From a vantage point very close to the track at a stock car race, you hear the sound emitted by a moving car. You detect a frequency that is 0.86 times as small as that emitted by the car when it is stationary. The speed of sound is 343 m/s. what is the speed of the car?
To solve this problem, we can use the concept of the Doppler effect. The Doppler effect is the change in frequency or wavelength of a wave perceived by an observer when the source of the wave is moving relative to the observer.
In this case, we are given that the frequency detected when the car is moving is 0.86 times smaller than the frequency emitted when the car is stationary. Let's denote the emitted frequency as f₀ and the detected frequency as f.
The Doppler effect formula for frequency is given by:
f = (v + v₀) / (v + vs) * f₀
Where:
- f is the detected frequency
- f₀ is the emitted frequency
- v is the speed of sound
- v₀ is the speed of the observer (car)
- vs is the speed of the source (moving car)
In this problem, we are trying to find the speed of the car (vs). We are given the following values:
- f = 0.86 * f₀
- v = 343 m/s (speed of sound)
- v₀ = 0 m/s (the observer is at rest)
Plugging in these values into the Doppler effect formula, we get:
0.86 * f₀ = (343 + 0) / (343 + vs) * f₀
Simplifying the equation, we can cancel out f₀:
0.86 = 343 / (343 + vs)
Cross-multiplying, we get:
0.86 * (343 + vs) = 343
Dividing by 0.86, we can solve for vs:
343 + vs = 343 / 0.86
vs = (343 / 0.86) - 343
Simplifying this equation, we find:
vs = 343 * (1 - 1 / 0.86)
After evaluating the right side of the equation, we can determine the value of vs, which represents the speed of the moving car relative to the observer.