The half-life of a particle that decays according to
is defined as the time such that . Therefore, if half of the muons live to reach the detector, then the muons must have experienced a time of between their generation and their detection.
We use the time dilation formula to find the time between the generation and the detection of the muons in the laboratory frame:
Squaring both sides and solving for gives:
Using the fact that the invariant interval between the generation and the detection of the muons must have the same value in both reference frames, we have
Therefore, the velocity in the lab frame is
Therefore, answer (C) is correct.