Hi,
here comes some math and code...
Most of this stuff is an extract of one of
my more "professional" physic books
The basic formulars for calculating travel
times of an accelarated body of constant
mass become much more mathematically handy, if
we perform some simple substitutions resulting
in some variables without any dimension.
V=v/c v=velocity(m/s)
T=(a*t)/c t=time(s)
X=(a*x)/c^2 x=distance(m) and a=accel(m/s^2)
The basic relation between time and velocity
is given by
v = c * sqr( (g*t/c)^2 / (1 + (g*t/c)^2))
After applying the substitutions you get
V = sqr( T^2 / (1+T^2) )
In order to get the distance you have to
integrate V(T) resulting in
X = sqr(1+T^2) - 1
resolved by T
T = sqr( (X+1)^2 - 1 )
These are values in the destinations reference
frame.
The time passing in the accelarated frame is
calculated with:
T' = arcsin T
In order to get values in usable units you have
to revert substitutions....
O.k., this put in some code:
Substitution ddist > dLJDist (x > X)
dLJDist = (dDist * 9.81 * tbG) / (c ^ 2)
Calculation of dTime (T)
dTime = Sqr((dLJDist + 1) ^ 2 - 1)
Revert substitution to get dTimeS (t) in seconds
dTimeS = ((dTime * c) / (9.81 * tbG))
Thats the time needed to move a distance of ddist
in accelerated way.
Calculate speed (V)
dRelV = c * Sqr(dTime ^ 2 / (1 + dTime ^ 2))
Calculate time in accelerated frame (T')
dRelTime = Log(dTime + Sqr(dTime * dTime + 1))
Thats the same as dRelTime arcsinh dTime but
most languages do not provide this function
directly.
Revert substitution to get time in seconds
dRelTimeS = ((dRelTime * c) / (9.81 * tbG))
This the time passing in the accelerated frame.
So, thats all You need.
If you want to calc mid-turn trips, just divide
original distance by two, do calcs and finally
double the "time" results. Just as usual...
A little bit more complicate is to calc travel
times with a speed limit set.
Here you would have to subdivide travel into
accelarated sections and constant speed sections.
In order to get the time dilation effect for
constant speed travel you could use
dRelTimeS = dTimeS * Sqr(1 - (V ^ 2) / (c ^ 2))
I hope, this stuff is ok that way
If anybody should find some flaws I am quite sure
these will be presented in this board
Trusting these calcs, relativistic effects seem
to be neglegtable for most of the standard Traveller
travel procedures anyway.....
Regards,
Mert
