Using Newton's law of gravitation, the acceleration due to gravity at a certain distance from the center of the Earth can be calculated using the formula:
\[g = \frac{G \cdot M}{r^2}\]
Where:
- \(g\) is the acceleration due to gravity,
- \(G\) is the gravitational constant (\(6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2\)),
- \(M\) is the mass of the Earth (\(6 \times 10^{24} \, \text{kg}\)),
- \(r\) is the distance from the center of the Earth.
We want to find the height where the acceleration due to gravity becomes \(4 \, \text{m/s}^2\), so we can rearrange the formula to solve for \(r\):
\[r = \sqrt{\frac{G \cdot M}{g}}\]
Now we plug in the given values:
- \(M = 6 \times 10^{24} \, \text{kg}\),
- \(g = 4 \, \text{m/s}^2\),
- \(G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2\).
Calculate:
\[r = \sqrt{\frac{(6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2) \cdot (6 \times 10^{24} \, \text{kg})}{4 \, \text{m/s}^2}\]
The result of this calculation will be the distance from the center of the Earth to the height where the acceleration due to gravity is \(4 \, \text{m/s}^2\). However, keep in mind that this value will be in meters. If you want the answer in kilometers, you should convert the result from meters to kilometers by dividing by \(1000\) since \(1 \, \text{km} = 1000 \, \text{m}\).