ELECTROMAGNETISM & RELATIVITY - Chapter 8, Exercise 2 Solution
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Length Contraction at 0.6c

PROBLEM
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A spacecraft has a proper length of 80 m and travels at 0.6c. Using
the same Lorentz factor from Exercise 1, calculate its contracted
length as measured by a stationary observer.

SOLUTION
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Using the Lorentz factor already found in Exercise 1 (gamma = 1.25
at v = 0.6c), and the length contraction formula:

    L' = L / gamma

Substitute the given values:

    L = 80 m (proper length, measured in the spacecraft's own rest
        frame)
    gamma = 1.25

    L' = 80 / 1.25
    L' = 64 m

ANSWER: The stationary observer measures the spacecraft's length as
64 m - shorter than its own 80 m proper length.

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WHY THIS WORKS AS AN ANSWER

Length contraction and time dilation share the exact same Lorentz
factor, which is why Exercise 1's own gamma value (1.25) could be
reused directly here without recalculating it - a genuine, real
structural link between the two effects, both consequences of the
same single formula derived from Einstein's two postulates. The
spacecraft's own crew, in their own rest frame, still measures their
ship at its full, unchanged 80 m - it is only the STATIONARY
observer, watching the spacecraft move past at 0.6c, who measures the
shorter 64 m. Both measurements are equally real and equally
"correct" within each observer's own reference frame - this is the
genuine, non-illusory nature of relativistic length contraction the
chapter's own worked example describes.