Understanding Acceleration Due to Gravity
The acceleration due to gravity (g) at the Earth's surface is approximately 9.81 m/s². As you move away from the Earth's surface, this value decreases. To find the height at which g reduces by 10%, we need to understand the relationship between height and gravity.
Formula for Acceleration Due to Gravity
- The formula for gravitational acceleration at a height (h) is:
g' = g * (R / (R + h))²
- Where:
- g' = acceleration due to gravity at height h
- g = acceleration due to gravity at the surface (9.81 m/s²)
- R = radius of the Earth (approximately 6,371 km)
Calculating 10% Reduction
- A 10% reduction in g means:
g' = 0.9 * g
- Using the formula:
0.9 * g = g * (R / (R + h))²
- Simplifying gives:
0.9 = (R / (R + h))²
- Taking the square root:
√0.9 = R / (R + h)
- Rearranging this equation:
R + h = R / √0.9
- Thus, h = (R / √0.9) - R
Plugging in Values
- Substitute R = 6,371 km:
- Calculating yields:
h ≈ 0.54 * R
- Therefore, h ≈ 0.54 * 6,371 km ≈ 3,440 km
Conclusion
The height at which the acceleration due to gravity reduces by 10% compared to its surface value is approximately 3,440 km above the Earth's surface. This illustrates how gravity weakens with distance from the Earth.