Question

A sound source is located somewhere along the x-axis. Experiments show that the same wave front simultaneously reaches listeners at x = -7.0 m and x = +3.0 m.
part a) What is the x-coordinate of the source?
part b) A third listener is positioned along the positive y-axis. What is her y-coordinate if the same wave front reaches her at the same instant it does the first two listeners?

Answer 1

Answer:

a)   x₀ = - 2 m  , b)     y = 4.47 m

Explanation:

A wave travels in the middle with constant speed, let's use the equation of uniform motion

    v = d / t

    t = d / v

The distance to the first listeners, see attached

    d₁ = x₀-x

     t = (x₀ +7) / v

The distance to the second listener

    d₂ = x - x₀

     t = (+ 3- x₀) / v

As the wave arrives at the same time, we can equal the two equations

     (x₀ +7) / v = (3 -x₀) / v

      x₀ + 7 = 3 - x₀

      2 x₀ = 3 - 7

      x₀ = -4/2

      x₀ = - 2 m

b) The time it takes for the wave to reach the listeners of the x-axis, where the speed of sound is 340 m / s

          t = 5/340

          t = 0.0147 s

Let's look for the distance the wave travels for the listener axis and

       v = d₃ / t

       d₃ = v.t

       d₃ = 340 * 0.0147

       d₃ = 5 m

For the distance component we use the Pythagorean triangle

      d₃² = x₀² + y²

      y² = d₃² - x₀²

     y = √ (d₃² -4)

      y = √ (5² -4)

     y = 4.47 m