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/m2 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/m2. How far do you need to go from the airport in kilometers to accomplish this?
To determine the distance you need to be from the airport to reduce the airplane noise to below the threshold of conversation, we can use the inverse square law for sound intensity. The inverse square law states that the intensity of a sound wave decreases inversely with the square of the distance from the source.
The formula for the inverse square law is:
I₁/I₂ = (r₂/r₁)²
Where:
I₁ is the initial intensity (100 W/m²),
I₂ is the desired intensity for conversation (10⁻⁶ W/m²),
r₁ is the initial distance (30 m), and
r₂ is the desired distance.
Let's plug in the given values into the formula and solve for r₂:
(100 W/m²) / (10⁻⁶ W/m²) = (r₂/30 m)²
Simplifying the equation:
10⁸ = (r₂/30)²
Taking the square root of both sides:
√(10⁸) = r₂/30
10⁴ = r₂/30
Multiplying both sides by 30:
r₂ = 10⁴ × 30
r₂ = 300,000 meters
Since the answer is requested in kilometers, we convert meters to kilometers by dividing by 1000:
r₂ = 300,000 meters ÷ 1000
r₂ = 300 kilometers
Therefore, you need to go approximately 300 kilometers away from the airport to reduce the airplane noise to a level below the threshold of conversation.