Question

Two cars are approaching each other at an equal speed of 7.2 km/hr. When they see each other, both blow horns having frequency of 676 Hz. The beat frequency heard by each driver will be ______ Hz. [Velocity of sound in air is 340 m/s.]

Hint:

For low speeds $(v \ll c)$, use the approximation $\Delta f \approx f \cdot \frac{2v_{rel}}{c}$. Here $2 \times 676 \times \frac{4}{340} \approx 7.95 \approx 8 \text{ Hz}$.

Answer 1

Correct Answer: 8

To calculate the beat frequency heard by each driver, we start by understanding the Doppler Effect, which affects the frequency of a sound as the source and observer move relative to each other.

Each car is moving towards the other with a speed of \(7.2 \text{ km/hr}\). First, convert this speed to meters per second:

\[7.2 \text{ km/hr} = \frac{7.2 \times 1000}{3600} \text{ m/s} = 2 \text{ m/s}\]

Step 1: Calculate the frequency heard by each driver due to the approaching car.

The formula for the frequency observed by a moving observer or source is given by:

\[\text{Observed Frequency, } f' = \frac{v + v_o}{v - v_s} \times f\]

Where:

• \(v\) = velocity of sound in air = 340 m/s
• \(v_o\) = velocity of the observer = 2 m/s (towards the source)
• \(v_s\) = velocity of the source = 2 m/s (towards the observer)
• \(f\) = original frequency = 676 Hz

Substituting into the formula:

\[f' = \frac{340 + 2}{340 - 2} \times 676 = \frac{342}{338} \times 676\]

Calculate:

\[f' \approx 1.0118 \times 676 \approx 684.36 \text{ Hz}\]

Step 2: Determine the beat frequency.

The beat frequency is the difference between the observed frequencies:

\[\text{Beat Frequency} = |f' - f| = |684.36 \text{ Hz} - 676 \text{ Hz}|\]

\[\text{Beat Frequency} \approx 8.36 \text{ Hz}\]

Checking against the expected range (8,8), the beat frequency, approximately 8 Hz, lies within the specified range.

Therefore, the beat frequency heard by each driver is 8 Hz.