# Variable acceleration relativistic rocket

I was wondering how a realistic fusion-or-something rocket would fly... More specifically, I wonder about the time dilation that would be experienced (say in the classic example of two twins).

So, of course I started by searching the web for some nice calculators... There are two that do basically the same thing, this is one of them:

http://nathangeffen.webfactional.com/spacetravel/spacetravel.php

It assumes some acceleration a for half of the trip and the same a deceleration for the other half. But for large distances your spaceship's velocity climbs really fast to over 95% of c, at which point I'd assume the spaceship will start getting thermalized by blue-shifted cosmic rays and dust... So I'd like to calculate time dilation for a trip like that:

1) launch, accelerate at a

2) reach a given max v and stop the thrust

3) decelerate at a, land

Would that be the same as summing up (1), (3) with the calculator above, with (2) from a calculator that assumes constant velocity? Or if not, what would be a formula for variable relativistic acceleration?

Alternatively, is there a way to calculated a, based on a set max v and distance?

[I'm interested in is the final age difference of twins, one observer, the other on a relativistic rocket with a max-v limit, given the distance and max-v or a.]

Edit: If I wasn't clear before, I'm asking for a given v max, NOT 0.95c or anything near it. For fusion, probably up to 0.2c? But that's not the point - I'm looking to calculate for any reasonably relativistic maximum v.

Edit #2: So I went ahead and tried to do the math... Can someone check if it is conceptually ok? I'm still not sure I can sum time intervals that way... Also, I'm ignoring any jerk from (1)->(2) and (2)->(3).

Indexes refer to which part of the trip I'm talking about. c is speed of light, v, t, d are measured based on stationary observer. T is time on-board. ch and th are hyperbolic function. Acceleration formulas from: http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html Time dilation from memory.

$$T_{tot} = T_1 + T_2+T_3 = 2T_1 + T_2$$

$$v_f = c*th(a*T_1/c) => T_1 = c*th^{-1}(v_f/c)/a$$

$$T_2 = t_2*sqrt(1-(v_f/c)^2) = d_2*sqrt(1-(v_f/c)^2)/v_f$$

$$d_2 = d_{tot} - 2*d_1 = d_{tot} - 2*c^2*(ch(a*T_1/c)-1)/a$$

Hence:

$$T = 2*c*th^{-1}(v_f/c)/a + sqrt(1-(v_f/c)^2)/v_f*[d_{tot} - 2*c^2*(ch(a*T_1/c)-1)/a]$$

$$T = 2*c*th^{-1}(v_f/c)/a + sqrt(1-(v_f/c)^2)/v_f*[d_{tot} + 2*c^2/a - 2*c^2/(a*sqrt(1-(v_f/c)^2)]$$

$$T = 2*c*ch^{-1}(1/sqrt(1-(v_f/c)^2))/a + d_{tot}*sqrt(1-(v_f/c)^2)/v_f - 2*c^2*(1-sqrt(1-(v_f/c)^2))/(a*v_f)$$

So that's the time on-board T as a function of total distance, acceleration and the top velocity which you can reach without starting to burn... Please tell me if that's correct, and if not, how to correct it?