the siren of a burglar alarm system has a frequency of 960 Hz. During a patrol, a security officer, travelling in his car, hears the siren of the alarm of a house and approaches the house at a constant velocity.A detector in his car registers the frequency of the sound as 1000 Hz. the speedof sound in air is 340 m.s^-1
1. calculate the spped at which the patrol car approaches the house. use the speed of sound in air as 340 m.s^-1.
First, we need to use the Doppler effect equation to calculate the speed of the patrol car approaching the house:
f' = f(v + vo) / (v + vs)
Where:
f' = observed frequency (in Hz) = 1000 Hz
f = actual frequency of the siren (in Hz) = 960 Hz
v = speed of sound in air = 340 m/s
vo = speed of the patrol car
vs = speed of the siren
Now, let's plug in the values:
1000 = 960(340 + vo) / (340 + vs)
Solving for vo:
1000(340 + vs) = 960(340 + vo)
340000 + 1000vs = 326400 + 960vo
1000vs = 13600 + 960vo
vs = 13.6 + 0.96vo
Now, let's calculate the relative speed vs of the siren:
vs = vo - v
13.6 + 0.96vo = vo - 340
13.6 = 0.04vo + 340
0.04vo = 326.4
vo ≈ 8160 m/s
Therefore, the speed at which the patrol car approaches the house is approximately 8160 m/s.