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peteshimmon
2007-Jan-13, 02:15 PM
I described a thought experiment on how a
traveller going at half the speed of light
experiances about 14 minutes time dilation
doing a distance of one light hour. But even
as such an experiment I find it unsatisfactory
as the instant change to half c is never
possible. So I have used basic stuff to find
it takes 182.8 hours to accelerate at 10 metres
per second per second, turning round and
decelerating at the half way stage to travel
a distance of one light hour. And I note the
maximum velocity reached was about 1% of light
speed. Does anyone have a simple formula for
the total time dilation? It will be some kind
of integral I suppose. With this I can find
how long it takes for someone to reach varous
distances as they measure their own time. And
I would like such a formula for 9 metre/sec
squared for the really decadent traveller:)

grant hutchison
2007-Jan-13, 03:11 PM
At constant acceleration a' for a period of on-board time t', the elapsed time as seen by stay-at-home colleagues is:

t = c/a'.sinh(a't'/c)

The distance travelled is

z = c2/a'.[cosh(a't'/c) - 1]

And the velocity reached (as a proportion of lightspeed) is

β = tanh(a't'/c)

For the scenario you describe, in which the traveller accelerates and then decelerates, you use these formulae for the half-journey time, and then double to find the total elapsed time and the distance covered.

Grant Hutchison

peteshimmon
2007-Jan-13, 06:23 PM
Thanks. I never did quite understand those
hyperwhotsit functions!

grant hutchison
2007-Jan-13, 06:53 PM
I never did quite understand those hyperwhotsit functions!Well, you can always assemble them from exponentials, if necessary.

sinh(x) = (ex - e-x)/2

cosh(x) = (ex + e-x)/2

tanh(x) = sinh(x)/cosh(x)

Grant Hutchison