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Help w/ Relativity Problem?

I am a high school teacher teaching an AP Level B physics class for the
second time. Last year my "freshamn status" as an AP teacher led me to
the common problem of running out of time at the end of the year before I
had gotten to do much special relativity or modern physics. This year,
having learned from last year, I have planned a lot better and I am able
to get a lot more detail on the relativity chapter. Although I am glad
to have given myself (and the students) the time to study this topic, it
has exposed some of my weaknesses in terms of my understanding.

(Sorry for the long intro!) Here's the bottom line, I was hoping I could
get some help regarding a certain problem. The problem reads as follows:

A spaceship has a length of 200m in its own reference
frame. It is traveling at 0.95c relative to Earth.
Suppose that the tail of the spaceship emits a flash
of light. (a)In the reference frame of the spaceship,
how long does the light take to reach the nose?
(b)In the reference frame of the Earth, how long does
this take? Calculate the time directly from the motions of
the spaceship and the flash of light, and explain
why you cannot obtain the answer by applying the
time-dilation factor to the result from Part (a).
(from Ohanian's Principles of Physics)

The answers listed in the book are (a)6.67E-7s and (b)2.13E-6s I have no
difficulty with (a) as being simple v=d/t using c and the
non-length-contracted 200m since the observers are at rest relative to
the ship. It is part (b) that I am having trouble with. As I understand
it, observers that see the ship fly by will see its length contracted.
(It is actually contracted to 62.4m) So, since the speed of light is the
same in all reference frames, I figured this observer would see the light
travel 62.4m at a speed of c. Using v=d/t this gave me 2.08E-7s.
WRONG! I though about it, and I figured that this was wrong because not
only does the light pulse have to travel the length of the ship, but also
the distance that the ship travels while it is headed towards the nose.
Setting this up (which I think I set it up right) and solving still gave
me a wrong answer (4.16E-6).

I have tried several other calculations (including length-contracted the
distance travelled by the ship) but I am afraid that I am trying to fit
the calculation to give me the desired result. I would rather have the
understanding of the problem and work from there.

The crazy thing is, although the book says "explain why you cannot obtain
the answer by applying the time-dilation factor to the result of Part
(a)", that is exactly what you have to do to get the answer they list!?
What's going on here? Am I missing some basic step or do I understand
even less than I previously thought!? Any insight or advice would be
greatly appreciated!

Thanks in advance!

Jonathan Gillis
Enloe High School
Raleigh, NC