The horn of a car emits sound with a dominant frequency of 2400 Hz. What will be the apparent dominant frequency heard by a person standing on the road in front of the car if the car is approaching at `18.0 km h^-1`? Speed of sound in air` = 340 m s^-1`

Explanation

To solve the problem of determining the apparent dominant frequency heard by a person standing on the road in front of a car approaching at 18.0 km/h with a horn emitting a frequency of 2400 Hz, we will use the Doppler effect formula. Here are the steps to find the solution:

### Step 1: Convert the speed of the car from km/h to m/s
The speed of the car is given as 18.0 km/h. We need to convert this to meters per second (m/s) using the conversion factor \(1 \text{ km/h} = \frac{1}{3.6} \text{ m/s}\).

\[ \text{Speed of the car} = 18.0 \text{ km/h} \times \frac{1 \text{ m/s}}{3.6 \text{ km/h}} = 5.0 \text{ m/s} \]

### Step 2: Identify the known values
- Frequency of the sound emitted by the car (\(f_0\)): 2400 Hz
- Speed of sound in air (\(v\)): 340 m/s
- Speed of the source (car) (\(v_s\)): 5.0 m/s
- Speed of the observer (\(v_o\)): 0 m/s (since the observer is stationary)

### Step 3: Use the Doppler effect formula
The formula for the apparent frequency (\(f\)) when the source is moving towards a stationary observer is:

\[ f = f_0 \left( \frac{v + v_o}{v - v_s} \right) \]

Substituting the known values into the formula:

\[ f = 2400 \left( \frac{340 + 0}{340 - 5} \right) \]

### Step 4: Simplify the equation
Calculate the denominator and numerator:

\[ f = 2400 \left( \frac{340}{335} \right) \]

### Step 5: Calculate the apparent frequency
Now, compute the value:

\[ f = 2400 \times 1.0149253731343284 \approx 2435.4 \text{ Hz} \]

Rounding this value gives us:

\[ f \approx 2436 \text{ Hz} \]

### Final Answer
The apparent dominant frequency heard by the person standing on the road is approximately **2436 Hz**.

---