##
Special Relativity in 14 Easy (Hyper)steps

**12. Paradox explained**
This one's easy to sort out if we include the tunnel clocks
in the drawing
showing the train's frame. Recall
that clocks which are synchronized in one frame will appear out of synch
in another frame. In particular, the "forward" clock is set to an earlier
time than the "aft" clock.

Recall that
where v is 0.8 c and L is the **rest-frame separation**
between the clocks. Plugging
in reveals that the left tunnel clock reads -640 nsec (according to observers
on the train) when the right tunnel clock reads zero. Keep in mind
that the train sees the tunnel clocks ticking slowly,
so the left
clock won't reach zero until (640 / 0.6) nsec (about 1007 nsec)
after the right clock does.

Here's another diagram, including the tunnel clocks:

The right door slams when its clock reads zero.
The train crashes
through it and continues moving. By the time the left tunnel clock finally
reads zero (~1007 nsec after the right door slams, in the train's rest
frame), the left door of the tunnel will have moved an additional 853.3
feet, past the left end of the train.