Question

The speed of sound is 340 m/s and a train whistle with a frequency of 410 Hz is emitted from a train coming toward you at 30.0 m/s. What frequency will you hear?
Group of answer choices

410 Hz

170 Hz

580 Hz

450 Hz

Answer 1

To find the frequency you will hear from the train whistle, we need to use the Doppler effect formula. In this case, since the source of the sound (the train whistle) is moving toward the observer (you), the formula for the observed frequency \( f' \) when the source is moving towards a stationary observer is given by:

\[
f' = f \left( \frac{v + v_o}{v - v_s} \right)
\]

Where:
- \( f' \) is the observed frequency,
- \( f \) is the emitted frequency (410 Hz),
- \( v \) is the speed of sound in air (340 m/s),
- \( v_o \) is the speed of the observer (0 m/s since you are stationary),
- \( v_s \) is the speed of the source (30.0 m/s).

Plugging in the values:

\[
f' = 410 \, \text{Hz} \left( \frac{340 \, \text{m/s} + 0 \, \text{m/s}}{340 \, \text{m/s} - 30.0 \, \text{m/s}} \right)
\]

\[
f' = 410 \, \text{Hz} \left( \frac{340}{310} \right)
\]

Now we calculate the fraction:

\[
\frac{340}{310} \approx 1.09677
\]

Now substituting this back into the equation:

\[
f' \approx 410 \, \text{Hz} \times 1.09677 \approx 449.69 \, \text{Hz}
\]

Rounding this value gives us approximately 450 Hz. Therefore, the observed frequency you will hear is approximately **450 Hz**.

Thus, the correct answer among the choices is **450 Hz**.