A car horn has a frequency of 100 Hz. How fast does the car need to be
going in order to sound like its 110 Hz?
•Assume speed of sound is 300 m/s:

This problem involves the **Doppler effect**, which describes how the observed frequency of a wave changes when the source of the wave is moving relative to the observer. When a sound source moves towards the observer, the observed frequency increases.

The formula for the Doppler effect when the source is moving towards the observer is:

\[
f' = \frac{f}{1 - \frac{v_s}{v}}
\]

Where:
- \( f' \) is the observed frequency (110 Hz in this case),
- \( f \) is the actual frequency (100 Hz),
- \( v_s \) is the speed of the source (the car, which we need to find),
- \( v \) is the speed of sound (300 m/s in this case).

### Step 1: Rearrange the equation to solve for \( v_s \)
\[
f' = \frac{f}{1 - \frac{v_s}{v}}
\]
Multiplying both sides by \( 1 - \frac{v_s}{v} \):
\[
f' \left( 1 - \frac{v_s}{v} \right) = f
\]
Now, divide both sides by \( f' \):
\[
1 - \frac{v_s}{v} = \frac{f}{f'}
\]
Solving for \( v_s \):
\[
\frac{v_s}{v} = 1 - \frac{f}{f'}
\]
Finally:
\[
v_s = v \left( 1 - \frac{f}{f'} \right)
\]

### Step 2: Plug in the values
\[
v_s = 300 \left( 1 - \frac{100}{110} \right)
\]
\[
v_s = 300 \left( 1 - 0.909 \right)
\]
\[
v_s = 300 \times 0.091
\]
\[
v_s \approx 27.3 \, \text{m/s}
\]

### Conclusion:
The car needs to be moving at approximately **27.3 m/s** (about 98.3 km/h or 61.1 mph) for its horn to sound like 110 Hz.

Would you like further details or have any other questions?

Here are 5 related questions to expand your knowledge:
1. How does the Doppler effect apply to light waves?
2. What happens to the observed frequency if the car is moving away from the observer?
3. Can the Doppler effect be experienced in everyday life other than with sound?
4. How would the speed of sound affect the observed frequency?
5. What would the observed frequency be if the car were moving at 15 m/s instead?

**Tip:** The Doppler effect is also used in astronomy to measure the speed of distant stars and galaxies by observing shifts in the frequency of light waves.

A car horn has a frequency of 100 Hz. How fast does the car need to be going in order to sound like its 110 Hz? Assume speed of sound is 300 m/s.

Find out how fast a car needs to travel for its horn (100 Hz) to sound like 110 Hz using the Doppler effect formula. This detailed explanation uses the speed of sound (300 m/s) and provides step-by-step calculations for high school physics students.

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