A friend passes by you in a spacecraft traveling at a high...

A friend passes by you in a spacecraft traveling at a high speed. He tells you that his craft is $20.0 \mathrm{m}$ long and that the identically constructed craft you are sitting in is $19.0 \mathrm{m}$ long. According to your observations, (a) how long is your spacecraft, (b) how long is your friend's craft, and (c) what is the speed of your friend's craft?

Key Concepts

Proper Length

The proper length is the length of an object as measured in its own rest frame, where the object is not moving. It represents the maximum length that can be recorded for the object, and any moving observer would record a length that is contracted relative to the proper length.

Length Contraction

Length contraction is a phenomenon predicted by special relativity in which the length of an object measured by an observer in motion relative to the object is shorter than its proper length. This contraction occurs only along the direction of motion and becomes significant at speeds close to the speed of light.

Lorentz Factor

The Lorentz factor, often denoted by ? (gamma), quantifies the effects of time dilation and length contraction in special relativity. It is defined as ? = 1/?(1 - v²/c²), where v is the relative velocity between frames and c is the speed of light. The factor determines how measurements of time and space differ between inertial frames.

Relative Motion

Relative motion refers to the observation of different inertial frames moving with respect to each other. In the context of special relativity, the effects such as time dilation and length contraction depend on the relative velocity between the observer and the moving object, making the measurements of spatial and temporal intervals frame-dependent.

Inertial Frame

An inertial frame is a reference frame in which an observer is either at rest or moving with constant velocity, meaning no acceleration is present. The laws of physics, including the principles of special relativity, hold in their standard form in these frames, allowing for consistent application of formulas like length contraction and time dilation.

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