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

a) x₀ = - 2 m , b) y = 4.47 m

Step-by-step 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