Height of geostationary satellite is

Options

A. 1000 km
B. 32000 km
C. 36000 km
D. 850 km

Explanation

To find the height of a geostationary satellite, we can follow these steps:

### Step 1: Understand the concept of a geostationary satellite
A geostationary satellite is one that orbits the Earth at a height where its orbital period matches the Earth's rotation period, which is approximately 24 hours.

### Step 2: Use the formula for the orbital period
The orbital period \( T \) of a satellite is given by the formula:
\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]
where:
- \( T \) is the orbital period,
- \( r \) is the distance from the center of the Earth to the satellite,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth.

### Step 3: Set the orbital period for a geostationary satellite
For a geostationary satellite, the orbital period \( T \) is 24 hours, which we convert to seconds:
\[ T = 24 \times 60 \times 60 = 86400 \text{ seconds} \]

### Step 4: Rearrange the formula to find \( r \)
We can rearrange the formula to solve for \( r \):
\[ T^2 = 4\pi^2 \frac{r^3}{GM} \]
\[ r^3 = \frac{T^2 GM}{4\pi^2} \]
\[ r = \left(\frac{T^2 GM}{4\pi^2}\right)^{1/3} \]

### Step 5: Substitute known values
We know:
- \( G \approx 6.674 \times 10^{-11} \, \text{m}^3/\text{kg s}^2 \)
- \( M \approx 5.972 \times 10^{24} \, \text{kg} \)

Substituting these values into the equation:
\[ r = \left(\frac{(86400)^2 \times (6.674 \times 10^{-11}) \times (5.972 \times 10^{24})}{4\pi^2}\right)^{1/3} \]

### Step 6: Calculate \( r \)
Calculating the above expression gives us the value of \( r \).

### Step 7: Calculate the height \( h \)
The height \( h \) of the satellite above the Earth's surface is given by:
\[ h = r - R \]
where \( R \) is the radius of the Earth (approximately \( 6400 \, \text{km} \)).

### Step 8: Final calculation
After performing the calculations, we find that the height \( h \) of a geostationary satellite is approximately \( 36,000 \, \text{km} \).

### Conclusion
Thus, the height of a geostationary satellite is approximately **36,000 kilometers** above the Earth's surface.

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