Use frame-dependent lengths and momentum while avoiding photon and mass misconceptions.
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Method and worked examples
Proper length L0 is measured in the object’s rest frame. The length parallel to relative motion is L=γL0; perpendicular dimensions are unchanged. Moving length requires simultaneous endpoint positions in the measuring frame. Contraction describes frame-dependent measurement rather than ordinary mechanical crushing.
For nonzero rest mass, p=γmv, total energy E=γmc2 and kinetic energy K=(γ−1)mc2. At low speed γ≈1, giving p≈ mv. As v approaches c, energy and momentum grow without bound, so finite energy cannot accelerate a massive particle to c. Do not apply γmv to photons at v=c; use p=cE.
Example 1
A spacecraft has proper length 120m and γ=1.50. Find its length along motion in the observer frame.
L=γL0=1.50120=80.0m.
Example 2
A particle has rest mass 2.00×10−27kg,v=0.600c and c=3.00×108ms−1. Find momentum.