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Answer :
To determine the frequency at which the signal will be received on Earth, we need to consider the effect of the relative motion between the alien civilization and Earth.
When an object is moving relative to an observer, the observed frequency of a wave is shifted due to the Doppler effect.
In the case where the alien civilization is moving away from us at 0.4 times the speed of light (0.4 c), the observed frequency will be lower than the transmitted frequency. The formula for calculating the observed frequency (f') is given by:
f' = f * (1 - v/c),
where f is the transmitted frequency, v is the relative velocity, and c is the speed of light.
Plugging in the values, we have:
f' = 99.1 MHz * (1 - 0.4) = 59.46 MHz.
Therefore, the signal will be received on Earth at a frequency of approximately 59.46 MHz.
If the alien civilization were instead moving towards Earth at 0.4 c, the observed frequency would be higher than the transmitted frequency. Using the same formula, we have:
f' = 99.1 MHz * (1 + 0.4) = 138.74 MHz.
In this scenario, the signal would be received on Earth at a frequency of approximately 138.74 MHz.
These calculations demonstrate how the relative motion between the source and the observer affects the observed frequency of a signal due to the Doppler effect.
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