Answered

The Doppler effect causes a sound moving towards you to sound higher pitched and a sound moving away from you to sound lower pitched.

The equation for this is [tex]f=\left(\frac{c}{c+v}\right) f_0[/tex], where:
- [tex]f[/tex] is the observed frequency (pitch) of the sound, in Hertz,
- [tex]c[/tex] is the speed of sound, about 761 miles per hour,
- [tex]v[/tex] is the speed of the sound source, in this case the car, in miles per hour,
- [tex]f_0[/tex] is the frequency of the sound when stationary.

Using a spectrum analyzer, we determined the observed frequency when the car was driving away was 240 Hz and the stationary frequency was 255 Hz.

Use this information and the equation to estimate the speed of the car in miles per hour, to 1 decimal place.



Answer :

GinnyAnswer
Certainly! Let's go through the problem step-by-step.

We are given the following details:
- The observed frequency when the car is driving away, [tex]\( f = 240 \)[/tex] Hz.
- The stationary frequency, [tex]\( f_0 = 255 \)[/tex] Hz.
- The speed of sound, [tex]\( c = 761 \)[/tex] miles per hour.

We are to find the speed of the car [tex]\( v \)[/tex] using the Doppler effect formula for a source moving away from the observer:

[tex]\[ f = \left(\frac{c}{c + v}\right) f_0 \][/tex]

To solve for [tex]\( v \)[/tex], we first rearrange the equation:

1. Multiply both sides by [tex]\( (c + v) \)[/tex] to clear the fraction:
[tex]\[ f \cdot (c + v) = c \cdot f_0 \][/tex]

2. Distribute [tex]\( f \)[/tex] on the left side:
[tex]\[ f \cdot c + f \cdot v = c \cdot f_0 \][/tex]

3. Isolate the term involving [tex]\( v \)[/tex]:
[tex]\[ f \cdot v = c \cdot f_0 - f \cdot c \][/tex]

4. Factor [tex]\( c \)[/tex] out of the right side:
[tex]\[ f \cdot v = c \cdot (f_0 - f) \][/tex]

5. Finally, solve for [tex]\( v \)[/tex]:
[tex]\[ v = \frac{c \cdot (f_0 - f)}{f} \][/tex]

Now substitute the given values into the formula:

[tex]\[ v = \frac{761 \cdot (255 - 240)}{240} \][/tex]

Calculate the difference in frequencies:
[tex]\[ 255 - 240 = 15 \][/tex]

Multiply this difference by the speed of sound:
[tex]\[ 761 \cdot 15 = 11415 \][/tex]

Divide by the observed frequency:
[tex]\[ v = \frac{11415}{240} \][/tex]

[tex]\[ v = 47.5625 \][/tex]

Round to one decimal place:
[tex]\[ v = 47.6 \][/tex]

Therefore, the estimated speed of the car is [tex]\( 47.6 \)[/tex] miles per hour.