in the previous question, the change in potential energy.

Options

A. 1. `(GM_(E)m)/(2R_(E))`
B. 2. `(GM_(E)m)/(4R_(E))`
C. 3. `(GM_(E)m)/(8R_(E))`
D. 4. `(GM_(E)m)/(16R_(E))`

Explanation

To find the change in potential energy when a satellite moves from an orbit of radius \(2R_e\) to \(4R_e\), we can follow these steps:

### Step 1: Understand the formula for gravitational potential energy
The gravitational potential energy \(U\) of a satellite in orbit is given by the formula:
\[ U = -\frac{G M m}{r} \]
where:
- \(G\) is the gravitational constant,
- \(M\) is the mass of the Earth,
- \(m\) is the mass of the satellite,
- \(r\) is the distance from the center of the Earth to the satellite.

### Step 2: Calculate potential energy at \(2R_e\)
For the satellite at radius \(2R_e\):
\[ U_{2R_e} = -\frac{G M m}{2R_e} \]

### Step 3: Calculate potential energy at \(4R_e\)
For the satellite at radius \(4R_e\):
\[ U_{4R_e} = -\frac{G M m}{4R_e} \]

### Step 4: Calculate the change in potential energy
The change in potential energy \(\Delta U\) when the satellite moves from \(2R_e\) to \(4R_e\) is given by:
\[ \Delta U = U_{4R_e} - U_{2R_e} \]
Substituting the values we calculated:
\[ \Delta U = \left(-\frac{G M m}{4R_e}\right) - \left(-\frac{G M m}{2R_e}\right) \]

### Step 5: Simplify the expression
This simplifies to:
\[ \Delta U = -\frac{G M m}{4R_e} + \frac{G M m}{2R_e} \]
To combine the terms, we can express \(\frac{G M m}{2R_e}\) with a common denominator:
\[ \Delta U = -\frac{G M m}{4R_e} + \frac{2G M m}{4R_e} = \frac{G M m}{4R_e} \]

### Final Result
Thus, the change in potential energy when the satellite moves from \(2R_e\) to \(4R_e\) is:
\[ \Delta U = \frac{G M m}{4R_e} \]