SpiderET said:
Lets assume we have a starship which is flying from Earth to star XY which is in distance for example 100 lightyears. The computer of that ship is programmed that way, that it maintains acceleration 1 g. After some time the speed of ship reaches some significant part of speed of light and to maintain the acceleration, the engins must burn more and more fuel to maintain 1 g. But there is also the effect of time dilation, contraction and increase of mass.
With the right analysis, you can get rid of a lot of these factors. Interestingly enough, though, people seem to resist doing this, even when they are told how it can be accomplished.
How do you do this? Well, the first key observation is this. The value of the coordinate acceleration of the ship depends on the observer, or the observers frame of reference. This is probably the most important step to working the problem. The notion of "proper acceleration" would also be useful to understand the problem, but it appears to be unfamiliar to many readers.
Now I'll start some analysis using a 'question and answer' format.
1) We noted that the acceleration of the ship depends on the observer. For w hat observer, in what frame of reference, is the coordinate acceleration of the ship equal to specified value of 1g.
answer: this frame of reference is the reference frame of the ship. Note that this is the only frame where the proper acceleration is equal to the more familiar coordinate acceleration.
2) What is the "relativistic mass" of the ship in this frame.
answer: it is equal to the rest mass of the ship.
3) Doesn't this make the relativistic mass pretty much irrelevant to solving the problem
answer yes
4) Why do people go on and on about the relativistic mass, then, if it's really not that useful in solving the problem
answer beats me :-). If you find out, let me know. I do suspect that the usual reason for this focus on relativistic mass is related to a belief that saying "it requires infinite energy to accelerate close to the speed of light" (true) is the best way to describe why objects can't reach the speed of light (probably false).5) Wait - this is getting a bit off topic.
answer: sorry about that, couldn't resist.
6) OK, let's get back on track. We've determined that we find the acceleration of the ship in its own frame, where we don't need to worry about relativistic mass, and we only need its non-relativistic mass and it's thrust. How do we go about finding the acceleration (and velocity, and position) of the ship versus time in a wholly inertial reference frame - i.e the initial inertial reference frame the ship was in before it started accelerating?
answer: This starts to get a bit mathematical, but the math required doesn't need any dynamics at all, only kinematics. The kinematics required are the Lorentz transform and perhaps the velocity addition formula. See
http://en.wikipedia.org/wiki/Velocity-addition_formula
In some amount of ship time, (also called proper time), ##\tau##, we know that the ships velocity increases by a multipled by ##\tau## in the ship frame. The velocity equation in general says that we add two velocities v1 and v2 using the relativistic formula
v_tot = (v1 + v2) / (1 + v1 v2 / c^2)
In this case we have v1 = ##v_{ship}## and v2 = ##a \tau## so we get
<br />
\frac {v_{ship} + a \, \tau}{1 + v_{ship} a \tau / c^2}<br />
where## v_{ship} ## is the current velocity of the ship
a is the proper acceleration of the ship (measured in the ship frame)
##\tau## is the proper time of the ship (measured in the ship frame)
c is the speed of light
If you then consider ## v_{ship}## to be a function of ##\tau##, you can write a differential equation and solve it as a function of ##\tau##. You would still need to do some work to express ##\tau## in terms of what you probably want, which would be "t", the amount of coordinate time in the initial inertial reference frame of the ship before it started accelerating.
You can use the well known time dilation equation ##d\tau = {dt}{\sqrt{1-v^2/c^2}}## to relate t and ##\tau##
This gets detailed enough that I'll give a link to the answer instead of trying to wade through it - see "The Relativistic Rocket" http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html
The end result for the distance, velocity, and acceleration of the ship in an inertial where the ships initial velocity at t=0 is 0 is:
distance = ( (c2/a) (sqrt[1 + (at/c)2] - 1)
velocity = at / sqrt[1 + (at/c)2]
The reference doesn't give acceleration, we can derive it easily enough by differentiating velocity with respect to t
acceleration = a / [1 + (at/c)^2)] ^ (3/2)
Note that while the proper acceleration a of the ship in its own frame remains constant, the coordinate acceleration drops off as time increases, becoming lower and lower.
