Question

What frequency is received by a mouse just before being dispatched by a hawk flying at it at $25.0\mathrm{m}/\mathrm{s}$and emitting a screech of frequency$3500\mathrm{H}\mathrm{z}$? Take the speed of sound to be$331\mathrm{m}/\mathrm{s}$.

Short answer

The apparent frequency is$3785.94\mathrm{H}\mathrm{z}$.

Step-by-step solution

The apparent Frequency

The apparent frequency of a moving source depends on the relative speed between the source and the listener.

Given Data

The frequency is$f=3500\mathrm{H}\mathrm{z}$.

The speed of the ambulance is$v_s=25.0\mathrm{m}/\mathrm{s}$.

The speed of sound is$v=331\mathrm{m}/\mathrm{s}$.

The speed of the listener is zero.

Calculation of the apparent frequency

The Doppler Effect tells the apparent frequency is,

$f^{\prime }=f\left(\frac{v-v_o}{v-v_s}\right)$

Now, the apparent frequency is,

$\begin{array}{c}{f}^{\prime }=3500\left(\frac{331-0}{331-25}\right)\\ =3500\times \frac{331}{306}\\ =3785.94\mathrm{H}\mathrm{z}\end{array}$

Calculation of the frequency of the jet moving away the observer

Now, the apparent frequency of the passing away ambulance is,

$\begin{array}{c}{f}^{″}=3500\left(\frac{342-0}{342-\left(-333.3\right)}\right)\\ =3500\times \frac{1140}{2251}\\ =1772.5\mathrm{H}\mathrm{z}\end{array}$

Hence, the apparent frequencies are $3785.94\mathrm{H}\mathrm{z}$and $1772.5\mathrm{H}\mathrm{z}$.