This problem involves the concept of length contraction in special relativity. According to the theory of relativity, an object moving at a high velocity will appear shorter in the direction of motion when measured by an observer at rest relative to the object. The formula for length contraction is:
L' = L / γ
where L is the length of the object in its rest frame, L' is the length as measured by an observer in motion relative to the object, and γ is the Lorentz factor, given by:
γ = 1 / sqrt(1 - v^2/c^2)
where v is the velocity of the object and c is the speed of light.
In this problem, the length of the rocket ship as measured by the ground-based observer is L = 160 m, and the velocity of the rocket is v = 0.82c, where c is the speed of light. We want to find the length of the rocket when it is brought to rest, which corresponds to the rest frame of the rocket. In the rest frame, the rocket's velocity is zero, so we have:
γ = 1 / sqrt(1 - 0^2/c^2) = 1
Therefore, the length of the rocket in its rest frame is:
L_rest = L / γ = L = 160 m
So the length of the rocket when brought to rest is the same as its length when measured by the ground-based observer, which is 160 m.