A car approaches you with its horn blowing. The observed frequency is 503.7 Hz. Assume the speed of sound is 1100 ft/s. The car's horn has a frequency of 450 Hz. How fast is the car going? Express your answer in mph. mph
Question
A car approaches you with its horn blowing. The observed frequency is 503.7 Hz. Assume the speed of sound is 1100 ft/s. The car's horn has a frequency of 450 Hz. How fast is the car going? Express your answer in mph. mph
Solution
This problem can be solved using the Doppler effect formula for sound:
f' = f * (v + vo) / (v + vs)
where: f' is the observed frequency (503.7 Hz), f is the source frequency (450 Hz), v is the speed of sound (1100 ft/s), vo is the observer's speed (0 ft/s because the observer is stationary), vs is the source's speed (the car's speed, which we are trying to find).
Rearranging the formula to solve for vs gives:
vs = ((f' / f) * (v + vo)) - v
Substituting the given values into the formula gives:
vs = ((503.7 Hz / 450 Hz) * (1100 ft/s + 0 ft/s)) - 1100 ft/s
Solving this gives a speed in feet per second. To convert this to miles per hour (mph), multiply by 0.681818 (since 1 ft/s is approximately 0.681818 mph).
So, the car's speed in mph is:
vs_mph = vs * 0.681818
This will give you the car's speed in mph.
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