Airplanes are noisy things. For example, if you stand really close to a runway, a military jet can produce a sound intensity of about 100~W/m^2 at a distance of 30 m from the plane (this is beyond painful noise). You live near a military airport and want to work on physics with your friend. Since you don't want to shout, you decide to find a spot far enough away from the airport that the airplane noise is below the threshold of conversation (10^{-6}~W/m^2. How far do you need to go from the airport in kilometers to accomplish this?
To determine how far you need to go from the airport to reduce the airplane noise below the threshold of conversation, we can use the inverse square law for sound.
The inverse square law states that the intensity of sound decreases inversely proportional to the square of the distance from the source of the sound. Mathematically, this can be expressed as:
I₁/I₂ = (d₂/d₁)²
where:
I₁ is the initial sound intensity (100 W/m²)
I₂ is the desired sound intensity (10^-6 W/m²)
d₁ is the initial distance from the source (30 m)
d₂ is the distance from the source we need to find
Let's solve for d₂ by rearranging the equation:
I₂ = (I₁ * d₁²) / d₂²
Substituting the given values:
10^-6 = (100 * 30²) / d₂²
Simplifying the equation:
10^-6 * d₂² = 100 * 900
d₂² = (100 * 900) / 10^-6
d₂ = √((100 * 900) / 10^-6)
d₂ ≈ √(900,000,000,000)
d₂ ≈ 94868.63 meters
As the question asks for the distance in kilometers, we convert meters to kilometers:
d₂ ≈ 94.87 kilometers
Therefore, you would need to go approximately 94.87 kilometers away from the airport to reduce the airplane noise below the threshold of conversation.