To solve the problem, we need to calculate the kinetic energy (K.E.), potential energy (P.E.), and total energy (T.E.) of a satellite orbiting the Earth.
### Given Data:
- Mass of the satellite (m) = 50 kg
- Height of the satellite (h) = 100 km = 100,000 m
- Radius of the Earth (R) = 6400 km = 6,400,000 m
- Gravitational constant (G) = \(6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2\)
- Mass of the Earth (M) = \(6 \times 10^{24} \, \text{kg}\)
### Step 1: Calculate the distance from the center of the Earth to the satellite
The total distance (r) from the center of the Earth to the satellite is given by:
\[
r = R + h = 6,400,000 \, \text{m} + 100,000 \, \text{m} = 6,500,000 \, \text{m}
\]
### Step 2: Calculate the Kinetic Energy (K.E.)
The formula for the kinetic energy of a satellite in orbit is given by:
\[
K.E. = \frac{G \cdot M \cdot m}{2r}
\]
Substituting the values:
\[
K.E. = \frac{(6.67 \times 10^{-11}) \cdot (6 \times 10^{24}) \cdot (50)}{2 \cdot (6,500,000)}
\]
Calculating the above expression:
\[
K.E. = \frac{(6.67 \times 10^{-11}) \cdot (3 \times 10^{26})}{13,000,000}
\]
\[
K.E. = \frac{2.001 \times 10^{16}}{13,000,000} \approx 15.39 \times 10^{8} \, \text{J}
\]
### Step 3: Calculate the Potential Energy (P.E.)
The formula for gravitational potential energy is:
\[
P.E. = -\frac{G \cdot M \cdot m}{r}
\]
Substituting the values:
\[
P.E. = -\frac{(6.67 \times 10^{-11}) \cdot (6 \times 10^{24}) \cdot (50)}{6,500,000}
\]
Calculating the above expression:
\[
P.E. = -\frac{(6.67 \times 10^{-11}) \cdot (3 \times 10^{26})}{6,500,000}
\]
\[
P.E. = -\frac{2.001 \times 10^{16}}{6,500,000} \approx -30.77 \times 10^{8} \, \text{J}
\]
### Step 4: Calculate the Total Energy (T.E.)
The total energy is given by the sum of kinetic and potential energy:
\[
T.E. = K.E. + P.E.
\]
Substituting the values:
\[
T.E. = (15.39 \times 10^{8}) + (-30.77 \times 10^{8}) = -15.38 \times 10^{8} \, \text{J}
\]
### Final Results:
- Kinetic Energy (K.E.) = \(15.39 \times 10^{8} \, \text{J}\)
- Potential Energy (P.E.) = \(-30.77 \times 10^{8} \, \text{J}\)
- Total Energy (T.E.) = \(-15.38 \times 10^{8} \, \text{J}\)
