To solve the problem step by step, we will calculate the initial velocity of the satellite and its time period of rotation.
### Step 1: Understand the Parameters
- Height (h) of the satellite above the Earth's surface = 150 km = 150,000 m
- Radius of the Earth (R) = 6400 km = 6,400,000 m
- Mass of the Earth (M) = \(6 \times 10^{24}\) kg
- Gravitational constant (G) = \(6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)
### Step 2: Calculate the Distance from the Center of the Earth
The total distance (R + h) from the center of the Earth to the satellite is:
\[
R + h = 6,400,000 \, \text{m} + 150,000 \, \text{m} = 6,550,000 \, \text{m}
\]
### Step 3: Use the Formula for Orbital Velocity
The formula for the orbital velocity \(V_0\) of a satellite in a circular orbit is given by:
\[
V_0 = \sqrt{\frac{GM}{R + h}}
\]
Substituting the known values:
\[
V_0 = \sqrt{\frac{(6.67 \times 10^{-11}) \times (6 \times 10^{24})}{6,550,000}}
\]
### Step 4: Calculate the Orbital Velocity
Calculating the above expression:
1. Calculate \(GM\):
\[
GM = 6.67 \times 10^{-11} \times 6 \times 10^{24} = 4.002 \times 10^{14} \, \text{m}^3/\text{s}^2
\]
2. Now substitute this into the velocity formula:
\[
V_0 = \sqrt{\frac{4.002 \times 10^{14}}{6,550,000}} \approx \sqrt{6.11 \times 10^{7}} \approx 7.81 \times 10^{3} \, \text{m/s}
\]
### Step 5: Calculate the Time Period of Rotation
The time period \(T\) of the satellite can be calculated using the formula:
\[
T = \frac{2\pi(R + h)}{V_0}
\]
Substituting the values:
\[
T = \frac{2\pi \times 6,550,000}{7.81 \times 10^{3}}
\]
### Step 6: Calculate the Time Period
Calculating the above expression:
1. Calculate the circumference:
\[
2\pi \times 6,550,000 \approx 41,116,000 \, \text{m}
\]
2. Now substitute this into the time period formula:
\[
T \approx \frac{41,116,000}{7.81 \times 10^{3}} \approx 5,256 \, \text{s}
\]
### Step 7: Convert Time Period to Minutes
To convert seconds into minutes:
\[
T \approx \frac{5256}{60} \approx 87.6 \, \text{minutes}
\]
This can be expressed as:
\[
1 \, \text{hour} + 27 \, \text{minutes}
\]
### Final Answer
- Initial velocity \(V_0 \approx 7.81 \times 10^{3} \, \text{m/s}\)
- Time period \(T \approx 1 \, \text{hour} \, 27 \, \text{minutes}\)
