Question

Many towns have tornado sirens, large elevated sirens used to warn locals of imminent tornados. In one small town, a siren is elevated \(100 . \mathrm{m}\) off the ground. A car is being driven at \(100 . \mathrm{km} / \mathrm{h}\) directly away from this siren while it is emitting a \(440 .-\mathrm{Hz}\) sound. What is the frequency of the sound heard by the driver as a function of the distance from the siren at which he starts? Plot this frequency as a function of the car's position up to \(1000 . \mathrm{m} .\) Explain this plot in terms of the Doppler effect.

Short answer

Answer: The observed frequency of the sound heard by the driver is given by the function: \(f_{observed}(L) = \frac{440(340)}{(340-27.8)}\times\frac{L}{\sqrt{L^2+100^2}}\), where L is the horizontal distance from the siren at which the car starts. The plot of this function shows a decreasing trend, indicating that the frequency of the sound heard by the driver decreases as they travel farther away from the siren. This behavior is in accordance with the Doppler effect, which states that the observed frequency will be lower than the source frequency when the observer (car) is moving away from the source (siren), causing the frequency to decrease on the plot.

Step-by-step solution

Recall the Doppler Effect formula for sound

We will use the Doppler effect formula for sound, which is given by: \(f_{observed}=\frac{f_{source}(c\pm v_o)}{(c\mp v_s)}\) Where \(f_{observed}\) is the observed frequency, \(f_{source}\) is the source frequency, \(c\) is the speed of sound (\(340\frac{\text{m}}{\text{s}}\)), \(v_o\) is the speed of the observer (the car), and \(v_s\) is the speed of the source(the siren). Since the car is moving away from the siren, we will use the "minus" sign in the numerator, and as the siren is stationary, we will use the "plus" sign in the denominator.

Convert the car's speed to meters per second

As the car's speed is given in km/h, and the speed of sound is given in m/s, we need to convert the car's speed to m/s. The conversion factor is: \(1\frac{\text{km}}{\text{h}}\approx0.278\frac{\text{m}}{\text{s}}\) So, the car's speed in m/s is: \(v_o=100\frac{\text{km}}{\text{h}}\times0.278\frac{\text{m}}{\text{s}}=27.8\frac{\text{m}}{\text{s}}\)

Calculate the actual distance between siren and car

The siren is elevated 100m off the ground. We need to find the actual distance between the siren and the car, considering both the horizontal distance and the vertical distance. This distance, let's call it \(d\), can be found using the Pythagorean theorem: \(d=\sqrt{L^2+100^2}\) Where \(L\) is the horizontal distance from the siren at which the car starts.

Calculate the observed frequency as a function of L

Now, we can plug in the values of \(f_{source}\), \(v_o\), and the actual distance \(d\) into the Doppler effect formula to find the observed frequency as a function of L: \(f_{observed}(L) = \frac{440(340)}{(340-27.8)}\times\frac{L}{\sqrt{L^2+100^2}}\)

Create the plot for the observed frequency as a function of L

We need to plot the observed frequency as a function of the car's position (L) up to 1000m. You can use a graphing calculator or software to plot the function \(f_{observed}(L)\) obtained in step 4 for the range of \(0 \le L \le 1000\). This plot will show a decreasing trend indicating that the frequency of the sound heard by the driver decreases as they travel farther away from the siren. This behavior is in accordance with the Doppler effect: since the car is moving away from the source, the observed frequency will be lower than the source frequency, causing the frequency to decrease on the plot.