SCSA Physics Science as a Human Endeavour

The relativistic speed of the mesons indicates they experience time dilation relative to the observed time over the distance travelled. This means the mesons appear to travel for a longer period of time and hence cover a larger distance.

In the muon's reference frame, the distance travelled is less than that observed by a stationery observer on Earth's surface, due the effects of length contraction. This shortened distance means that they will get further than would be expected by an observer observing the rest length.

From the Earth's frame of reference, the time dilation means that the half-life of the muon is dilated compared to the half-life measured in the rest frame of the muon. This greater time allows more muons to reach the ground than would otherwise be expected.

The two postulates of special relativity are: the speed of light in a vacuum is an absolute constant, and all inertial frames of reference are equivalent.

The consequences of these postulates are time dilation, length contraction and relativistic momentum dilation. Time dilation means that the measured time between events increases as the speed of an observer increases. Length contraction means that the length of an object decreases as the speed of an observer increases, and relativistic momentum dilation means that objects (with mass) cannot travel at c as it takes increasing amounts of energy to accelerate objects as they approach the speed of light.

Length contraction and time dilation can both be seen in observations of muons at Earth's surface. The short half-life of muons suggests that there should be very few of them that arrive at Earth's surface from their point of origin high in Earth's atmosphere, but the muon flux is greater than suggested by half-life and distance travelled alone. However, when the distance travelled is considered from the muon's frame of reference, it is length-contracted according to $l = l_0 \sqrt{\left(1 - \frac{v^2}{c^2}\right)}$, giving an effective shorter distance for them to travel. From the perspective of an observer on Earth's surface, the time taken for them to decay is dilated, giving an effective greater amount of time for them to travel before decaying.

The effects of relativistic momentum can be seen in particle accelerators in terms of the increasing amounts of energy to required to accelerate the particles as they approach the speed of light – it is though the objects are much more massive than they are.

Since muons are short-lived particles, they shouldn't reach the Earth's surface according to Newtonian physics.
But muons travel at close to the speed of light. As muons enter the atmosphere, an observer external to the particles' frame of reference will witness them existing longer than expected due to time dilation. Muons experience length contraction, so they can travel further than predicted by Newtonian physics.