A policeman in a stationary car measures the speed of approaching cars by means of an ultrasonic device that emits a sound with a frequency of 41.2 kHz. A car is approaching him at a speed of 33.0 m/s. The wave is reflected by the car and interferes with the emitted sound producing beats. What is the frequency of the beats? The speed of sound in air is 330 m/s
Isn't there a formula for this?
Two of them :)
To find the frequency of the beats, we need to find the difference between the frequency of the emitted sound and the frequency of the reflected sound.
Step 1: Find the frequency of the reflected sound.
The frequency of the reflected sound can be calculated using the Doppler effect equation:
f' = f * (v + v_r) / (v + v_s)
Where:
f' = frequency of the reflected sound
f = frequency of the emitted sound (41.2 kHz)
v = speed of sound in air (330 m/s)
v_r = velocity of the reflecting object (in this case, the car) towards the stationary car (-33.0 m/s)
v_s = velocity of the stationary car (0 m/s)
Calculating the frequency of the reflected sound:
f' = 41.2 kHz * (330 m/s + (-33.0 m/s)) / (330 m/s + 0 m/s)
f' = 41.2 kHz * (297 m/s) / (330 m/s)
f' = 37.0 kHz
Step 2: Calculate the frequency of the beats.
The beats are produced when two sound waves interfere constructively and destructively, causing periodic variations in the resultant amplitude. The frequency of the beats is the difference between the frequencies of the emitted and reflected sounds.
frequency of the beats = |f' - f|
frequency of the beats = |37.0 kHz - 41.2 kHz|
frequency of the beats = 4.2 kHz
Therefore, the frequency of the beats is 4.2 kHz.
To find the frequency of the beats produced when the ultrasonic wave is reflected by the car, we need to calculate the frequency difference between the emitted sound and the reflected sound.
The frequency of the emitted sound is given as 41.2 kHz, which is equivalent to 41,200 Hz.
The frequency of the reflected sound can be determined using the Doppler effect formula:
f' = f * (v + vr) / (v + vs)
Where:
f' is the frequency observed (the frequency of the beats)
f is the frequency emitted by the source (41,200 Hz)
v is the speed of sound in air (330 m/s)
vr is the speed of the receiver (the car in this case), which is equal to the speed of the car (-33.0 m/s, since it is approaching the stationary car)
vs is the speed of the source (the stationary car), which is zero.
Plugging in the values, we get:
f' = 41,200 * (330 - 33.0) / (330 + 0)
f' = 41,200 * (297) / (330)
f' ≈ 37,200 Hz
Therefore, the frequency of the beats produced when the ultrasonic wave is reflected by the car is approximately 37,200 Hz.