Certainly! Let's solve the given problem step-by-step using the concepts from the Doppler effect.
### Step 1: Understand the Problem
We are given:
- Speed of the train (\(v_s\)): \(23 \, \text{m/s}\) (moving towards the east)
- Frequency of the horn (\(f\)): \(164 \, \text{Hz}\)
- Speed of the car (\(v_o\)): \(15 \, \text{m/s}\) (moving towards the east, parallel to the train)
- Speed of sound (\(v\)): \(343 \, \text{m/s}\)
The objective is to find the frequency heard by the driver of the car. Both the car and the train are moving towards each other.
### Step 2: Doppler Effect Formula
The Doppler effect formula for a source moving towards a stationary observer is given by:
[tex]\[ f' = f \cdot \frac{v + v_o}{v + v_s} \][/tex]
where:
- \( f' \) is the perceived frequency
- \( f \) is the emitted frequency
- \( v \) is the speed of sound
- \( v_o \) is the speed of the observer (car)
- \( v_s \) is the speed of the source (train)
In this case, both the observer and the source are moving towards each other, so we use the formula as is.
### Step 3: Plug in the Values
Given:
[tex]\[ f = 164 \, \text{Hz} \][/tex]
[tex]\[ v = 343 \, \text{m/s} \][/tex]
[tex]\[ v_o = 15 \, \text{m/s} \][/tex]
[tex]\[ v_s = 23 \, \text{m/s} \][/tex]
Substitute these values into the formula:
[tex]\[ f' = 164 \cdot \frac{343 + 15}{343 + 23} \][/tex]
### Step 4: Simplify the Expression
First, we add the values in the numerator and the denominator:
[tex]\[ f' = 164 \cdot \frac{358}{366} \][/tex]
Next, simplify the fraction:
[tex]\[ \frac{358}{366} \approx 0.978 \][/tex]
### Step 5: Calculate the Perceived Frequency
Now, multiply this with the original frequency to get the perceived frequency:
[tex]\[ f' = 164 \cdot 0.978 \][/tex]
[tex]\[ f' \approx 160.415 \, \text{Hz} \][/tex]
### Conclusion
The frequency heard by the driver of the car is approximately [tex]\(160.415 \, \text{Hz}\)[/tex].
