Question

A car travelling at 100.0 km/h sounds its horn as it approaches a hiker standing on the    highway. If the car’s horn has a frequency of 440 Hz and the temperature of the air is 0°C, what is the frequency of the sound waves reaching the hiker 

(a) as the car approaches? 

(b) after it has passed the hiker?

Answer 1

Let's first find the speed of the sound in the air at 0°C temperature:

$$v=(331.3+0.606\times T)\ \dfrac{m}{s},$$ $$v=(331.3+0.606\times0^{\circ})\ \dfrac{m}{s}=331.3\ \dfrac{m}{s}.$$

Let's also convert km/h to m/s:

$$v_s=100\ \dfrac{km}{h}\times\dfrac{1000\ m}{1\ km}\times\dfrac{1\ h}{3600\ s}=27.78\ \dfrac{m}{s}.$$

(a) We can find the frequency observed by the hiker as the car approaches him from the Doppler Shift formula:

$$f_o=f_s(\dfrac{v}{v-v_s}),$$ $$f_o=440\ Hz\times(\dfrac{331.3\ \dfrac{m}{s}}{331.3\ \dfrac{m}{s}-27.78\ \dfrac{m}{s}})=480\ Hz.$$

(b) We can find the frequency observed by the hiker after the car has passed the hiker from the Doppler Shift formula:

$$f_o=f_s(\dfrac{v}{v+v_s}),$$ $$f_o=440\ Hz\times(\dfrac{331.3\ \dfrac{m}{s}}{331.3\ \dfrac{m}{s}+27.78\ \dfrac{m}{s}})=406\ Hz.$$