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

Answer:

The truck will reach there in 250 seconds.

Explanation:

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

$f_{obv}$=$\frac{v}{v-v_{s} } f$

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

$\frac{v}{v-v_{0} } f$ = 460------------------------------------------( 1 )

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

$f_{obv}$=$\frac{v}{v+v_{s} } f$

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

$\frac{v}{v+v_{0} } f$ = 410-------------------------------------------( 2 )

Dividing ( 1 ) by ( 2 )

$\frac{v+v_{s} }{v-v_{s} } =\frac{460}{410}$

$\frac{v+v_{s} }{v-v_{s} } =\frac{46}{41}$

41v + 41$v_{s}$ = 46v  - 46$v_{s}$

87$v_{s}$= 5v

$v_{s}$=$\frac{5}{87}v$

Velocity of Sound (v)= 348 m/s

$v_{s}$=20 m/s

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

$Time=\frac{Distance}{Time}$

Distance= 5000 m

Time=$\frac{5000}{20}$

Time= 250 s

Time = 4 min 10 sec