Relativistic Uniform Accelerated Motion

[VVS]

Homework Statement

Hi I am supposed to calculate the distance traveled by a particle that is uniformly accelerated with acceleration equal to the Earth's gravity (i.e. a=9.81m/s^2) after 1, 10 and 100 years proper time.

Homework Equations

We derived in class the four vector which describes the position of the particle:
t(s)=\frac{c}{a}sinh\left(\frac{as}{c}\right)
x(s)=\frac{c^2}{a}cosh\left(\frac{as}{c}\right)

The Attempt at a Solution

So far all I did was to substitute in the second equation the values of 1,10,100years. The first result is reasonably since it yields 0.6 lightyears. However the other results yield values which are way above 10 lightyears and 100 lightyears respectively. This can't be possible, because the particle can never exceed the speed of light. So I should get a value which is below 10 and 100 light years.
I am wondering whether I have to do something else. Do I have to find the proper time s through the first equation and then substitute that value in the second equation?

 

[Orodruin]

It does not have to exceed the speed of light to travel over 10 lightyears in 10 years of proper time. If you have something moving close to the speed of light, it will move essentially 10 lightyears in 10 of your years. However, time dilation means that essentially no time will have elapsed for the traveller.

Edit: Of course, these 10 light years are in the frame of an external observer in an inertial frame.