Question

A whistle produces sound at 200 Hz and travels at 1000 ft/s. If a person runs away from the source at 100 ft/s. what wavelength do they perceive?

Answer 1

To find the wavelength perceived by a person running away from a sound source, we can use the Doppler effect. The formula for the observed frequency \( f' \) when the source is stationary and the observer is moving away is given by:

\[
f' = f \frac{v}{v + v_o}
\]

where:
- \( f \) is the frequency of the source (200 Hz),
- \( v \) is the speed of sound (1000 ft/s),
- \( v_o \) is the speed of the observer moving away from the source (100 ft/s).

Plugging in the values:

\[
f' = 200 \, \text{Hz} \cdot \frac{1000 \, \text{ft/s}}{1000 \, \text{ft/s} + 100 \, \text{ft/s}} = 200 \, \text{Hz} \cdot \frac{1000}{1100}
\]

Calculating this:

\[
f' = 200 \, \text{Hz} \cdot \frac{10}{11} \approx 181.82 \, \text{Hz}
\]

Now, to find the observed wavelength \( \lambda' \), we use the relationship between speed, frequency, and wavelength:

\[
\lambda' = \frac{v}{f'}
\]

Substituting \( v = 1000 \) ft/s and \( f' \approx 181.82 \, \text{Hz} \):

\[
\lambda' = \frac{1000 \, \text{ft/s}}{181.82 \, \text{Hz}} \approx 5.5 \, \text{ft}
\]

Thus, the wavelength that the person perceives while running away from the source of the whistle is approximately:

\[
\lambda' \approx 5.5 \, \text{ft}
\]