A car is parked 23.0 m directly south of a railroad crossing. A train is approaching the crossing from the west, heading directly east at a speed of 59.0 m/s. The train sounds a short blast of its 340-Hz horn when it reaches a point 23.0 m west of the crossing. What frequency does the car's driver hear when the horn blast reaches the car? The speed of sound in air is 343 m/s. (Hint: Assume that only the component of the train's velocity that is directed towards the car affects the frequency heard by the driver.)
To find the frequency heard by the car's driver, we need to consider the Doppler effect. The Doppler effect is the change in frequency of a waveform (such as sound waves) due to the relative motion between the source of the wave (the train) and the observer (the car).
The formula for the Doppler effect in the case of sound waves is given by:
f' = f * (v + vo) / (v + vs),
where:
- f' is the observed frequency
- f is the actual frequency
- v is the speed of sound in air
- vo is the velocity of the observer (car)
- vs is the velocity of the source (train)
In this case:
- f is the known frequency of the train's horn, 340 Hz
- v is the speed of sound in air, 343 m/s
- vo is the velocity of the observer (car), which is 0 since the car is stationary
- vs is the component of the velocity of the source (train) that is directed towards the car, which is the same as the train's velocity, 59.0 m/s.
Plugging in the values into the equation, we get:
f' = 340 Hz * (343 m/s + 0 m/s) / (343 m/s + 59.0 m/s).
Calculating this, we find:
f' = 340 Hz * 343 m/s / 402 m/s.
Simplifying further:
f' = (232,220 Hz*m) / 402 m/s.
Finally:
f' ≈ 577 Hz.
Therefore, the car's driver hears a frequency of approximately 577 Hz when the horn blast reaches the car.