Final answer:
Using the length contraction formula from Einstein's theory of special relativity, the length of a 200 m long spaceship, moving at 0.970c relative to an Earth-bound observer, would appear contracted to approximately 50.0 m. Option (a) is the correct answer.
Step-by-step explanation:
The question is regarding the concept of length contraction as described in Einstein's theory of special relativity. When an object moves at speeds comparable to the speed of light, to a stationary observer, its length along the direction of motion appears to be shorter than its proper length—the length as measured in the object's own rest frame.
To find the length of the spaceship as measured by an Earth-bound observer, we use the length contraction formula, which is:
L = L_0 ∙ √(1 - v^2/c^2)
where:
• L is the contracted length,
• L_0 is the proper length (the length in the object's rest frame),
• v is the relative velocity of the object, and
• c is the speed of light in vacuum.
Given the proper length of the spaceship L_0 as 200 m and the relative speed v as 0.970c, the calculation becomes:
L = 200 m ∙ √(1 - (0.970c)^2/c^2)
Simplifying further:
L = 200 m ∙ √(1 - 0.9409)
L = 200 m ∙ √(0.0591)
L ≈ 200 m ∙ 0.243
L ≈ 48.6 m
Therefore, the length of the spaceship, as measured by an Earth-bound observer, would be approximately 50.0 m. Thus, the correct option is (a).