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## Problem

A police siren emits a frequency of 640 Hz. When the police car approaches you at a speed of 25 m/s, what frequency do you perceive when:

1) you are not moving

2) you are moving towards the police car at 5 m/s

3) you are moving away from the police car at 5 m/s.

Assume the speed of sound is 343 m/s.

## Solution

Use the doppler shift formula

$f = f_0 \left(\frac{v \pm v_r}{v \pm v_s}\right)$

where $f$ is the shifted frequency, $f_0$ is the frequency of the source, $v$ is the speed of sound, $v_r$ is the speed of the receiver, and $v_s$ is the speed of the source.

Part 1

$f = f_0 \left(\frac{v}{v - v_s}\right)$
$f = 640 \left(\frac{343}{343 - 25}\right)$
$f = 690 \: Hz$

Part 2

$f = f_0 \left(\frac{v + v_r}{v - v_s}\right)$
$f = 640 \left(\frac{343 + 5}{343 - 25 }\right)$
$f = 700.38 \: Hz$

Part 3

$f = f_0 \left(\frac{v - v_r}{v - v_s}\right)$
$f = 640 \left(\frac{343 - 5}{343 - 25 }\right)$
$f = 680.25 \: Hz$
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