an automobile is travelling towards you at a speed 25.0m/s. When you measure the frequency of its horn, you obtain a value of 26Hz. If the actual frequency of the horn is known to be 240Hz, calculate speed of sound in air?
answer is 325m/s
Vs = 25 m/s = Velocity of the source.
F = 260 Hz(NOT 26Hz).
Fo = 240 Hz.
Vr = Velocity of the receiver(person)
F = (V+Vr)/(V-Vs)*Fo = 260 Hz.
(V+0)/(V-25)*240 = 260
240V/(V-25) = 260
240V = 260V - 6500
-20V = -6500
V = 325 m/s.
To solve this problem, we can use the Doppler effect equation for sound. The equation is given by:
f' = (v + v₀) / (v + vᵢ) * fᵢ
Where:
f' = observed frequency
fᵢ = source frequency
v = speed of sound in air
v₀ = velocity of the observer (you) relative to the medium
vᵢ = velocity of the source (automobile) relative to the medium
In this case, the observed frequency is 26 Hz, the source frequency is 240 Hz, and the speed of the automobile is 25.0 m/s. We need to find the speed of sound in air, v.
Rearranging the equation, we get:
v = [(f' / fᵢ) - 1] * (v + vᵢ)
Plugging in the values we have:
v = [(26 Hz / 240 Hz) - 1] * (25.0 m/s + 0 m/s)
= [-0.8917] * (25.0 m/s)
= -22.29 m/s
The negative sign indicates that sound travels in the opposite direction of the automobile. However, since the question is asking for the speed of sound in air, we take its magnitude, which is 22.29 m/s.
Therefore, the speed of sound in air is 22.29 m/s.
To calculate the speed of sound in air, you can use the Doppler effect equation. The Doppler effect describes the change in frequency of a wave when there is relative motion between the source of the wave and the observer.
The equation for the Doppler effect is as follows:
f' = f * (v + v₀) / (v + v₁)
Where:
f' is the observed frequency
f is the actual frequency of the horn (240 Hz)
v is the speed of sound in air (the value we want to find)
v₀ is the speed of the observer (you, in this case)
v₁ is the speed of the source (the automobile)
Given:
f' = 26 Hz (the observed frequency)
f = 240 Hz (the actual frequency)
v₁ = 25.0 m/s (the speed of the automobile)
Now we can rearrange the equation to solve for v (speed of sound in air):
v = ((f' - f) * (v + v₁)) / (f' + f)
Plugging in the given values:
v = ((26 Hz - 240 Hz) * (v + 25.0 m/s)) / (26 Hz + 240 Hz)
Simplifying:
v = (-214 Hz * v - 5350 m/s) / 266 Hz
To isolate v, we can multiply both sides by 266 Hz:
266 Hz * v = -214 Hz * v - 5350 m/s
Add 214 Hz * v to both sides:
480 Hz * v = -5350 m/s
Divide both sides by 480 Hz:
v = -5350 m/s / 480 Hz
v ≈ -11.14583 m/s
It's important to note that the negative sign indicates that either the direction of the sound wave or the direction of motion of the automobile is opposite to the direction of the observer. However, speed is always a positive quantity, so we take the absolute value of v:
v ≈ 11.14583 m/s
Therefore, the speed of sound in air is approximately 11.14583 m/s.