The frequency heard by the observer at the crossing is different from the frequency at which the whistle actually sounded due to the Doppler effect. The Doppler effect is the change in frequency or pitch of a wave (sound, light, etc.) as perceived by an observer when there is relative motion between the source of the wave and the observer.
The formula for calculating the apparent frequency (f') heard by the observer due to the Doppler effect when the source and observer are in motion relative to each other is:
f' = f * (v + v_obs) / (v + v_source)
Where: f = frequency of the source (whistle) as measured by someone stationary relative to the source. v = speed of sound in the medium (approximately 343 m/s in air at room temperature). v_obs = velocity of the observer relative to the medium (positive when the observer and source are approaching each other, negative when moving away). v_source = velocity of the source (whistle) relative to the medium (positive when the source is approaching the observer, negative when moving away).
In this case, let's assume the velocity of the observer and the train are in the same direction, so their signs are positive.
Given: f' = 275 cycles per second (heard frequency by the observer) v = 343 m/s (speed of sound in air) v_obs = 30.0 m/s (velocity of the observer, which is the speed of the train) v_source = unknown (velocity of the whistle, which we need to find)
Let's plug the values into the formula:
275 = f * (343 + 30) / (343 + v_source)
Now, solve for v_source:
275 * (343 + v_source) = f * (343 + 30) 275 * 343 + 275 * v_source = f * 373
Now, we need to know the value of the original frequency f at which the whistle sounded. If you provide that value, we can calculate the velocity of the whistle (v_source) relative to the medium (air).