The speed of the train is calculated using the observer's perceived frequency change due to the Doppler effect. By solving the Doppler equation, the train's speed is found to be approximately 19.82 meters per second.
When dealing with a situation where a train is moving at a constant speed and is passing a stationary observer, we use the Doppler effect to determine the train's velocity. The observer notices a change in the frequency of the sound being produced by a source on the train, in this case, a flute player continually playing concert A (f = 440 Hz). After the train has passed, the frequency heard by the observer is 415 Hz.
To calculate the speed (vs) of the train, we use the formula derived from the Doppler effect:
fobs = fs * (v / (v + vs))
Here, fobs is the observed frequency, fs is the source frequency, v is the speed of sound in air which is given as 343 m/s, and vs is the speed of the train we want to find. Plugging the observed and source frequencies into the formula, we can solve for vs.
415 Hz = 440 Hz * (343 m/s / (343 m/s + vs))
Multiplying both sides by (343 m/s + vs) / 343 m/s and simplifying, we get:
415 Hz * (343 m/s + vs) = 440 Hz * 343 m/s
Now let's solve for vs.
415 Hz * 343 m/s + 415 Hz * vs = 440 Hz * 343 m/s
415 Hz * vs = 440 Hz * 343 m/s - 415 Hz * 343 m/s
vs = (440 Hz * 343 m/s - 415 Hz * 343 m/s) / 415 Hz
vs ≈ ((440 - 415) Hz * 343 m/s) / 415 Hz
vs ≈ 25 Hz * 343 m/s / 415 Hz
vs ≈ 8225 m/s / 415
vs ≈ 19.82 m/s
So, the speed of the train is approximately 19.82 meters per second.
The complete question is- A train moving at a constant speed is passing a stationary observer on a platform. On one of the train cars, a flute player is continually playing the note known as concert A (f = 440 Hz). After the flute has passed, the observer hears the sound with a frequency of 415 Hz. What is the speed of the train? The speed of sound in air is 343 m/s.