A spaceship communicates with the Earth via 75 MHz radio waves. If the spaceship is traveling away from the Earth at a speed of 0.966c, what frequency should a receiver be tuned to on the Earth?
Question
A spaceship communicates with the Earth via 75 MHz radio waves. If the spaceship is traveling away from the Earth at a speed of 0.966c, what frequency should a receiver be tuned to on the Earth?
Solution
The Doppler effect for electromagnetic waves, such as radio waves, is given by the formula:
f' = f * sqrt((1 + v/c) / (1 - v/c))
where: f' is the observed frequency, f is the source frequency, v is the velocity of the source, c is the speed of light.
Given: f = 75 MHz (the frequency of the radio waves), v = 0.966c (the speed of the spaceship), c = 1 (when we use c as the unit for v, c becomes 1 in these units).
Substituting these values into the formula, we get:
f' = 75 MHz * sqrt((1 + 0.966) / (1 - 0.966)) f' = 75 MHz * sqrt(1.966 / 0.034) f' = 75 MHz * sqrt(57.8235) f' = 75 MHz * 7.601
Therefore, a receiver on the Earth should be tuned to approximately 570.08 MHz to receive the communication from the spaceship.
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