Here is my solution to part (b) of the posted problem which was 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)
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.