Final answer:
The speed of the train can be calculated using the formula for the Doppler effect. Therefore, the frequency heard by the observer as the train moves away is approximately 4.314 times the original frequency.
Step-by-step explanation:
When a train approaches a crossing, the frequency of the sound heard by an observer changes due to the Doppler effect. The Doppler effect describes the change in frequency and wavelength of a wave as the source and observer move relative to each other.
(a) To find the speed of the train, we can use the formula for the Doppler effect:
Δf/f = v/(v±vs)
Where Δf is the change in frequency, f is the original frequency, v is the velocity of sound, and vs is the velocity of the source (train). Rearranging the formula to solve for vs:
vs = v(1 - f/Δf)
Substituting the given values:
vs = 343(1 - 208/888)
vs ≈ 343(1 - 0.234)
vs ≈ 343(0.766)
vs ≈ 263.438 m/s
Therefore, the speed of the train is approximately 263.438 m/s.
(b) To find the frequency heard by the observer as the train moves away, we can use the same formula, but with a negative sign:
Δf/f = v/(v ± vs)
Substituting the given values:
Δf/f = 343/(343 ± 263.438)
Δf/f ≈ 343/79.562
Δf/f ≈ 4.314
Δf ≈ 4.314f
Therefore, the frequency heard by the observer as the train moves away is approximately 4.314 times the original frequency.