A spacecraft of the Trade Federation flies past the planet...

A spacecraft of the Trade Federation flies past the planet Coruscant at a speed of $0.600 c$. A scientist on Coruscant measures the length of the moving spacecraft to be $74.0 \mathrm{~m}$. The spacecraft later lands on Coruscant, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

Key Concepts

Special Relativity

Special relativity is a theory established by Einstein that describes how measurements of space and time vary for observers in different inertial frames of reference, particularly when relative velocities approach the speed of light. It introduces concepts such as time dilation and length contraction, emphasizing that the observed properties of objects (like length, time, and mass) depend on the relative motion between the observer and the object.

Length Contraction

Length contraction is a phenomenon predicted by special relativity, where an object in motion relative to an observer appears shorter along the direction of motion compared to its proper length (the length measured in the object’s rest frame). This contraction only becomes significant at speeds close to the speed of light and is calculated using a specific formula that involves the object's velocity relative to the observer.

Lorentz Factor

The Lorentz factor, often denoted as ? (gamma), quantifies the amount by which time, length, and relativistic mass change for an object moving at a relative velocity. It is defined as ? = 1/?(1 - v²/c²), where v is the relative velocity and c is the speed of light. This factor is essential in calculating phenomena like length contraction and time dilation, illustrating how the effects become more pronounced as the relative speed increases.

Proper Length

Proper length is the length of an object measured in the frame of reference in which the object is at rest. It serves as a reference measurement that remains unchanged regardless of the relative motion of the observer. When an object moves at relativistic speeds, the observed length will be contracted relative to its proper length, as described by the length contraction phenomenon.

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