to an observer the pitch of the stationary source of the sound appears to be reduced by 20% if speed of sound is 340m/s then sped and direction of 5he observer is
options
1.68 m/s away from the observer
2.96 m/s away from the observer
{f(observed) - f(source)}/f(observed) = 20/100, u = 340 m/s
At source approaching
f(observed) = [v/(v-u)] •f(source),
v =[f(obs)-f(sour)] •u/ f(obs) = 20% •340/100%= 68 m/s.
To answer this question, we can use the formula for the Doppler effect:
f' = (v + vo) / (v + vs) * fo
Where:
f' = observed frequency
fo = actual frequency (pitch)
v = speed of sound
vo = speed of the observer
vs = speed of the source
Given that the observed frequency is reduced by 20% (or 0.2), we can set up the equation:
(1 - 0.2) = (v + vo) / (v + vs)
Simplifying:
0.8 = (v + vo) / (v + vs)
We are given that the speed of sound (v) is 340 m/s. Now, let's consider each option:
Option 1: If the speed and direction of the observer is 1.68 m/s away from the source, the equation becomes:
0.8 = (340 + 1.68) / (340 + vs)
0.8 = 341.68 / (340 + vs)
Solving for vs:
(340 + vs) = 341.68 / 0.8
vs = 1.68 / 0.8
vs = 2.1 m/s away from the observer
Since the speed of sound was assumed to be 340 m/s, Option 1 is not correct.
Option 2: If the speed and direction of the observer is 2.96 m/s away from the source, the equation becomes:
0.8 = (340 + 2.96) / (340 + vs)
0.8 = 342.96 / (340 + vs)
Solving for vs:
(340 + vs) = 342.96 / 0.8
vs = 2.96 / 0.8
vs = 3.7 m/s away from the observer
Since the speed of sound was assumed to be 340 m/s, Option 2 is correct.
Therefore, the answer is Option 2. The speed and direction of the observer is 2.96 m/s away from the observer.
To solve this problem, we can use the formula for the Doppler effect:
f' = f (v + vo) / (v + vs)
where:
f' is the observed frequency,
f is the actual frequency,
v is the speed of sound,
vo is the speed of the observer, and
vs is the speed of the source.
Given that the pitch of the stationary source appears to be reduced by 20%, we can say that the observed frequency f' is 80% (or 0.8) of the actual frequency f.
So we have the equation:
0.8f = f (v + vo) / (v + vs)
Now, we use the given values: the speed of sound v = 340 m/s and the observed frequency reduction factor of 0.8.
0.8 = (v + vo) / (v + vs)
Next, we can find the ratio of speeds by rearranging the equation:
(v + vo) / (v + vs) = 0.8
Cross-multiplying gives us:
0.8(v + vs) = (v + vo)
Expanding the equation:
0.8v + 0.8vs = v + vo
To isolate vo, we can rearrange the equation:
0.8vs - vo = 0.2v
Now, we can substitute the given speed of sound value into the equation:
0.8vs - vo = 0.2(340)
0.8vs - vo = 68
Divide each term by 0.8:
vs - (vo / 0.8) = 85
vs = vo / 0.8 + 85
Since the observer and the source are stationary (i.e., not moving), the speed of the source vs is 0. Therefore, the equation becomes:
0 = vo / 0.8 + 85
vo / 0.8 = -85
vo = -85 * 0.8
vo = -68 m/s
The negative sign indicates that the observer is moving away from the source. Therefore, the speed and direction of the observer is 68 m/s away from the source. Among the options, the closest value is 2.96 m/s away from the observer, option 2.