At the Indianapolis 500, you can measure the speed of cars just by listening to the difference in pitch of the engine noise between approaching and receding cars. Suppose the sound of a certain car drops by a factor of 2.30 as it goes by on the straightaway. How fast is it going? (Take the speed of sound to be 343 m/s.)
Answer in km/h
To solve this question, we can use the Doppler effect formula. The Doppler effect describes the change in frequency (or pitch) of a wave when there is relative motion between the source of the wave and the observer.
The formula for the Doppler effect is as follows:
f' = (v + v₀) / (v - vₛ) * f₀
where:
f' is the observed frequency,
v is the velocity of sound,
v₀ is the velocity of the observer,
vₛ is the velocity of the source, and
f₀ is the original frequency.
In this case, the frequency drops by a factor of 2.30 as the car goes by. This means that f' is 1/2.30 times f₀. We can substitute these values into the formula and solve for vₛ, the velocity of the source (car).
1/2.30 = (343 + 0) / (343 - vₛ)
Simplifying the equation further, we get:
1/2.30 = 343 / (343 - vₛ)
To isolate vₛ, we can cross-multiply:
(343 - vₛ) = 343 * 1/2.30
Now, let's solve for vₛ:
vₛ = 343 - (343 * 1/2.30)
Calculating this, we find that vₛ ≈ 83.92 m/s.
The final step is to convert this to km/h. There are 1000 meters in 1 kilometer and 3600 seconds in 1 hour. Therefore, to convert m/s to km/h, we can multiply by 3.6:
vₛ ≈ 83.92 m/s * 3.6 km/h
vₛ ≈ 301.99 km/h
Hence, the car is going approximately 301.99 km/h.