Worked example - muon time dilation. Cosmic-ray muons are created ~10 km up with a mean lifetime of 2.2 microseconds; classically they travel only ~660 m before decaying and should never reach the ground. Step 1: observed muons arrive with v = 0.998c, so gamma = 1/sqrt(1 - 0.998^2) = 15.8. Step 2: in Earth's frame their lifetime is dilated: tau = gamma x 2.2 us = 34.8 us, letting them travel 0.998c x 34.8 us = 10.4 km - they reach the ground. Step 3: equivalently, in the muon's frame the atmosphere is length-contracted: 10 km/15.8 = 633 m, crossed in 2.1 us of proper time. Both viewpoints agree - this 1941 Rossi-Hall measurement was among the first direct confirmations of relativity.