Question

A person standing between two parallel hills fires a gun. He hears the first echo after 3/2 s, and second echo after 5/2 s. If the speed of sound is 332 m/s, calculate the distance between the hills. When will you hear the third echo?

Answer 1

Hint: An echo is heard after the sound covers twice the initial distance between source and obstacle and the distance is speed multiplied by time.
we can use the Formula: $2d = v \times t$

Step by step answer: A person standing between two parallel hills fires a gun. Let d₁ be the distance between the first hill and person and d₂ be the distance between the second hill and person. Speed of sound = 332 m/s.
The following steps are:
1.The echo produced for the first hill:
The total distance = 2d₁ [ Since the sound produced by the gun strikes the first hill and comes back to the person]
The given formula is $2{d_1} = v \times {t_1}$
 $2{d_1} = 332 \times \dfrac{3}{2}$[time taken for the first echo]
 ${d_1} = \dfrac{{498}}{2} = 249m$
2.The echo produced for the second hill:
The total distance = 2d₂ [Since the sound from the first wall strikes the second wall and comes back to the person]
The given formula is $2{d_2} = v \times {t_2}$
 $\Rightarrow$ $2{d_2} = 332 \times \dfrac{5}{2}$ [time taken for the second echo]
 $\Rightarrow$ ${d_2} = \dfrac{{830}}{2}$
 $\Rightarrow$ ${d_2} = 415$ m
3.Total distance between the hills=${d_1} + {d_2}$ $\therefore$ =$249 + 415 = 664m$
4.The third echo will be heard when the sound comes back to the person after reflection from the first hill and goes towards the second hill and reflects back to the person again.
The total time taken for the third echo = ${t_1} + {t_2}$ = $\dfrac{3}{2} + \dfrac{5}{2}$ = $\dfrac{8}{2} = 4$ s

Note: The total distance between the obstacle (hill) and the source (gun) is always taken twice the original distance. The time for the third echo is the sum of the first and second duration of echo.