FrontierScience / f4681d73-b08d-450c-89d3-85b9d1250419 / Consider a vehicle with a proper length of `L` in the `x`-direction.

Consider a vehicle with a proper length of `L` in the `x`-direction.
The rear end of the vehicle moves with a uniform proper acceleration `(a)→=a(x)^`, with `a>0`, i.e., in the inertial instantaneous rest frame of the vehicle's rear end, the vehicle's rear end has an acceleration of `(a)→`. 
At a certain moment, the velocity of the rear end of this vehicle relative to the laboratory frame, a fixed inertial frame, is `(v)→=v(x)^`, with `v>0`.
Find the length `L_(lab)` of the vehicle body measured in the laboratory frame at this moment.
You need to consider special relativistic effects, with the speed of light being `c`.
Please observe that, so as for the vehicle to not disintegrate during the acceleration process, we enforce that, for any constituent atom of the vehicle, when viewed within the instantaneous rest frame of that atom, the relative distance between that atom and any other atom of the vehicle does not change over time.
Give your answer in terms of `L,a,v` and `c`.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

Consider a vehicle with a proper length of `\( L \)` in the `\(x\)`-direction.
The rear end of the vehicle moves with a uniform proper acceleration `\( \vec{a} = a \hat{x} \)`, with `\( a > 0 \)`, i.e., in the inertial instantaneous rest frame of the vehicle's rear end, the vehicle's rear end has an acceleration of `\( \vec{a} \)`. 
At a certain moment, the velocity of the rear end of this vehicle relative to the laboratory frame, a fixed inertial frame, is `\( \vec{v} = v \hat{x}\)`, with `\( v > 0\)`.
Find the length `\( L_{\text{lab}} \)` of the vehicle body measured in the laboratory frame at this moment.
You need to consider special relativistic effects, with the speed of light being `\( c \)`.
Please observe that, so as for the vehicle to not disintegrate during the acceleration process, we enforce that, for any constituent atom of the vehicle, when viewed within the instantaneous rest frame of that atom, the relative distance between that atom and any other atom of the vehicle does not change over time.
Give your answer in terms of `\(L, a, v\)` and `\(c\)`.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.