How far from the center of the earth must a person be to experience acceleration due to gravity that is one half the acceleration at the surface?
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To find the distance from the center of the Earth where a person would experience acceleration due to gravity that is one half the acceleration at the surface, we can use the concept of exponential decay.
The acceleration due to gravity follows an inverse-square law, which means it decreases as the square of the distance from the center of the Earth. From this, we can say that the acceleration (a) at a certain distance (r) from the center of the Earth is proportional to 1/r^2.
Let's denote the acceleration at the Earth's surface as g (9.8 m/s^2). We want to find the distance where the acceleration is half of g.
Using the inverse-square law, we can set up the following equation:
g / (r^2) = (1/2)g
Here, g cancels out:
1 / (r^2) = 1/2
To solve for r, we can take the reciprocal of both sides of the equation:
r^2 = 2
Taking the square root of both sides gives:
r ≈ √2
So, the distance from the center of the Earth where a person would experience half the acceleration due to gravity is approximately equal to the square root of 2 times the radius of the Earth.
To determine the exact value, we need to know the radius of the Earth. The mean radius of the Earth is about 6,371 kilometers (or 6,371,000 meters). Therefore, the distance is approximately:
r ≈ √2 × 6,371,000 ≈ 8,993,000 meters
Hence, a person must be approximately 8,993 kilometers from the center of the Earth to experience acceleration due to gravity that is one half the acceleration at the surface.