 So the resolution to this paradox is that in the muon's frame of view, the atmosphere is length contracted because the atmosphere, like the Earth, is flying towards it at speed V. And so this only has length L on gamma. So now the condition for the muon to reach the surface is for tau to be greater than L on gamma on V or gamma tau greater than L on V. So now we see that the condition for the muon to reach the surface is the same in the Earth's frame or the particle's frame. This is the general theme of paradoxes and special relativity. It seems like looking at the situation from two different points of view, you get two different physical outcomes. But when you take everything into account and think clearly about the physical situation, you find out that there is no real problem and everything works out.