Richard stands on the flatbed car of a moving train playing an a on his horn. the note has a fundamental frequency of 220 Hz. Calculate weather or not the train could move fast enough for a stationary observer on the ground to hear the first overtone of the horn as the train passes

To determine whether a stationary observer on the ground can hear the first overtone of the horn as the train passes, we need to consider the Doppler effect.

The Doppler effect describes how the frequency of a sound wave changes relative to an observer when the source of the sound is in motion. When a moving source approaches an observer, the frequency appears higher (a higher pitch), and when it moves away, the frequency appears lower (a lower pitch).

In this scenario, the train is moving, and we want to know if the stationary observer on the ground can hear the first overtone of the horn. The first overtone, also known as the second harmonic, has a frequency that is twice the fundamental frequency.

Now let's calculate:

1. The fundamental frequency of the horn is given as 220 Hz.

2. The formula for the Doppler effect (moving source toward an observer) is:
F_observed = (v_sound + v_observer) / (v_sound + v_source) * F_source,

where:
- F_observed is the frequency observed by the observer,
- v_sound is the speed of sound,
- v_observer is the speed of the observer relative to the medium (in this case, zero for the stationary observer),
- v_source is the speed of the source relative to the medium (in this case, the speed of the train),
- F_source is the frequency emitted by the source.

3. To calculate whether the observer hears the first overtone, we need to determine if the observed frequency matches the first overtone frequency. The first overtone frequency is twice the fundamental frequency, which means 440 Hz (2 * 220 Hz).

4. Rearrange the Doppler effect formula to calculate the speed of the moving source (train) required for the observed frequency to be 440 Hz (first overtone frequency).

v_source = (F_observed * (v_sound + v_observer)) / (F_source - F_observed).

Substituting the known values:
F_observed = 440 Hz,
v_sound = speed of sound (approximated to 343 m/s),
v_observer = 0 m/s (stationary observer),
F_source = 220 Hz,

v_source = (440 Hz * (343 m/s + 0 m/s)) / (220 Hz - 440 Hz).

5. Calculate v_source:
v_source = (440 Hz * 343 m/s) / (-220 Hz).

Simplifying the equation:
v_source = -677.6 m/s.

The negative sign indicates that the source (train) is moving away from the stationary observer at a speed of approximately 677.6 m/s. Since this negative velocity is unrealistic, it means that the observer on the ground will not hear the first overtone of the horn as the train passes by.

171.5 m/s