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 determine the total energy of a satellite in orbit, we can follow these steps:

### Step 1: Understand the Components of Total Energy
The total energy (E) of a satellite in orbit is the sum of its kinetic energy (K) and potential energy (U).

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

### Step 3: Determine the Orbital Velocity
The orbital velocity \( v \) of the satellite can be derived from the gravitational force acting on it. The formula for 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 4: Substitute the Velocity into Kinetic Energy
Substituting the expression for \( v \) into the kinetic energy formula gives:
\[ K = \frac{1}{2} m \left(\sqrt{\frac{GM}{r}}\right)^2 \]
\[ K = \frac{1}{2} m \frac{GM}{r} \]
\[ K = \frac{GMm}{2r} \]

### Step 5: Calculate the Potential Energy
The gravitational potential energy (U) of the satellite is given by:
\[ U = -\frac{GMm}{r} \]

### Step 6: Find the Total Energy
Now, we can find the total energy (E) by adding kinetic and potential energy:
\[ E = K + U \]
\[ E = \frac{GMm}{2r} - \frac{GMm}{r} \]
\[ E = \frac{GMm}{2r} - \frac{2GMm}{2r} \]
\[ E = -\frac{GMm}{2r} \]

### Step 7: Analyze the Result
The total energy \( E \) is negative, which indicates that the satellite is in a bound state in its orbit.

### Conclusion
Thus, the total energy of a satellite in orbit is always negative.

### Final Answer
The total energy of a satellite is always negative.

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