Why Does Time Dilation Not Apply When Rocket A's and Earth's Tails Meet?

[eok20]

Suppose two relativistic rockets A and B are headed towards each other. When the noses meet, rocket A and an earthbound observer set x=t=0. Let T be the time, relative to the earthbound observer, that the tails of the rockets meet. Why is it not the case that the time that the tails meet relative to rocket A is $$\gamma$$T ?

 

[Doc Al]

That time dilation formula applies to a single moving clock. Do the two events (noses meet; tails meet) occur at the same position in the earthbound frame (and thus the time difference is recorded by a single earthbound clock)? Not necessarily.