A spaceship moves at a constant velocity of 0.40 c relative...

A spaceship moves at a constant velocity of $0.40 c$ relative to an Earth observer. The pilot of the spaceship is holding a rod, which he measures to be $1.0 \mathrm{m}$ long. (a) The rod is held perpendicular to the direction of motion of the spaceship. How long is the rod according to the Earth observer? (b) After the pilot rotates the rod and holds it parallel to the direction of motion of the spaceship, how long is it according to the Earth observer?

Key Concepts

Orientation Effects

The orientation of an object relative to the direction of motion plays a key role in determining the observed relativistic effects. When an object is oriented parallel to the motion, its length appears contracted to a stationary observer, while an object oriented perpendicular to the motion remains unchanged. This concept is central in understanding how observers in different inertial frames measure dimensions differently based on spatial alignment relative to motion.

Lorentz Factor

The Lorentz factor, denoted by gamma (?), is a crucial element in the equations of special relativity and is defined as ? = 1/?(1 - (v²/c²)). It quantifies the magnitude of time dilation and length contraction. In problems involving high velocities (a significant fraction of the speed of light), calculating gamma is essential for determining how much an object's dimensions or temporal intervals change when observed from different inertial frames.

Proper Length

This refers to the length of an object measured in the frame in which the object is at rest. In the context of the problem, the spaceship pilot measures the proper length of the rod. It is fundamental because it serves as the baseline from which relativistic effects such as length contraction are computed in different frames.

Length Contraction

Length contraction is a phenomenon in special relativity where the length of an object moving relative to an observer is measured as shorter along the direction of its motion. It only affects dimensions parallel to the direction of travel, meaning that an object oriented perpendicular to the motion will not contract in that direction, highlighting the directional dependence of the effect.

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