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 .

Part A)

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

Answer 1

According to the Doppler effect, the apparent frequency as heard by a listener depends on the original frequency and the relative motion between the listener and the source. the general equation for the apparent frequency of sound as heard by a listener is given by

where,

Here, you are the observer and you are at rest, so your velocity,

Let the velocity of sound,

Now for the above general equation,

• If the observer is approaching the source, then the plus sign will be taken in the numerator.
• If the observer is moving away from the source, then the minus sign will be taken in the numerator.
• If the source is approaching the observer, then the minus sign will be taken in the denominator.
• If the source is moving away from the observer, then the plus sign will be taken in the denominator.

CASE-1 - The truck is approaching you

Since the truck is approaching you the velocity of the source in the denominator is negative.

Given,

The apparent frequency heard by you is,

Therefore the equation for apparent frequency in this case is given by,

Substituting the values, we get,

CASE-2 - The truck is moving away from you

Since the truck is moving away from you the velocity of the source in the denominator is positive.

Given,

The apparent frequency heard by you is,

Therefore the equation for apparent frequency, in this case, is given by,

Substituting the values, we get,

Now divide equation (1) by equation (2)

Solve the above equation for

So the velocity with which the truck is moving is 23.386 m/s

Given,

Distance traveled by truck, S = 6km = 6000m

Therefore time taken to cover this distance with the constant velocity is given by,

Therefore the time required for the truck to move from your position to the fire is 256.564s