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Question: The frequency of the sound produced in the horn of a vehicle is \(680\,Hz\) . When the vehicle appro...

The frequency of the sound produced in the horn of a vehicle is 680Hz680\,Hz . When the vehicle approaches a stationary observer, the frequency of the sound heard by the observer is 700Hz700\,Hz . Find the speed of the vehicle.

Explanation

Solution

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=VVVsF0F = \dfrac{V}{{V - {V_s}}}{F_0}
Where FF is the frequency of the horn after it reaches the observer, F0{F_0} is the frequency of the horn at a long distance to observer, VV is the speed of the sound in air and Vs{V_s} is the speed of the vehicle.

Complete step by step solution:
It is given that the frequency of the horn sound, F0=680Hz{F_0} = 680\,Hz
The frequency of the horn sound heard by the observer, F=700HzF = 700\,Hz
By using the formula of the frequency,
F=VVVsF0F = \dfrac{V}{{V - {V_s}}}{F_0}
Substituting the known values in the above formula.
700=350350Vs×680700 = \dfrac{{350}}{{350 - {V_s}}} \times 680
In the above step, the velocity of the air is considered as the 350ms1350\,m{s^{ - 1}}. By simplifying the above step, we get
700680=350350Vs\dfrac{{700}}{{680}} = \dfrac{{350}}{{350 - {V_s}}}
By performing basic arithmetic operation,
350=1.029(350Vs)350 = 1.029\left( {350 - {V_s}} \right)
By further simplifying the above step,
Vs=3503501.029{V_s} = 350 - \dfrac{{350}}{{1.029}}
By doing the basic arithmetic operation,
Vs=6.86ms1{V_s} = 6.86\,m{s^{ - 1}}

Hence the velocity of the vehicle is obtained as 6.86ms16.86\,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.