Question

A person standing between two walls as shown in the figure shouts out and hears three echoes after $1s\text{,}2s\text{,}3s$ respectively. Determine values of x and y. Take the speed of sound to be $330m/s$.

Answer 1

Step 1: Analyze the given situation and extract information-

1. Firstly, understand that echoes are nothing but the reflection of sound waves. Echoes listened after some time mean that the sound waves reflected from the source reach back to the source after that particular time.
2. We are given that echoes of sound reached back to the person after $1s,2s,3s$.
3. So, the sound wave reflected after $1$ second has reached back to the person earliest, which means the sound waves must have travelled the least.
4. From the figure, it is seen that the person is standing nearer to wall $A$ than wall $B$. So echo listened after 1s is reflected sound wave from wall$A$.
5. Similarly, the echo listened after $2s$ must have reflected from wall $B$.
6. The third echo needs to be analyzed carefully. It must be the third earliest possible reflection of sound waves reaching the person. In this case, the sound wave should have travelled to wall $A$, then to the person, and then after it goes to wall $B$ and again after reflecting, it again reaches the person. So, the distance travelled will be $x+x+y+y=2(x+y)$.
7. If we take a path where the sound wave reaches wall $B$ first then go to the person and wall $A$ and again reach the source, in that case also the distance travelled by the sound wave will be$y+y+x+x=2(x+y)$.
8. We are also given the speed of sound is $330m/s$.

Step 2: Formulate equations-

For the first echo, sound waves have travelled to wall $A$ and come back to the person. So the total distance travelled by sound waves will be $2x$.

$\therefore 1=\frac{2x}{330}$.

Similarly, for echo coming from wall $B$, the equation will be $2=\frac{2y}{330}$.

For echo listened after $3 s$, the equation will be $3=\frac{2(x+y)}{330}$.

We can see that this equation can be derived by adding the above two equations. So it is anyway not giving any new information about the situation so it is a redundant equation that can be eliminated for further usage.

Step 3: Solve the above equations to find $x$ and $y$-

$2x=330$ or, $x=155m$.

$2y=660$ or, $y=330m$.

Hence, value of $xis155m$ and value $yis330m$.