I'm having trouble reconciling the quantitative and conceptual aspects of length contraction. This example is taken out of a book:
Say a particle is moving toward us at 0.99c, relative to us. If at rest, this particle typically decays within 0.2 microseconds, then we will measure 1.4 microseconds (I understand this part). Then, relative to us, the particle travels 1.4 * 0.99c ≈ 420 meters. One way to find the distance relative to the particle is to simply do: 0.2 * 0.99c ≈ 60 meters. This doesn't make sense to me. Shouldn't what "we" measure be the smaller length? For this situation to mathematically work out we have to do L = Lo/γ, where L = 60 and Lo = 420. Shouldn't it be the other way around? Isn't the distance measured in the rest frame 60 (the smaller value), not 420?
This is not a homework question. I'm just reviewing these concepts and was stumped by this one.