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

d = 5 m

Step-by-step 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

Step-by-step 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:

`(I_(1))/(I_(2))=(r_(2)^(2))/(r_(1)^(2))`

Isolating
`r_(2)`:

`r_(2)=r_(1)\sqrt{(I_(1))/(I_(2))}`

`r_(2)=7.5 m\sqrt{(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`