A policeman with a very good ear and a good understanding of the Doppler effect stands on the shoulder of a freeway assisting a crew in a 40 -mph work zone. He notices a car approaching that is honking its horn. As the car gets closer, the policeman hears the sound of the horn as a distinct \(\mathrm{B} 4\) tone \((494 \mathrm{~Hz}) .\) The instant the car passes by, he hears the sound as a distinct \(\mathrm{A} 4\) tone \((440 \mathrm{~Hz}) .\) He immediately jumps on his motorcycle, stops the car, and gives the motorist a speeding ticket. Explain his reasoning.

The Doppler effect formula is given by: \(f' = f \frac{v + v_{0}}{v + v_{s}}\) where \(f'\) is the frequency observed by the listener (policeman), \(f\) is the source frequency (car's honk), \(v\) is the speed of sound, \(v_{0}\) is the speed of the listener relative to the air (usually 0 for a stationary listener), and \(v_{s}\) is the speed of the source relative to the air.

Now, we can set up two separate equations for when the car is approaching and when it passes the policeman. As the car approaches, the observed frequency is higher (494 Hz), and as it passes, the observed frequency is lower (440 Hz). For approaching: \(f' = f \frac{v}{v - v_{s}}\) For passing: \(f' = f \frac{v}{v + v_{s}}\)

Now, we can solve for \(v_s\). First, we will need to find the ratio of approaching to passing frequencies, which is \(\frac{494}{440}\). Then, we can write the following equation: \(\frac{v}{v - v_{s}} = \frac{494}{440}\frac{v}{v + v_{s}}\) Next, we will solve this equation for \(v_s\). First, multiply both sides by \((v-v_s)(v+v_s)\) to get: \(v(v + v_{s})(v - v_{s}) = \frac{494}{440}(v + v_{s})(v - v_{s})(v)\) Now we can simplify this equation to: \(v_s^2 = \frac{494}{440}v^2 - v^2\) Let \(v\) be the typical speed of sound, about 343 m/s: \(v_s^2 = \frac{494}{440}(343^2) - 343^2\) Solving for \(v_s\): \(v_s = \sqrt{(\frac{494}{440})(343^2) - 343^2} = 56.62~\mathrm{m/s}\) Convert this to mph: \(v_s = 56.62 \cdot \frac{3600}{1609.34} = 126.62~\mathrm{mph}\)

Now that we've calculated the speed of the car 126.62 mph, we can compare it to the speed limit in the work zone, which is 40 mph. Since the car's speed is much higher than the speed limit, the policeman's decision to give the motorist a speeding ticket is justified.