a source emits 1700Hz frequency travelling at a speed of 70m/s. A detector is placed 200m away. What is the frequency detected by the detector when the source is closest to it?
Also find the distance between them when the detector detects the frequency to be 1200Hz. Velocity of sound in air is 340m/s.
To find the frequency detected by the detector when the source is closest to it, we need to consider the concept of Doppler Effect.
The Doppler Effect states that the frequency heard by an observer is different from the frequency emitted by a source when there is relative motion between them. The formula to calculate the perceived frequency (f') is:
f' = (v + v_det) / (v + v_src) * f
where:
f = emitted frequency of the source
v = velocity of sound in air
v_det = velocity of the detector
v_src = velocity of the source
In this case, the source is emitting a frequency of 1700Hz and is traveling at a velocity of 70m/s. The detector is stationary. The velocity of sound in air is 340m/s.
Given:
f = 1700Hz
v_src = 70m/s
v = 340m/s
v_det = 0m/s (as the detector is stationary)
Substituting the given values into the formula, we have:
f' = (v + v_det) / (v + v_src) * f
f' = (340 + 0) / (340 + 70) * 1700
f' = 340 / 410 * 1700
f' = 1.832 * 1700
f' ≈ 3092Hz
So, the frequency detected by the detector when the source is closest to it is approximately 3092Hz.
To find the distance between the source and the detector when the detector detects a frequency of 1200Hz, we can rearrange the formula to solve for distance (d):
d = v * (f_src - f_det) / (f_src + f_det)
where:
d = distance between source and detector
f_src = emitted frequency of the source
f_det = detected frequency by the detector
v = velocity of sound in air
Given:
f_src = 1700Hz
f_det = 1200Hz
v = 340m/s
Substituting the given values into the formula, we have:
d = 340 * (1700 - 1200) / (1700 + 1200)
d = 340 * 500 / 2900
d ≈ 58.62m
Therefore, the distance between the source and the detector when the detector detects a frequency of 1200Hz is approximately 58.62 meters.