Use frame-dependent lengths and momentum while avoiding photon and mass misconceptions.
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.