The total energy of a satellite is

Options

A. always positive
B. always negative
C. always zero
D. positive or negative depending upon radius of orbit

Answer

To find the total energy of a satellite revolving around the Earth, we need to calculate both its kinetic energy (KE) and potential energy (PE). Here are the steps to derive the total energy:

### Step 1: Calculate the Kinetic Energy (KE)
The kinetic energy of the satellite can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the satellite and \( v \) is its orbital velocity.

### Step 2: Determine the Orbital Velocity (v)
The orbital velocity of a satellite in a circular orbit can be derived from the gravitational force acting as the centripetal force. The formula for the orbital velocity is:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth to the satellite.

### Step 3: Substitute the Velocity into the Kinetic Energy Formula
Now, substituting the expression for \( v \) into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\sqrt{\frac{GM}{r}}\right)^2 \]
This simplifies to:
\[ KE = \frac{1}{2} m \frac{GM}{r} = \frac{GMm}{2r} \]

### Step 4: Calculate the Potential Energy (PE)
The gravitational potential energy of the satellite in orbit is given by:
\[ PE = -\frac{GMm}{r} \]
where the negative sign indicates that the gravitational force is attractive.

### Step 5: Calculate the Total Energy (E)
The total energy of the satellite is the sum of its kinetic and potential energy:
\[ E = KE + PE \]
Substituting the expressions for KE and PE:
\[ E = \frac{GMm}{2r} - \frac{GMm}{r} \]
Now, simplifying this expression:
\[ E = \frac{GMm}{2r} - \frac{2GMm}{2r} = -\frac{GMm}{2r} \]

### Conclusion
The total energy of the satellite is:
\[ E = -\frac{GMm}{2r} \]
This result indicates that the total energy of the satellite is negative, which implies that the satellite is in a bound (stable) orbit.