of an outside observer.

I'd like to know which equations I'll need to use to make this calculations, preferably in a form that I can use to repeat the calculations given other acceleration rates, distances and top speeds. I do not need the results to be extremely precise, a reasonable approximation will do (if that matters).

Please note that I only have a basic understanding of physics and math, so I would appreciate if any answers are kept as simple as possible. Though an answer I may not understand is better than no answer :)

 

[Passionflower]

Sounds like a homework exercise.

Strictly out of curiosity, why do you want to give the readers of your book all these technical details, especially about acceleration?

A ball park number you get by simply ignoring the acceleration parts. Calculate the Lorentz factor, you have 0.8c for the spaceship, do you know how to get the Lorentz factor from that? From there you can calculate the time it takes for the traveler, the time it takes for an outside observer is trivial if you ignore acceleration.

 

[lordl]

I suppose it does sound a bit like a homework question, but it's not, I finished school quite some time ago :)

I don't actually want to give the readers all the technical details, but I'd like to know them myself. The thing is that my book will follow a technological development where the propulsions systems of the spacecraft become more efficient over time, allowing for higher and higher top speeds. As I'm not to keen on the idea of FTL travel, my space ships will instead rely on time dilation to get the passengers to their destinations in a survivable timeframe. Accordingly, I need to know how much subjective time passes by for them at different speeds.

The lorentz factor is easy, at 0.8 c it should be about 1.66. Are you saying I can just ignore the acceleration and deceleration phases of the trip, and still get a good approximation of the subjective time? So that given a 100 light year distance, the trip would take about 75 years subjectively?
Or am I misunderstanding something?

 

[Passionflower]

It takes about a year for a spaceship with a proper acceleration of 1g to reach 0.8c. So on a 75 year during trip that is not that much.

But is 0.8c dramatic enough for your sci-fi scenario?

 

[lordl]

0.8c is just the beginning, so to speak. The idea is that later generation ships will have higher top speeds, perhaps up to about 0.99c and above (cue, drama). I haven't really decided on how far I'll take it yet, but I'd like to follow an incremental technological development.

I should add that 100 light years was just an arbitrary example. What if the ship was going to Proxima Centauri, about 4.2 light years away. Could I still make a good approximation of the subjective time on board the ship using only the Lorentz factor?

 

[Passionflower]

0.8c is just the beginning, so to speak. The idea is that later generation ships will have higher top speeds, perhaps up to about 0.99c and above (cue, drama). I haven't really decided on how far I'll take it yet, but I'd like to follow an incremental technological development.

I should add that 100 light years was just an arbitrary example. What if the ship was going to Proxima Centauri, about 4.2 light years away. Could I still make a good approximation of the subjective time on board the ship using only the Lorentz factor?

Well since the spaceship needs about 2 years to accelerate and decelerate it is no longer very accurate to ignore acceleration. To calculate time, velocity and distance during acceleration you need hyperbolic functions (e.g. sinh, cosh and tanh).

 

[lordl]

All right, thanks very much for your help!

 

[DrGreg]

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