To solve the problem, we need to find the time taken by the winning car in the race based on the frequencies detected and the original frequency of the sound emitted by both cars.
### Step-by-Step Solution:
1. **Identify Given Data**:
- Original frequency of both cars, \( f_0 = 300 \, \text{Hz} \)
- Observed frequency from car A, \( f_1 = 330 \, \text{Hz} \)
- Observed frequency from car B, \( f_2 = 360 \, \text{Hz} \)
- Velocity of sound, \( v = 330 \, \text{m/s} \)
- Separation between the cars at the end of the race, \( d = 100 \, \text{m} \)
2. **Use the Doppler Effect Formula**:
The apparent frequency observed can be calculated using the Doppler effect formula for a source moving towards a stationary observer:
\[
f' = f_0 \frac{v}{v - v_s}
\]
Where:
- \( f' \) is the observed frequency,
- \( f_0 \) is the original frequency,
- \( v \) is the speed of sound,
- \( v_s \) is the speed of the source (car).
3. **Calculate the Speed of Car A**:
For car A, using the observed frequency \( f_1 = 330 \, \text{Hz} \):
\[
330 = 300 \frac{330}{330 - v_1}
\]
Rearranging gives:
\[
330 - v_1 = 300 \cdot \frac{330}{330}
\]
Simplifying:
\[
330 - v_1 = 300
\]
Thus:
\[
v_1 = 330 - 300 = 30 \, \text{m/s}
\]
4. **Calculate the Speed of Car B**:
For car B, using the observed frequency \( f_2 = 360 \, \text{Hz} \):
\[
360 = 300 \frac{330}{330 - v_2}
\]
Rearranging gives:
\[
330 - v_2 = 300 \cdot \frac{330}{360}
\]
Simplifying:
\[
330 - v_2 = 275
\]
Thus:
\[
v_2 = 330 - 275 = 55 \, \text{m/s}
\]
5. **Set Up the Equation for Distance**:
The distance between the two cars is 100 m when car B reaches the endpoint. The time taken \( t \) can be calculated using the relative speeds of the two cars:
\[
d = (v_2 - v_1) \cdot t
\]
Substituting the values:
\[
100 = (55 - 30) \cdot t
\]
Simplifying gives:
\[
100 = 25t
\]
Thus:
\[
t = \frac{100}{25} = 4 \, \text{s}
\]
### Final Answer:
The time taken by the winning car is **4 seconds**.
