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The problem involves the Doppler effect, where the frequency of a sound changes due to the relative motion between the source (the fire truck) and the observer (the stationary detector).

Given:
- Frequency emitted by the fire truck (\( f_s \)) = 450 Hz
- Difference in frequency measured by the detector (\( \Delta f \)) = 58 Hz
- Speed of sound (\( v_{\text{sound}} \)) = 345 m/s

We need to determine the speed of the fire truck (\( v_{\text{truck}} \)).

### Step-by-Step Solution:
The Doppler effect formula for sound is:

\[
f_{\text{detector}} = f_s \left( \frac{v_{\text{sound}}}{v_{\text{sound}} - v_{\text{truck}}} \right)
\]

Where:
- \( f_{\text{detector}} \) is the frequency detected,
- \( f_s \) is the frequency of the source,
- \( v_{\text{sound}} \) is the speed of sound,
- \( v_{\text{truck}} \) is the speed of the fire truck.

The frequency shift is given by:

\[
\Delta f = f_{\text{detector}} - f_s
\]

Substituting the Doppler formula for \( f_{\text{detector}} \):

\[
\Delta f = f_s \left( \frac{v_{\text{sound}}}{v_{\text{sound}} - v_{\text{truck}}} \right) - f_s
\]

Now solving for \( v_{\text{truck}} \). Let me calculate the result.The calculated speed of the fire truck comes out as approximately \(-39.39 \, \text{m/s}\), which is negative. This negative value indicates that the truck is moving **towards** the stationary detector, as expected in the Doppler effect when the frequency increases.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions for further exploration:
1. What happens to the detected frequency if the truck moves away from the detector?
2. How would the calculation change if the detector were also moving?
3. What would the detected frequency be if the fire truck were stationary?
4. How does temperature affect the speed of sound and the frequency shift?
5. Can the Doppler effect be observed with light, and how does it differ from sound?

**Tip:** Always check the sign of the relative velocity in Doppler problems. Negative velocity typically means the source is moving towards the observer.

A fire truck emitting a 450 Hz signal passes by a stationary detector. The difference in frequency measured by the detector is 58 Hz. If the speed of sound is 345 m/s, how fast is the fire truck moving?

Solve a Doppler Effect problem involving a fire truck emitting a 450 Hz signal and a stationary detector that records a frequency shift of 58 Hz. Learn how to calculate the speed of the fire truck using the Doppler Effect formula, given the speed of sound as 345 m/s. Ideal for high school and college-level students.

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