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15 January, 15:53

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

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  1. 15 January, 18:24
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    851 m

    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.
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