Answer to Question #163242 in Physics for Earl

Question

A train moving at a speed of 40m/s sounds its horn which has a frequency of 5 x 10² Hz. Determine the frequency heard by a stationary observer as the train approachea the observer. The ambient temperature is 30°C.

Answer 1

Let's first find the speed of sound at air temperature of $30\ ^{\circ}C$:

$$v=\sqrt{\dfrac{\gamma RT}{M}},$$ $$v=\sqrt{\dfrac{1.4\cdot8.314\ \dfrac{J}{mol\cdot K}\cdot303.15\ K}{0.032\ \dfrac{kg}{mol}}}=332\ \dfrac{m}{s}.$$

Finally, we can find the frequency of the sound heard by the stationary observer from the Doppler Shift formula:

$$f_o=f_s(\dfrac{v}{v-v_s}),$$ $$f_o=5\cdot10^2\ Hz\cdot(\dfrac{332\ \dfrac{m}{s}}{332\ \dfrac{m}{s}-40\ \dfrac{m}{s}})=568.5\ Hz.$$