Given below are two statements :
statement `I`:If `E` be the total energy of a satellite moving around the earth ,then its potential energy will be `E/2`.
statement `II`: The kinetic energy of a satellite revolving in an orbit is equal to be half the magnitude of total energy E.
In the light of the above statement ,choose the most appropriate answer from the options given below

Options

A. Both Statement `I` and statement `II` are incorrect
B. statement `I` is incorrect but statement `II` is correct
C. statement `I` is correct but statement `II` is incorrect
D. Both Statement `I` and statement `II` are correct

Answer

To solve the problem, we need to analyze the two statements regarding the energy of a satellite in orbit around the Earth.

**Step 1: Understand the Energy Relationships for a Satellite**
The total mechanical energy (E) of a satellite in a stable orbit is given by the equation:
\[ E = K + U \]
where \( K \) is the kinetic energy and \( U \) is the potential energy.

**Step 2: Express Potential Energy**
The gravitational potential energy (U) of a satellite at a distance \( r \) from the center of the Earth is given by:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the satellite.

**Step 3: Express Kinetic Energy**
The kinetic energy (K) of the satellite can be derived from the centripetal force requirement for circular motion:
\[ K = \frac{1}{2}mv^2 = \frac{GMm}{2r} \]

**Step 4: Calculate Total Energy**
The total energy (E) can be expressed as:
\[ E = K + U = \frac{GMm}{2r} - \frac{GMm}{r} \]
\[ E = \frac{GMm}{2r} - \frac{2GMm}{2r} = -\frac{GMm}{2r} \]

**Step 5: Relate Kinetic and Potential Energy**
From the total energy expression, we can see that:
\[ U = 2E \]
Thus, the potential energy is:
\[ U = -\frac{GMm}{r} = 2 \left(-\frac{GMm}{2r}\right) = 2E \]

**Step 6: Analyze the Statements**
- **Statement I**: "If E be the total energy of a satellite moving around the earth, then its potential energy will be E/2." This is incorrect because the potential energy is actually \( 2E \).

- **Statement II**: "The kinetic energy of a satellite revolving in an orbit is equal to half the magnitude of total energy E." This is also incorrect because the kinetic energy is actually equal to \( -\frac{1}{2}E \) (which is not the same as half the magnitude of total energy).

**Conclusion:**
Both statements are incorrect.

**Final Answer:**
Both statements are incorrect.

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