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Q & A: accelerating near c

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Most recent answer: 10/22/2007
Q:
In space, accelerating at one gravity, how long would it take to attain 90 % the speed of light? That is, if a ship had the ability to do this. The starting point would be earth orbit.
- Richard (age 64)
upland, ca
A:

This question reminds me of some of the homework problems from college, so it's good to work out to see if the old brain is still functioning. I worked out the following answer, which I hope agrees with what my colleague gets.

I assume by 90% of the speed of light you mean speed relative to a collection of things which were initially at rest with respect to you.

If by 'how long' you mean 'how long according to the person accelerating', then I get the following answer (with c = 3x10^8 m/s and g = 10 m/s^2):

tau= (c/g) arctanh(0.9). This comes out to be about 1.47 c/g or about 4.4 x10^7 sec. That's around 1.4 years.

If by 'how long' you mean 'how long according to the initial collection of stuff' I get
tau= (c/g) 0.9/sqrt(1-0.9^2) or around 2.06 c/g, about 6.2 x 10^7 or around 2 years.

Notice that for travel times of around a year, the traveler and the home base are already disagreeing quite a bit about time intervals. It rapidly gets more extreme. For a round trip of 20 traveller years, at this comfortable acceleration, I get around 20,000 years passing on Earth. This would be an expensive way to find out more World Series outcomes than possible for anyone staying home, so in that sense you waould have an extended life, even though the total number of heartbeats, meals etc would be normal.

Mike W.

Yup, I agree with what Mike gets. This is also worked out in Landau and Lifshitz, "The Classical Theory of Fields", on pages 22-23 of the Fourth Edition.

Tom

(published on 10/22/2007)

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