Question

A boy stands on ground between two vertical mountains. He is not in the midway between them. He fires a bullet. The 2nd echo is heard 1.92 seconds after the first and 1.47 seconds before the third. ( Speed of sound is 350 )

Find the distance between the hills

Answer 1

851 m

Step-by-step explanation:

Let's say the distance to one mountain is x, and the distance to the other mountain is y, such that x < y.

First, the wave moves a distance x, reflects off the nearer mountain, and travels another distance x back to the boy. This is the first echo. The time is:

t₁ = 2x / c

Meanwhile, the wave also moves a distance y in the opposite direction, reflects off the further mountain, and travels another distance y back to the boy. This is the second echo. The time is:

t₂ = 2y / c

Finally, each wave reflects off the opposite mountain and travels a total distance of 2x + 2y. This is the third echo. The time is:

t₃ = (2x + 2y) / c

Which means:

t₃ = t₁ + t₂

We are given the differences between the times:

t₂ − t₁ = 1.92 s

t₃ − t₂ = 1.47 s

Since t₃ = t₁ + t₂:

t₁ + t₂ − t₂ = 1.47 s

t₁ = 1.47 s

Which means:

t₂ − 1.47 s = 1.92 s

t₂ = 3.39 s

And:

t₃ = 1.47 s + 3.39 s

t₃ = 4.86 s

Therefore:

4.86 = (2x + 2y) / c

4.86c = 2x + 2y

2.43c = x + y

Given c = 350 m/s:

x + y = (2.43 s) (350 m/s)

x + y = 850.5 m

Rounding to three significant figures, the distance between the mountains is 851 m.