A car B, with a hom of frequency 700 Hz moves with a velocity `90 kmh^(-1).` It approaches another car A, coming towards it with a velocity `72 kmh^(-1).` Calculate the apparent frequency of the sound heard by the driver in the car A (a) before crossing (b) after crossing ? Velocity of sound `=350 ms^(-1)`.

Answer

To solve the problem of calculating the apparent frequency of the sound heard by the driver in car A before and after crossing car B, we will use the Doppler effect formula.

### Given Data:
- Frequency of the horn (f) = 700 Hz
- Velocity of car B (source, Vs) = 90 km/h
- Velocity of car A (observer, V0) = 72 km/h
- Velocity of sound (V) = 350 m/s

### Step 1: Convert velocities from km/h to m/s
To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \).

- For car B (Vs):
\[ Vs = 90 \times \frac{5}{18} = 25 \text{ m/s} \]

- For car A (V0):
\[ V0 = 72 \times \frac{5}{18} = 20 \text{ m/s} \]

### Step 2: Calculate apparent frequency before crossing
When both cars are approaching each other, the formula for apparent frequency (f') is given by:
\[ f' = f \frac{V + V0}{V - Vs} \]
Where:
- f = frequency of the source
- V = speed of sound
- V0 = speed of the observer (car A)
- Vs = speed of the source (car B)

Substituting the values:
\[ f' = 700 \frac{350 + 20}{350 - 25} \]

Calculating the numerator and denominator:
\[ f' = 700 \frac{370}{325} \]

Calculating further:
\[ f' = 700 \times 1.13846 \approx 796.92 \text{ Hz} \]

### Step 3: Calculate apparent frequency after crossing
After crossing, the formula for apparent frequency (f') changes as the observer and source are moving away from each other:
\[ f' = f \frac{V - V0}{V + Vs} \]

Substituting the values:
\[ f' = 700 \frac{350 - 20}{350 + 25} \]

Calculating the numerator and denominator:
\[ f' = 700 \frac{330}{375} \]

Calculating further:
\[ f' = 700 \times 0.88 \approx 616 \text{ Hz} \]

### Final Answers:
(a) The apparent frequency before crossing is approximately **796.92 Hz**.
(b) The apparent frequency after crossing is approximately **616 Hz**.

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