In really stormy weather and with the right conditions, the sky would dump a bunch of hail, making the roads rather hazardous. A choir director was driving his car in these conditions and going a tad fast under the circumstances. His car began to slip on the balls of ice in the road towards two music students walking in front of him, so he leaned on his horn to warn the students to jump out of the way. Since he has career in music, his car horn is set to emit a frequency at exactly Tenor C – 523 Hz. The students have perfect pitch and are able to determine that the frequency that they hear is now 554 Hz (a semitone sharp for any musicians in the room). Since they have also taken Physics with Algebra, they were able to determine the choir director’s speed. What was that speed?

Isn't there a standard formula for doppler , stationary listner, moving source?

Source moving towards observer:

f'=fo(vwave/(vwave-Vs)
and you are looking to compute vs

To determine the car's speed, we need to use the concept of the Doppler effect. The Doppler effect is the change in frequency or pitch perceived by an observer when there is relative motion between the source of the sound and the observer.

In this scenario, the students hear a frequency of 554 Hz, which is a semitone sharp from the original frequency of 523 Hz. This shift in frequency can be attributed to the relative motion between the car (the sound source) and the students (the observers).

We can use the formula for the Doppler effect to find the speed of the car:

v = (Δf * v_sound) / (f_source + f_sound),

where:
- v is the velocity of the car (the speed we want to find),
- Δf is the change in frequency (semitone sharp, which is equivalent to about a 1.06% increase in frequency),
- v_sound is the speed of sound in air (approximately 343 meters per second at room temperature),
- f_source is the frequency emitted by the car horn (523 Hz), and
- f_sound is the frequency observed by the students (554 Hz).

Substituting the given values into the formula:

v = ((1.06/100) * 343) / (523 + 554)

Calculating this expression, we can find the speed (v) of the car.