The time period of a geostationary satellite at a height 36000 km is 24 h a spy satellite orbits vergy close to earth surface (R=6400 km ) what will be its time period ?

Options

A. 4h
B. 1h
C. 2h
D. 1.5 h

Explanation

To find the time period of a spy satellite orbiting very close to the Earth's surface, we can use Kepler's third law of planetary motion, which states that the square of the time period (T) of a satellite is directly proportional to the cube of the semi-major axis (r) of its orbit.

### Step-by-Step Solution:

1. **Understanding Given Data:**
- Time period of geostationary satellite, \( T_1 = 24 \) hours.
- Height of geostationary satellite, \( h_1 = 36000 \) km.
- Radius of the Earth, \( R = 6400 \) km.
- For the spy satellite, which is very close to the Earth's surface, we can assume its height \( h_2 \approx 0 \) km.

2. **Calculate the Distance from the Center of the Earth:**
- For the geostationary satellite:
\[ r_1 = R + h_1 = 6400 \, \text{km} + 36000 \, \text{km} = 42400 \, \text{km} \]
- For the spy satellite:
\[ r_2 = R + h_2 = 6400 \, \text{km} + 0 \, \text{km} = 6400 \, \text{km} \]

3. **Applying Kepler's Third Law:**
According to Kepler's third law:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]
Rearranging gives:
\[ \frac{T_1}{T_2} = \sqrt{\frac{r_1^3}{r_2^3}} \]

4. **Substituting Values:**
Substitute \( T_1 = 24 \) hours, \( r_1 = 42400 \) km, and \( r_2 = 6400 \) km:
\[ \frac{24}{T_2} = \sqrt{\frac{(42400)^3}{(6400)^3}} \]

5. **Calculating the Ratio:**
Simplifying the right side:
\[ \frac{24}{T_2} = \sqrt{\left(\frac{42400}{6400}\right)^3} \]
Calculate \( \frac{42400}{6400} = 6.625 \):
\[ \frac{24}{T_2} = \sqrt{(6.625)^3} \]
Calculate \( (6.625)^3 \approx 291.6 \):
\[ \frac{24}{T_2} = \sqrt{291.6} \approx 17.05 \]

6. **Cross-Multiplying to Solve for \( T_2 \):**
\[ 24 = 17.05 \cdot T_2 \]
\[ T_2 = \frac{24}{17.05} \approx 1.407 \, \text{hours} \]

7. **Final Result:**
The time period of the spy satellite is approximately:
\[ T_2 \approx 1.407 \, \text{hours} \approx 1.5 \, \text{hours} \text{ (rounded)} \]