Question

Two fire trucks are moving toward each other at a rate of 25 m/s. If the first

fire truck emits a frequency of 10000Hz and the ambient temperature is 400C,

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 $40\ ^{\circ}C$:

$$v=\sqrt{\dfrac{\gamma RT}{M}},$$ $$v=\sqrt{\dfrac{1.4\cdot8.314\ \dfrac{J}{mol\cdot K}\cdot313.15\ K}{0.02896\ \dfrac{kg}{mol}}}=355\ \dfrac{m}{s}.$$

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

$$f_o=f_s(\dfrac{v+v_o}{v-v_s}),$$ $$f_o=10000\ Hz\cdot(\dfrac{355\ \dfrac{m}{s}+25\ \dfrac{m}{s}}{355\ \dfrac{m}{s}-25\ \dfrac{m}{s}})=11515\ Hz.$$