Distinguish proper time from laboratory time and calculate fast-particle lifetimes.
Method and worked examples
The Lorentz factor is γ=√1−c2v21. Proper time Δt0 is recorded in a frame where both events occur at one place. An observer who sees that clock moving records Δt=γΔt0. Identify the event pair and clock before substituting. γ is at least one and increases as v approaches c.
A fast unstable particle has a laboratory mean lifetime γτ0 and mean travel distance vγτ0. A mean lifetime is a statistical scale, not an exact decay time for every particle. The increased laboratory survival of fast atmospheric particles is a standard example of time dilation. Keep microseconds as 10−6s when calculating distances.
Example 1
Find γ at v=0.800c.
γ=√1−0.80021=0.6001=1.67.
Example 2
A particle has proper mean lifetime 2.00μs and speed 0.800c. Use c=3.00×108ms−1. Find its laboratory mean travel distance.