Question

An alert physics student stands beside the tracks as a train rolls slowly past. He notes that the frequency of the train whistle is 489 Hz when the train is approaching him and 473 Hz when the train is receding from him. Using these frequencies, he calculates the speed of the train. What value does he find? (Assume the speed of sound in air is 343 m/s.)

Answer 1

Answer on Question #73335, Physics / Molecular Physics

Question. An alert physics student stands beside the tracks as a train rolls slowly past. He notes that the frequency of the train whistle is $489Hz$ when the train is approaching him and $473Hz$ when the train is receding from him. Using these frequencies, he calculates the speed of the train.

What value does he find? (Assume the speed of sound in air is $343m/s$).

Given. $V_1 = 489Hz$; $V_2 = 473Hz$; $v = 343m/s$.

Find. $u - ?$

Solution.

According to the Doppler effect:

the train is approaching the student

$$V_1 = V_0 \frac{1}{1 - \frac{u}{v}}$$

the train is receding from the student

$$V_2 = V_0 \frac{1}{1 + \frac{u}{v}}.$$

We have

$$\frac{V_1}{V_2} = \frac{1 + \frac{u}{v}}{1 - \frac{u}{v}} \Rightarrow \frac{V_1}{V_2} = \frac{v + u}{v - u} \Rightarrow v(V_1 - V_2) = u(V_1 + V_2).$$

Finally

$$u = v \frac{V_1 - V_2}{V_1 + V_2} = 343 \cdot \frac{489 - 473}{489 + 473} = 5.7m/s.$$

Answer. $u = 5.7m/s$.

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