A student has a small battery-powered alarm buzzer. The student throws the buzzer toward a stationary wall (possibly due to frustration following a physics midterm examination). The buzzer emits sound with a frequency of 268 Hz, and the wall reflects the sound. If the buzzer is moving directly away from the student (and directly toward the wall) with a speed of 9.3 m/s, what beat frequency would the student hear?
The speed of sound is 340 m/s. The student stands still after throwing the buzzer.
To answer this question, we need to take into account the motion of the buzzer and the Doppler effect. The Doppler effect is the change in frequency or wavelength of a wave as observed by an observer moving relative to the source of the wave.
In this case, the buzzer is moving directly away from the student with a speed of 9.3 m/s. The buzzer emits sound with a frequency of 268 Hz. As the sound waves from the buzzer travel towards the wall, they will be stretched out (i.e., their wavelength will increase) due to the motion of the buzzer away from the student.
The formula for the Doppler effect in sound is:
f' = f * (v + vu) / (v - vs)
Where:
f' is the observed frequency
f is the original frequency emitted by the buzzer
v is the speed of sound (given as 340 m/s)
vu is the velocity of the observer (student)
vs is the velocity of the source (buzzer)
In this case, the student is standing still after throwing the buzzer, so the velocity of the observer (vu) is 0 m/s. The velocity of the source (vs) is 9.3 m/s.
Plugging in the values, we can calculate the observed frequency (f'):
f' = 268 Hz * (340 m/s + 0 m/s) / (340 m/s - 9.3 m/s)
Simplifying further:
f' = 268 Hz * 340 m/s / (330.7 m/s)
f' ≈ 276.11 Hz
Therefore, the student would hear a beat frequency of approximately 276.11 Hz when the buzzer is moving away from them towards the wall.