A student in a parked car honks the horn, which has a `proper' frequency of 320 Hz. An observer in an approaching vehicle measures the frequency of the sound to be 355 Hz. Calculate the speed of the approaching vehicle. Use 340 m/s as the speed of sound in air.
To solve this problem, we can use the Doppler Effect formula. The formula for the Doppler Effect when the source is moving towards the observer is:
f' = (v + vo) / (v + vs) * f
Where:
- f' is the observed frequency
- v is the speed of sound in air (given as 340 m/s)
- vo is the velocity of the observer (unknown)
- vs is the velocity of the source (unknown)
- f is the actual frequency of the source (given as 320 Hz)
In this problem, the observed frequency is 355 Hz. Substituting the known values into the formula, we get:
355 = (340 + vo) / (340 + vs) * 320
To solve for the speed of the approaching vehicle (vo), we need to isolate it in the equation. The equation can be rearranged as follows:
355(340 + vs) = 320(340 + vo)
Now, let's solve this equation step-by-step:
1. Distribute on the left-hand side of the equation:
355 * 340 + 355 * vs = 320(340 + vo)
2. Simplify:
120,700 + 355 * vs = 108,800 + 320 * vo
3. Rearrange terms, moving the known values to the right-hand side:
355 * vs = 320 * vo - 120,700 + 108,800
4. Simplify:
355 * vs = 320 * vo - 11,900
Now, we have an equation relating the velocity of the source (vs) and the velocity of the observer (vo). However, we are only interested in the velocity of the observer. So, let's solve for vo by isolating it:
5. Add 11,900 to both sides of the equation:
355 * vs + 11,900 = 320 * vo
6. Divide both sides of the equation by 320:
(355 * vs + 11,900) / 320 = vo
Now, we can calculate the speed of the approaching vehicle (vo) by plugging in the values:
vo = (355 * 0 + 11,900) / 320
Simplifying, we get:
vo = 11,900 / 320
vo = 37.1875 m/s
Therefore, the speed of the approaching vehicle is approximately 37.19 m/s.
To calculate the speed of the approaching vehicle, you can use the Doppler Effect equation for sound:
f' = (v + vo) / (v + vs) * f
where:
f' = observed frequency (355 Hz)
f = source frequency (320 Hz)
v = speed of sound in air (340 m/s)
vo = observer's velocity with respect to the medium (observer's velocity is 0 in this case since they are not moving, so vo = 0)
vs = source's velocity with respect to the medium (velocity of the approaching vehicle)
Rearranging the equation, we get:
vs = (f' - f) / (f' / v - 1)
Substituting the given values:
vs = (355 Hz - 320 Hz) / (355 Hz / 340 m/s - 1)
Now, let's calculate it step by step:
1. Calculate the denominator:
denominator = (355 Hz / 340 m/s - 1) = (355 / 340 - 1) = 1.0441
2. Calculate the numerator:
numerator = 355 Hz - 320 Hz = 35 Hz
3. Calculate the final result:
vs = numerator / denominator = 35 Hz / 1.0441 = 33.52 m/s
Therefore, the speed of the approaching vehicle is approximately 33.52 m/s.
(V+Vr)/(V-Vs) * Fo = 355 Hz
(340+0)/(340-Vs) * 320 = 355
108800/(340-Vs) = 355
355(340-Vs) = 108800
340-Vs = 306.48
Vs = 340-306.48 = 33.5 m/s. = Velocity of source or vehicle.
Vr = Velocity of receiver or person.