What is the approximate height of a geostationary satellite from the surface of the earth?

Options

A. 981 km
B. 15000 km
C. 35,000 km
D. 55,000 km

Answer

To find the approximate height of a geostationary satellite from the surface of the Earth, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Concept of Geostationary Satellites**:
- A geostationary satellite orbits the Earth at the same rotational speed as the Earth. This means it takes 24 hours to complete one orbit, which allows it to remain stationary over a specific point on the Earth's surface.

2. **Identify the Time Period (T)**:
- The time period (T) for a geostationary satellite is 24 hours. To use this in calculations, convert it into seconds:
\[ T = 24 \text{ hours} \times 3600 \text{ seconds/hour} = 86400 \text{ seconds} = 8.64 \times 10^4 \text{ seconds} \]

3. **Use the Formula for the Height of the Satellite**:
- The formula to calculate the height (h) of a satellite above the Earth's surface is derived from the gravitational force and centripetal force. The formula is:
\[ T^2 = \frac{4 \pi^2 (r + h)^3}{g r^2} \]
- Rearranging this gives:
\[ h = \left( \frac{T^2 g r^2}{4 \pi^2} \right)^{1/3} - r \]

4. **Substitute Known Values**:
- Here, \( g \) (acceleration due to gravity) is approximately \( 9.8 \, \text{m/s}^2 \).
- The radius of the Earth \( r \) is approximately \( 6.4 \times 10^6 \, \text{m} \).
- Substitute these values into the formula:
\[ h = \left( \frac{(8.64 \times 10^4)^2 \cdot 9.8 \cdot (6.4 \times 10^6)^2}{4 \pi^2} \right)^{1/3} - (6.4 \times 10^6) \]

5. **Calculate the Height**:
- First, calculate \( T^2 \):
\[ T^2 = (8.64 \times 10^4)^2 = 7.45 \times 10^9 \]
- Now, calculate \( g r^2 \):
\[ g r^2 = 9.8 \cdot (6.4 \times 10^6)^2 = 9.8 \cdot 4.096 \times 10^{13} \approx 4.01 \times 10^{14} \]
- Now, substitute these values back into the height formula:
\[ h = \left( \frac{7.45 \times 10^9 \cdot 4.01 \times 10^{14}}{4 \cdot (3.14)^2} \right)^{1/3} - (6.4 \times 10^6) \]
- After calculating, we find:
\[ h \approx 3.6 \times 10^7 \text{ m} \text{ or } 36,000 \text{ km} \]

6. **Conclusion**:
- The approximate height of a geostationary satellite from the surface of the Earth is **36,000 kilometers**.