Question

Hearing the siren of an approaching fire truck, you pull over to the side of the road and stop. As the truck approaches, you hear a tone of 470 Hz ; as the truck recedes, you hear a tone of 410 Hz .

How much time will it take for the truck to get from your position to the fire 6.0 km away, assuming it maintains a constant speed?

Answer 1

I am assuming speed of sound to be 340 m/s.

equate the observed frequencies to the frequency of the source

f_receding ( v + v_s) / v = f_approach ( v - v_s) / v

f_receding ( v + v_s) = f_approach ( v - v_s)

f_receding * v + f_receding * v_s =  f_approach * v - f_approach *v_s

Solving for v_s , we get

v_s = v(f_approach - f_receding ) / (f_approach + f_receding )

v_s = 340 (470 - 410) / 470 + 410

v_s = 23.18 m/s

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Now, we know the speed, and we have the distance

so,

time taken = distance / speed

time = 258.8 seconds or 4.31 minutes