When particles from space (cosmic rays) strike atomic nuclei of the air high in the atmosphere, they create particles called muons. These appear about km up and head for the ground at very nearly the speed of light.
Take the speed here as . The Lorentz factor is then .
Detectors on the ground find muons in large numbers. Yet a calculation from their lifetime says hardly any should arrive.
Even at rest, a muon decays into an electron and other particles after microseconds on average. These microseconds are its mean lifetime.
In one mean lifetime it covers microseconds, about m. Covering km takes about mean lifetimes.
After mean lifetimes, the fraction still undecayed is . is about one in several million, so Newtonian mechanics cannot explain why so many are found at the ground.
Seen from the ground, the muon moves at . The time it takes to decay is stretched by time dilation, by the factor .
The microseconds is the lifetime measured on a clock moving with the muon. On ground clocks it is microseconds, about microseconds.
In microseconds it covers microseconds, about km. The trip down takes about of the stretched lifetime, so , roughly percent, reach the ground.
In the frame moving with the muon, the muon is at rest and its lifetime stays microseconds. Instead, the Earth and its atmosphere rush up from below at .
A moving length contracts by the factor , so the km of air becomes km, about m thick.
The ground takes about of a mean lifetime to cover those m. Just as in the ground frame, , roughly percent, reach the ground.
From the ground the time stretched; from the muon the distance shrank. The reasoning differs, but the fraction reaching the ground is either way.
How many muons reached the detector is a single fact, settled by counting. What may differ between frames is the value of a time or a length, not the count.
In Frisch and Smith counted muons on a mountain top about m high and near sea level: about an hour at the top and about near sea level. Without the stretched lifetime, the count near sea level should have dropped to around . The agrees with the prediction that includes time dilation.
Time dilation and length contraction are not separate phenomena; they are the same phenomenon, written from the ground's frame or from the muon's.