A bullet passes past a person at a speed of 220 m s^{−1}. Find the fractional change in the frequency of the whistling sound heard by the person as the bullet crosses the person. Speed of sound in air = 330 m s^{−1}.

#### Solution

Given:

Velocity of bullet \[v_s\]= 220 ms^{−1}

Speed of sound in air *v* = 330 ms^{−1}

Let the frequency of the bullet be *f*.

Apparent frequency heard by the person \[\left( f_1 \right)\] before crossing the bullet is given by:

\[f_1 = \left( \frac{v}{v - v_s} \right) \times f\]

On substituting the values, we get :

\[f_1 = \left( \frac{330}{330 - 220} \right) \times f = 3f . . . . \left( 1 \right)\]

Apparent frequency heard by the person \[\left( f_2 \right)\] after crossing the bullet is given by :

\[f_2 = \left( \frac{v}{v + v_s} \right) \times f\]

On substituting the values, we get :

\[f_2 = \left( \frac{330}{330 + 220} \right) \times f = 0 . 6f . . . . . \left( 2 \right)\]

So,

\[ \left( \frac{f_2}{f_1} \right) = \frac{0 . 6f}{3f} = 0 . 2\]