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.]
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Correct Answer: 8
Solution and Explanation
To solve this problem, we need to find the beat frequency perceived by each driver as the two cars approach each other. The relevant concept is the Doppler Effect, which describes the change in frequency due to the relative motion between the source and the observer.
Step 1: Understand the Given Data.
Both cars travel at 7.2 km/h, which needs to be converted into meters per second:
7.2 km/h = (7,200 m/3,600 s) = 2 m/s.
Each horn has a frequency of \( f_0 = 676 \) Hz. The speed of sound in air is \( v = 340 \) m/s.
Step 2: Calculate the apparent frequency heard by each driver.
Since both cars are moving towards each other, the Doppler Effect causes an increase in the frequency heard by each driver. The formula for the observed frequency \( f' \) when both source and observer move towards each other is:
\( f' = f_0 \times \frac{v + v_o}{v - v_s} \)
where \( v_o \) is the observer's speed, and \( v_s \) is the source's speed. In this case, \( v_o = v_s = 2 \) m/s. Thus, the formula becomes:
\( f' = 676 \times \frac{340 + 2}{340 - 2} \)
\( f' = 676 \times \frac{342}{338} \)
Calculate \( f' \):
\( f' \approx 676 \times 1.0118 \approx 684.05 \) Hz.
Step 3: Determine the Beat Frequency.
The beat frequency \( f_b \) is given by the absolute difference between the observed frequency and the original frequency:
\( f_b = |f' - f_0| = |684.05 - 676| \)
\( f_b \approx 8.05 \) Hz.
Since the beat frequency should be an integer, round it to 8 Hz.
Final Answer:
The beat frequency heard by each driver is 8 Hz, which falls within the given range of 8 to 8 Hz.
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