E, U and K represent total mechanical energy. potential energy and kinetic energy respectively of a satellite revolving around a planet, then which of the following is correct?

Options

A. K = E
B. U= 2E
C. `K= abs(E)`
D. Both (2) and (3)

Answer

To solve the problem, we need to analyze the relationships between the total mechanical energy (E), potential energy (U), and kinetic energy (K) of a satellite revolving around a planet. Let's break it down step by step.

### Step 1: Define the energies involved
1. **Kinetic Energy (K)** of the satellite:
\[ K = \frac{GMm}{2(R + h)} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the satellite, \( R \) is the radius of the planet, and \( h \) is the height of the satellite above the planet's surface.

2. **Potential Energy (U)** of the satellite:
\[ U = -\frac{GMm}{R + h} \]

3. **Total Mechanical Energy (E)** of the satellite:
\[ E = K + U \]
Substituting the expressions for \( K \) and \( U \):
\[ E = \frac{GMm}{2(R + h)} - \frac{GMm}{R + h} \]
Simplifying this:
\[ E = -\frac{GMm}{2(R + h)} \]

### Step 2: Analyze the relationships
Now we can analyze the relationships between these energies:

1. **Relationship between K and E**:
From the expression for total energy \( E \):
\[ E = -\frac{GMm}{2(R + h)} \]
We can see that:
\[ K = -2E \]
This implies that:
\[ K = |E| \quad \text{(since E is negative, we take the absolute value)} \]

2. **Relationship between U and E**:
We also have:
\[ U = -\frac{GMm}{R + h} \]
And since \( E = -\frac{GMm}{2(R + h)} \), we can express \( U \) in terms of \( E \):
\[ U = 2E \]

### Step 3: Conclusion
Based on the relationships derived:
- Kinetic energy \( K \) is equal to the absolute value of total energy \( |E| \).
- Potential energy \( U \) is equal to \( 2E \).

### Final Answer
Thus, the correct options are:
- Potential energy is 2 times the total energy: \( U = 2E \)
- Kinetic energy is equal to the absolute value of total energy: \( K = |E| \)

Therefore, the answer is option 4: both statements 2 and 3 are correct.

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