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My apologies for my original misconstruction of Part (b) -- I agree with
unprimed frame = spaceship frame
primed frame = earth frame,
which moves at speed .95 in the negative x direction relative to the rocket
assume the two frames are coincident at t=t'=0 and the origin of either
frame is the location of the tail of the rocket at this time.
I'll measure time in units of meters (c=1) and convert to SI units at the
end of the calculation.
Event A = emission of light pulse at the tail of the rocket
Event B = detection of light pulse at the head of the rocket
We will use the idea of the invariant interval to solve the problem; i.e.
the interval between A and B is an invariant.
* denotes multiplication
^ denotes eponentiation
_ denotes a subscript
Interval_AB^2 = delta x ^2- delta t ^2 = delta x'^2 - delta t'^2 = 0
delta x = 200 meters , delta t = 200 meters , as discussed in
The Lorentz transformation says that:
delta x' = gamma * .95*delta t + gamma * delta x = gamma * (1.95)* 200
= 3.203*1.95*200=1249 meters
Note: gamma=1/sqrt(1-.95^2)= 3.203
substitute this into the expression for the interval and solve for delta t'
and you get
delta t' = 1249 meters (or just realize the interval is light-like)
which upon dividing by the speed of light in SI units gives
delta t' = 4.16 micro seconds.
In short I think the analysis in the original posting (second method) was
correct and uses more physical based reasoning then the above, but I wanted
to give the more mathematically based solution; and therefore the stated
answer in the book is incorrect.
I believe that you can not just use a time dilation factor (which the book
apparently did!!) because both dilation and and contraction effects are
occuring in this problem.