Question

Hearing the siren of an approaching fire truck, you pull over to side of the road and stop. As the truck approaches, you hear a tone of 460 Hz; as the truck recedes, you hear a tone of 410 Hz. How much time will it take to jet from your position to the fire 5.00 km away, assuming it maintains a constant speed?

Answer 1

The truck will reach there in 250 seconds.

Step-by-step explanation:

The frequency due to doppler effect, when the observer is stationary and the source is moving towards it is

`f_(obv)`=
`(v)/(v-v_(s) ) f`

where v= velocity of sound in air

`v_(s)`= velocity of source of sound

f= frequency of sound and

`f_(obv)`= frequency oberved due to Doppler effect

`(v)/(v-v_(0) ) f` = 460------------------------------------------( 1 )

The frequency due to doppler effect, when the observer is stationary and the source is moving away from it

`f_(obv)`=
`(v)/(v+v_(s) ) f`

where v= velocity of sound in air

`v_(s)`= velocity of source of sound

f= frequency of sound and

`f_(obv)`= frequency oberved due to Doppler effect

`(v)/(v+v_(0) ) f` = 410-------------------------------------------( 2 )

Dividing ( 1 ) by ( 2 )

`(v+v_(s) )/(v-v_(s) ) =(460)/(410)`

`(v+v_(s) )/(v-v_(s) ) =(46)/(41)`

41v + 41
`v_(s)` = 46v - 46
`v_(s)`

87
`v_(s)`= 5v

`v_(s)`=
`(5)/(87)v`

Velocity of Sound (v)= 348 m/s

`v_(s)`=20 m/s

Therefore, the truck is moving at 20 m/s.

`Time=(Distance)/(Time)`

Distance= 5000 m

Time=
`(5000)/(20)`

Time= 250 s

Time = 4 min 10 sec