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 #72986, Physics / Mechanics | Relativity

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 have for each of the police officers? Take the speed of sound equal to 340 m/s.

Solution:

We use the equation for the Doppler Effect:

$$f = \left(\frac {c + v _ {r}}{c + v _ {s}}\right) f _ {0}$$

where $c$ is the velocity of waves in the medium; $v_r$ is the velocity of the receiver relative to the medium; positive if the receiver is moving towards the source (and negative in the other direction); $v_s$ is the velocity of the source relative to the medium; positive if the source is moving away from the receiver (and negative in the other direction).

If the speeds $v_s$ and $v_r$ are small compared to the speed of the wave, the relationship between observed frequency $f$ and emitted frequency $f_0$ is approximately

$$f = \left(1 + \frac {\Delta v}{c}\right) f _ {0}$$

Police officer follows the ambulance

$$f = \left(1 + \frac {22.2 \, \text{m/s} - 25 \, \text{m/s}}{340 \, \text{m/s}}\right) \times 250 \, \text{Hz} = 247.9 \, \text{Hz}$$

Police officer moving towards the ambulance

$$f = \left(1 + \frac {25 \, \text{m/s} - 22.2 \, \text{m/s}}{340 \, \text{m/s}}\right) \times 250 \, \text{Hz} = 252 \, \text{Hz}$$

Answer: 247.9 Hz and 252 Hz

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