Question

Two cars are approaching each other at an equal speed of $7.2\mathrm{km}/\mathrm{hr}$. When they see each other, both blow horns having frequency of$676\mathrm{Hz}$. The beat frequency heard by each driver will be ________ Hz. [Velocity of sound in air is $340\mathrm{m}/\mathrm{s}$.]

Answer 1

Step 1. Given data

Speed of each car is $7.2\mathrm{km}/\mathrm{hr}$ which is equals to $2\mathrm{m}/\mathrm{s}$ which is shown in above figure and frequency is $676\mathrm{Hz}$ and velocity of sound is $340\mathrm{m}/\mathrm{s}$.

We have to find the apparent frequency.

Step 2. Formula to be used.

As we know that the formula of doppler effect.

The frequency of the sound heard by an observer can be affected by both the motion of the observer and the motion of the source of the sound.

So, we can calculate the observed sound frequency $\mathrm{f}\text{'}$ by using equation is,

$\mathrm{f}\text{'}={\mathrm{f}}_0\left(\frac{\mathrm{v}+{\mathrm{v}}_0}{\mathrm{v}-{\mathrm{v}}_{\mathrm{s}}}\right)$

Here, ${\mathrm{f}}_0$ is the actual frequency of sound waves, $\mathrm{f}\text{'}$ is observed or apparent frequency, $\mathrm{v}$ is speed of the sound waves, ${\mathrm{v}}_0$ is velocity of the observer and ${\mathrm{v}}_{\mathrm{s}}$ is the velocity of the source.

Step 3. Find the apparent frequency.

The doppler effect is,

$\mathrm{f}\text{'}={\mathrm{f}}_0\left(\frac{\mathrm{v}+{\mathrm{v}}_0}{\mathrm{v}-{\mathrm{v}}_{\mathrm{s}}}\right)$

Substitute the given values in the above fromula, we get,

$\mathrm{f}\text{'}=676\left(\frac{340+2}{340-2}\right)$

$=676\left(\frac{342}{338}\right)$

$=676\times 1.01183432$

$=684\mathrm{Hz}$

Now,

${\mathrm{f}}_{\mathrm{beat}}=\mathrm{f}\text{'}-{\mathrm{f}}_0$

$=684-676$

$=8\mathrm{Hz}$

Hence, when they see each other, both blow horns have a frequency of$676\mathrm{Hz}$, then the beat frequency heard by each driver will be $8\mathrm{Hz}$.