Care to show us that envelope, Nyrath? Worked examples of rearranging the formulae can be as informative as the formulae themselves for us struggling maths students.
Certainly!
From
http://www.projectrho.com/rocket/rocket3aj.html#relativity
v = c * Tanh[a*T/c]
where
v = final velocity in planet's frame of reference
(1.0 = lightspeed)
c = speed of light
(= 1.0)
a = acceleration
(1g = 1.03, 4g = 4.12)
T = duration of acceleration in crew's frame of reference
(years)
Tanh[x] = hyperbolic tangent of x
(in Windows calculator, check the "hyp" box and use Tan key)
{to make things simpler, the speed of light is set equal to 1, time is in units of years, and acceleration is in terms of light-years per year per year (ly/yr^2)}
Scout has four weeks worth of fuel. So it will use two weeks accelerating, coast to the destination, then use two weeks of deceleration to brake to a halt.
Two weeks = 0.03846 years
v = c * Tanh[a*T/c]
v = 1 * Tanh[4.12 * 0.03846 / 1]
v = Tanh[4.12 * 0.03846]
v = Tanh[0.1584552]
v = 0.157 c
How far did the ship travel in those two weeks?
d = (c^2/a) * (Cosh[a*T/c] - 1)
where
d = distance in light years
(3.26 light years = 1.0 parsec)
Cosh[x] = hyperbolic cosine of x
d = (c^2/a) * (Cosh[a*T/c] - 1)
d = (1^2 / 4.12) * (Cosh[4.12 * 0.03846 / 1] - 1)
d = (1 / 4.12) * (Cosh[4.12 * 0.03846] - 1)
d = 0.243 * (Cosh[0.1584552] - 1)
d = 0.243 * (1.0125803 - 1)
d = 0.243 * 0.0125803
d = 0.003 light-years = 0.0009 parsec
So ship burns half its fuel, accelerates at 4g for two weeks, travels 0.0009 parsec, and has an ending velocity of 0.157 c.
Now, at the destination, it will require another 0.0009 parsecs and another two weeks to brake to a halt. This means the coasting distance is 1.0 parsec - (0.0009 + 0.0009) = 0.9982 parsec.
How long will the coast phase be?
0.9982 parsec = 3.254132 light-years
Ship's coast speed is 0.157c
3.254132 / 0.157 = 20.7 years
So the mission is:
Accelerate at 4g for 2 weeks. Ship will then have traveled 0.0009 parsec and have a velocity of 0.157c.
Ship drifts for 20.7 years, traveling a further 0.9982 parsec
Ship decelerates at 4 g for 2 weeks. Ship will travel remaining 0.0009 parsec and have braked to a halt. All fuel has been expended.
Bottom line: 20.8 years to travel 1 parsec.
