Question

An ambulance siren has frequency 250 Hz. The ambulance is headed towards an
accident site with a speed of 90 km /h. Two police officers on separate motor
cycles head for the same accident site: one follows the ambulance with a speed of
80 km/h and the other approaches the accident site from the other direction with a
speed of 80 km/h. What frequency does ambulance siren has for each of the police
officers? Take the speed of sound equal to 340 m/s.

Answer 1

Answer on Question #74870 Physics / Other

An ambulance siren has frequency $f_0 = 250 \, \text{Hz}$. The ambulance is headed towards an accident site with a speed of $90 \, \text{km/h}$. Two police officers on separate motor cycles head for the same accident site: one (first) follows the ambulance with a speed of $u = 80 \, \text{km/h}$ and the other (second) approaches the accident site from the other direction with a speed of $v = 80 \, \text{km/h}$. What frequency does ambulance siren have for each of the police officers? Take the speed of sound equal to $c = 340 \, \text{m/s}$.

Solution:

The relative velocity

(For officer that follows the ambulance) $\Delta v_1 = u - v = 80 - 90 = -10\frac{\mathrm{km}}{\mathrm{h}} = -2.8\frac{\mathrm{m}}{\mathrm{s}}$

(For officer that moves towards the ambulance) $\Delta v_2 = u + v = 80 + 90 = 170\frac{\mathrm{km}}{\mathrm{h}} = 47.2\frac{\mathrm{m}}{\mathrm{s}}$

Doppler Effect states

$$f = f_0 \left(1 + \frac{\Delta v}{c}\right)$$

Thus

$$f_1 = f_0 \left(1 + \frac{\Delta v_1}{c}\right) = 250 \left(1 - \frac{2.8}{340}\right) = 248 \, \text{Hz}$$ $$f_2 = f_0 \left(1 + \frac{\Delta v_2}{c}\right) = 250 \left(1 + \frac{47.2}{340}\right) = 285 \, \text{Hz}$$

Answers:

First officer: $f_1 = 285 \, \text{Hz}$

Second officer: $f_2 = 248 \, \text{Hz}$

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