Muons are subatomic particles that sometimes strike Earth's surface.
One way they can be produced is via cosmic ray interactions in the
upper atmosphere.  They exist for only 0.0001 second before they decay.

Let's say some muons are produced at an altitude of 100 km - that is, 
100000 meters from Earth's surface.  They head toward the surface at
98% the speed of light.

(a) During the lifetime of a muon described above, what is the maximum
distance light can travel?  Since nothing travels faster than light,
why would this calculation lead you to conclude that no un-decayed
muons would ever reach Earth's surface from an altitude of 100 km?

(b) However, un-decayed muons produced at an altitude of 100 km HAVE been 
detected at Earth's surface.  The reason is that muons are subject to
special relativistic time dilation and length contraction.  Either point
of view explains the paradox!

Let's do time dilation first.  During the time the muon experiences
0.0001 second, how much time would you, standing on Earth's surface,
experience?  (Remember that t' = t * sqrt(1 - v^2/c^2) )

(c) Using the "dilated time" of you observing the muon, how far could
the muon go before it decays?  So, would the muon decay before or after
it reaches Earth's surface?

(d) Now let's do length contraction.  You, standing on Earth's surface,
see the muon being produced 100 km away.  How far away does Earth's
surface appear to the muon?  (Remember that L' = L * sqrt(1 - v^2/c^2) )

(e) Using the "contracted length" of the muon observing Earth, how much
time would it take for the muon to get to Earth's surface?  Will it decay
before or after it gets there?