Final answer:
Using the Doppler Effect formula, we calculate the speed of a train approaching a crossing with a 200-Hz horn that is observed to be 208 Hz as approximately 19.23 m/s. Consequently, as the train moves away, the observer would perceive a frequency of approximately 193 Hz. None of the provided answer choices align with these calculations.
Step-by-step explanation:
Determining the Speed of the Train
When a commuter train blows its 200-Hz horn as it approaches a crossing, and an observer receives a frequency of 208 Hz, we can determine the speed of the train using the Doppler Effect formula for a source moving towards an observer:
f' = f (v / (v - vs))
Where:
f' is the observed frequency (208 Hz)
f is the source frequency (200 Hz)
v is the speed of sound (335 m/s)
vs is the speed of the source (the train)
Rearranging the formula to solve for the speed of the source (vs), we get:
vs = v(f/f' - 1)
Plugging in the values, we find:
vs = 335 m/s (200 Hz / 208 Hz - 1)
After calculating, vs ≈ 19.23 m/s. None of the given options match this calculation.
Calculating the Observed Frequency as the Train Recedes
Using the Doppler Effect formula for a source moving away from an observer, the observed frequency can be calculated:
f'' = f (v / (v + vs))
Substituting vs with the speed found earlier (19.23 m/s) and the given variables, we get:
f'' = 200 Hz (335 m/s / (335 m/s + 19.23 m/s)) ≈ 192.5 Hz
Again, rounding this to the nearest whole number gives us approximately 193 Hz, which does not match any of the options provided.