A car is moving with `90 km h^(-1)` blows a horn of 150 Hz, towards a cliff. The frequency of the reflected sound heard by the driver will be (speed of sound in air is `340 ms^(-1)` )

Options

A. 100 Hz
B. 150 Hz
C. 170 Hz
D. 130 Hz

Answer

To solve the problem, we will use the Doppler effect formula to find the frequency of the reflected sound heard by the driver of the car. Here are the steps:

### Step 1: Convert the speed of the car from km/h to m/s
The speed of the car is given as 90 km/h. To convert this to meters per second (m/s), we use the conversion factor \( \frac{5}{18} \).

\[ \text{Speed of car (v)} = 90 \, \text{km/h} \times \frac{5}{18} = 25 \, \text{m/s} \]

**Hint:** Remember that to convert km/h to m/s, multiply by \( \frac{5}{18} \).

### Step 2: Identify the frequency of the sound emitted by the car
The frequency of the sound emitted by the car is given as 150 Hz.

\[ f_0 = 150 \, \text{Hz} \]

**Hint:** The emitted frequency is the frequency of the sound source before any Doppler effect is applied.

### Step 3: Use the Doppler effect formula to find the frequency of the sound reaching the cliff
The formula for the apparent frequency \( f' \) when the source is moving towards a stationary observer is given by:

\[ f' = f_0 \frac{v + v_o}{v - v_s} \]

Where:
- \( f' \) = frequency heard by the observer (cliff)
- \( f_0 \) = emitted frequency (150 Hz)
- \( v \) = speed of sound in air (340 m/s)
- \( v_o \) = speed of the observer (0 m/s, since the cliff is stationary)
- \( v_s \) = speed of the source (25 m/s)

Substituting the values:

\[ f' = 150 \, \text{Hz} \cdot \frac{340 + 0}{340 - 25} \]

Calculating the denominator:

\[ 340 - 25 = 315 \]

Thus, we have:

\[ f' = 150 \cdot \frac{340}{315} \]

**Hint:** Make sure to keep track of whether the source or observer is moving towards or away from each other when using the Doppler effect formula.

### Step 4: Calculate the frequency of the sound reflected back to the car
Now we need to find the frequency of the sound reflected back to the car. The cliff acts as a new source of sound emitting frequency \( f' \) towards the car. The car is moving towards the cliff, so we apply the Doppler effect again:

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

Where:
- \( f'' \) = frequency heard by the driver
- \( v_s \) = speed of the source (0 m/s, since the cliff is stationary)
- \( v_o \) = speed of the observer (25 m/s)

Substituting the values:

\[ f'' = f' \cdot \frac{340 + 25}{340 - 0} \]

Calculating:

\[ f'' = f' \cdot \frac{365}{340} \]

Substituting \( f' \):

\[ f'' = 150 \cdot \frac{340}{315} \cdot \frac{365}{340} \]

The \( 340 \) cancels out:

\[ f'' = 150 \cdot \frac{365}{315} \]

### Step 5: Final Calculation
Now we calculate the final frequency:

\[ f'' = 150 \cdot \frac{365}{315} \approx 173.81 \, \text{Hz} \]

Rounding this gives approximately 174 Hz. However, based on the options provided, the closest answer is 170 Hz.

### Final Answer:
The frequency of the reflected sound heard by the driver will be approximately **170 Hz**.

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