Question

The frequency of the sound produced in the horn of a vehicle is $680\,Hz$ . When the vehicle approaches a stationary observer, the frequency of the sound heard by the observer is $700\,Hz$ . Find the speed of the vehicle.

Answer 1

Hint:Use the formula of the frequency given below. Substitute the frequency of the horn in both distances and also substitute the value of the assumed speed of sound in air to find the value of the velocity of the vehicle which produces the horn.

Formula used:
The frequency of the horn sound is given by
$F = \dfrac{V}{{V - {V_s}}}{F_0}$
Where $F$ is the frequency of the horn after it reaches the observer, ${F_0}$ is the frequency of the horn at a long distance to observer, $V$ is the speed of the sound in air and ${V_s}$ is the speed of the vehicle.

Complete step by step solution:
It is given that the frequency of the horn sound, ${F_0} = 680\,Hz$
The frequency of the horn sound heard by the observer, $F = 700\,Hz$
By using the formula of the frequency,
$F = \dfrac{V}{{V - {V_s}}}{F_0}$
Substituting the known values in the above formula.
$700 = \dfrac{{350}}{{350 - {V_s}}} \times 680$
In the above step, the velocity of the air is considered as the $350\,m{s^{ - 1}}$. By simplifying the above step, we get
$\dfrac{{700}}{{680}} = \dfrac{{350}}{{350 - {V_s}}}$
By performing basic arithmetic operation,
$350 = 1.029\left( {350 - {V_s}} \right)$
By further simplifying the above step,
${V_s} = 350 - \dfrac{{350}}{{1.029}}$
By doing the basic arithmetic operation,
${V_s} = 6.86\,m{s^{ - 1}}$

Hence the velocity of the vehicle is obtained as $6.86\,m{s^{ - 1}}$ .

Note:Frequency is the number of times the waves vibrates per unit second. It is directly proportional to that of the velocity of the vehicle. Since the speed is equal to the product of the wavelength and the frequency of any sound waves.