Question

Two fire trucks is moving toward each other at a ratw of 25ml/s. If the first fire truck emits a frequency of 10000Hz and the ambient temperture is 40°C, What will be the observed frequency of the second fire truck?

Answer 1

Let's first find the speed of sound at air temperature of $40\ ^{\circ}C$:

$$v=\sqrt{\dfrac{\gamma RT}{M}},$$ $$v=\sqrt{\dfrac{1.4\cdot8.314\ \dfrac{J}{mol\cdot K}\cdot313.15\ K}{0.032\ \dfrac{kg}{mol}}}=337.5\ \dfrac{m}{s}.$$

Finally, we can find the frequency of the sound detected by the second fire truck from the Doppler Shift formula:

$$f_o=f_s(\dfrac{v+v_o}{v-v_s}),$$ $$f_o=10000\ Hz\cdot(\dfrac{337.5\ \dfrac{m}{s}+25\ \dfrac{m}{s}}{337.5\ \dfrac{m}{s}-25\ \dfrac{m}{s}})=11600\ Hz.$$