Two cars, one in front of the other, are traveling down the highway at 25 m/s. The car behind sounds its horn, which has a frequency of 500 Hz. What is the frequency heard by the driver of the lead car? (vsound = 340 m/s). I have the answer but I just don't know how to get it. Thanks!
To determine the frequency heard by the driver of the lead car, we need to apply the concept of the Doppler effect. The Doppler effect describes the change in frequency of a wave, such as sound or light, as perceived by an observer moving relative to the source of that wave.
In this scenario, we have two cars: one in front (lead car) and one behind (car sounding its horn). The lead car is stationary, while the car behind is moving towards it. The sound waves produced by the horn of the car behind will be compressed in front of the car and stretched as they move away from it.
The formula for the Doppler effect with sound is:
f' = (vsound ± vobserver) / (vsound ± vsource) * f
Where:
f' = frequency heard by the observer
vsound = speed of sound in the medium (given as 340 m/s)
vobserver = velocity of the observer relative to the medium (lead car, which is stationary, so its velocity is 0 m/s)
vsource = velocity of the source relative to the medium (car behind, which is moving towards the lead car)
Since the car behind is moving towards the lead car, the velocity of the source (vsource) will be negative (-25 m/s).
Plugging the values into the formula:
f' = (340 m/s + 0 m/s) / (340 m/s - (-25 m/s)) * 500 Hz
Simplifying:
f' = 340 m/s / (340 m/s + 25 m/s) * 500 Hz
f' = 340 m/s / 365 m/s * 500 Hz
Calculating:
f' = 500 Hz * 0.9315
f' ≈ 466 Hz
Therefore, the frequency heard by the driver of the lead car is approximately 466 Hz.