Question

An ambulance travels down a highway at a speed of 120 km/h, its siren emitting sound at a frequency of 400 Hz. A passenger in a car which is approaching the ambulance hears the sound at 470 Hz. If the air temperature during that time is 24^oC, at what speed the car travels?

Answer 1

Let's first find the speed of sound at air temperature of $24\ ^{\circ}C$:

$$v=\sqrt{\dfrac{\gamma RT}{M}},$$ $$v=\sqrt{\dfrac{1.4\cdot8.314\ \dfrac{J}{mol\cdot K}\cdot297.15\ K}{0.032\ \dfrac{kg}{mol}}}=329\ \dfrac{m}{s}.$$

Finally, we can find the speed of the car (or the speed of the observer) from the Doppler Shift formula:

$$f_o=f_s(\dfrac{v+v_o}{v-v_s}),$$ $$v_o=\dfrac{f_o}{f_s}(v-v_s)-v,$$ $$v_o=\dfrac{470\ Hz}{400\ Hz}(329\ \dfrac{m}{s}-33.3\ \dfrac{m}{s})-329\ \dfrac{m}{s}=18.45\ \dfrac{m}{s}.$$