A very fast-moving train car passes you, moving to the right...

A very fast-moving train car passes you, moving to the right at $0.50 c .$ You measure its length to be $12 \mathrm{m} .$ Your friend David flies past you to the right at a speed relative to you of $0.80 \mathrm{c}$. How long does David measure the train car to be?

Key Concepts

Inertial Reference Frames

Inertial reference frames are coordinate systems in which objects either remain at rest or move at constant velocity (i.e., there is no acceleration). In the context of special relativity, physical laws, including those governing the speed of light and the Lorentz transformations, are the same in all inertial frames, making it fundamental to understanding measurements like length contraction and time dilation.

Relativistic Length Contraction

Relativistic length contraction is a phenomenon predicted by special relativity whereby an object moving relative to an observer is measured to be shorter along the direction of motion than its proper length (the length measured in the object’s own rest frame). This contraction is not due to a physical compression but arises from the geometric nature of spacetime as described by Lorentz transformations.

Proper Length

Proper length is the intrinsic length of an object measured by an observer who is at rest relative to that object. It is an important concept in special relativity because measurements from any other frame (where the object is in motion) will yield a contracted length due to the effects of relative motion.

Lorentz Factor

The Lorentz factor, often denoted by ? (gamma), quantifies the amount by which time, length, and relativistic mass change for an object moving relative to an observer. It is defined as ? = 1/?(1 ? v²/c²), where v is the relative velocity and c is the speed of light. The Lorentz factor plays a central role in computing length contraction and time dilation.

Relativistic Velocity Addition

Relativistic velocity addition is the rule in special relativity used to combine velocities when objects are moving in different inertial frames. Unlike classical addition of velocities, the relativistic formula ensures that the resulting speed does not exceed the speed of light and accounts for the effects of time dilation and length contraction on measured velocities.

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