Here’s How to Crack It
- As measured on Earth, the time interval between the moment when the
muon is created and the moment it decays is
∆T 2 = γ • ∆T 1 = γ • (0.2 microsec)
In this case, the relativistic factor is
So, as measured by observers on Earth, the muon survives for
∆T 2 = γ • ∆T 1 ≈ 7 • (0.2 microsec) = 1.4 microsec
- Since the speed of the muon is 0.99c, and we measure its lifetime as 1.4
microseconds, the distance it travels, as measured by us, is
D = vT = (0.99c)(1.4 × 10−6 s) = (0.99)(4.2 × 10^2 m) ≈ 420 m
- In the reference frame of the muon, it’s the earth that’s rushing up toward
the muon at a speed of 0.99c. Since, in the muon’s frame of reference, it only
lives for 0.2 microsecond, the muon measures its distance of travel to be
d = vt = (0.99c)(0.2 × 10−6 s) = (0.99)(0.6 × 10^2 m) ≈ 60 m
Notice that since the muon is moving relative to the earth, it
measures distances as being shorter than we do. So, from our
point of view, the muon lives longer as it travels a proper
distance of 420 m. From the muon’s point of view, it lives its
ordinary, proper lifetime, 0.2 microsecond, but only travels 60
m. These points of view are both correct (remember, distances
and time are relative) and they’re compatible. In one point of
view, the time stretched by a factor of γ = 7, while in the other
point of view it was the distance that shortened by a factor of γ
= 7. (The time interval between two events is said to be proper
