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The Starship Enterprise approaches a planet at a speed of 0
The Starship Enterprise approaches a planet at a speed of 0.8 c. The planet is considered at rest. A Klingon fighter ship approaches from the opposite direction at 0.6 c.
(a) What speeds do people at each location see the other traveling?
(b) A light signal of 100 MHz is issued by Enterprise. What speed does each measure the signal as traveling and at what frequency?
In part a, I am assuming I should be using the Lorentz velocity transformation, but am unsure of how to do this since I don't know what direction (x, y) the rockets are traveling.
In part b, if I use the relativistic Doppler Effect, I have two unknowns:
f_O = (?1- v/c / ?1 + v/c) f_S
I need to find both the velocity (v) and the observed frequency (f_O). How do I do this?
Expert Solution
(a)
You are right to use the Lorentz transformation.
As to the axes note the text "... Klingon fighter ship approaches from the opposite direction ...".
This means the two ships move along the same straight line towards each other.
Therefore you can just choose this line to be the X-axis (or Y or Z whichever you like).
Suppose this axis is directed from Enterprise to Klingon. Then, with respect to that planet, the Entrerprise velocity is v1 = +0.8c and the Klingon velocity is v2 = -0.6c along the X-axis.
Therefore, the velocity of Klingon with respect to Enterprise is
vK = (v2-v1)/(1-v1*v2/c^2) = (-0.6c - 0.8c)/(1+0.8*0.6) = -0.9459c along X-axis
and the velocity of Enterprise with respect to Klingon is
vE = +0.9459c along X-axis
(b) The speed of light is the same in any frame, so both the spaceships measure it to be c
The frequency however is not the same.
Enterprize measures the same 100 MHz it issues, however the frequency measured by Klingon is
100 MHz *sqrt[(c+vE)/(c-vE)] = 100 MHz *sqrt[(c+0.9459c)/(c-0.9459c) =
= 100 MHz * sqrt(1.9459/0.0541) = 600 MHz
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