Answer :
Final answer:
The spacecraft must be moving away from the radio station to hear a lower frequency due to the Doppler Effect. Since the speed of light is not a critical factor for such a small change in frequency, the velocity of the spacecraft would be much less than the speed of light.
Explanation:
The problem presented involves Doppler Effect in waves, an essential concept in Physics, particularly in the study of wave motion and relativity. In this case, we're examining how the frequency of a radio wave changes due to relative motion between the source and the observer. Here's the breakdown:
1. The radio station on Earth is transmitting at 100 Hz.
2. The spacecraft wants to tune into the radio station at 99.9 Hz.
Since the frequency heard is lower than the transmitted frequency (99.9 Hz < 100 Hz), the spacecraft must be moving away from the radio station. This is known as the Doppler Effect, where the frequency decreases ('redshift') if the source and observer are moving away from each other.
To calculate the velocity, we would use the formula for the relativistic Doppler shift:
f' = f * sqrt((1 - v/c) / (1 + v/c))
Where:
- f' = observed frequency (99.9 Hz)
- f = emitted frequency (100 Hz)
- v = velocity of the receiver (spacecraft)
- c = speed of light (3 \u00d7 108 m/s)
However, the student's question is designed to check for conceptual understanding and does not require an exact computation. Since the change in frequency is small (0.1%), we can assume a non-relativistic scenario where the speed of light does not play a critical role, and therefore, the speed of the spacecraft will also be very small compared to the speed of light.
Hence, the correct answer choice is that the spacecraft would have to be moving away from the radio station (option a), and the velocity is much less than either 3 \u00d7 108 m/s (option c) or 3 \u00d7 106 m/s (option d).
