Question

You are investigating the report of a UFO landing in an isolated portion of New Mexico, and encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are 7.5 m from it, you measure its intensity to be 0.11W/m2.
An intensity of 1.0W/m2 is often used as the "threshold of pain". How much closer to the source can you move before the sound intensity reaches this threshold?

Answer 1

Answer:

d = 5 m

Explanation:

Given:

- The initial distance from UFO r_1 = 7.5 m

- The initial intensity I_1 = 0.11 W / m^2

- The intensity of "threshold of pain" I_2 = 1.0 W / m^2

Find:

How much closer to the source can you move before the sound intensity reaches this threshold?

Solution:

- For waves that spread out in 3 dimensions , its intensity I is inversely proportional to square of the distance from the source. The expression is given as :

                                    I = k / r^2

Where,

k: Proportionality constant

r: The distance from the source.

- Using the relation above the amount of distance r_2 from source that is required before I_1 --> I_2 is :

                                    I_1 / I_2 = (r_2 / r_1)^2

Re- arrange to get r_2:

                                    r_2 = r_1*sqrt(I_1 / I_2)

Plug in the given values:

                                    r_2 = 7.5*sqrt(0.11/1)

                                    r_2 = 2.5 m

- So the amount of distance from source is 2.5 m. So from initial position we have moved distance d:

                                    d = 7.5 - 2.5

                                    d = 5 m

Answer 2

Answer: 5 m

Explanation:

We have the following data:

$I_{1}=0.11 W/m^{2}$ is the intensity of the sound at 7.5 m from the source

$r_{1}=7.5 m$ is the distance at which the intensity $I_{1}$ was measured

$I_{2}=1 W/m^{2}$ is the intensity of the sound at $r_{2}$ from the source

We have to find $r_{2}$

Since the object is radiating the signal uniformly in all directions, we can use the Inverse Square Law for Intensity:

$\frac{I_{1}}{I_{2}}=\frac{r_{2}^{2}}{r_{1}^{2}}$

Isolating $r_{2}$:

$r_{2}=r_{1}\sqrt{\frac{I_{1}}{I_{2}}}$

$r_{2}=7.5 m\sqrt{\frac{0.11 W/m^{2}}{1 W/m^{2}}}$

$r_{2}=2.48 m$ This is the distance at which the intensity is the "threshold of pain"

Now, we have to substract this value to $r_{1}$ to find how much closer to the source can we move:

$r_{1}-r_{2}=7.5 m - 2.48 m=5.02 m \approx 5 m$