A spaceship is returning to Earth with its engine turned off. Consider only the gravitational field of Earth and let M be the mass of Earth, m be the mass of the spaceship, and R be the radius of Earth. In moving from position 1 to position 2 the kinetic energy of the spaceship increases by:

Options

A. `GMm//R_2`
B. `GMm//R_2^2`
C. `GMm(R_1-R_2)//R_1^2`
D. `GMm(R_1-R_2)//R_1R_2`

Explanation

To solve the problem of how the kinetic energy of a spaceship increases as it moves from position 1 to position 2 while returning to Earth, we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Understand the Conservation of Energy
The total mechanical energy (kinetic energy + potential energy) of the spaceship is conserved because gravitational force is a conservative force. Therefore, we can equate the total energy at position 1 and position 2.

### Step 2: Write the Energy Equations
Let:
- \( KE_1 \) = Kinetic energy at position 1
- \( PE_1 \) = Potential energy at position 1
- \( KE_2 \) = Kinetic energy at position 2
- \( PE_2 \) = Potential energy at position 2

According to the conservation of energy:
\[ KE_1 + PE_1 = KE_2 + PE_2 \]

### Step 3: Define the Potential Energy
The gravitational potential energy \( PE \) between two masses \( M \) (mass of Earth) and \( m \) (mass of the spaceship) at a distance \( r \) from the center of the Earth is given by:
\[ PE = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant.

### Step 4: Substitute Potential Energies
At position 1 (distance \( r_1 \) from the center of the Earth):
\[ PE_1 = -\frac{GMm}{r_1} \]

At position 2 (distance \( r_2 \) from the center of the Earth):
\[ PE_2 = -\frac{GMm}{r_2} \]

### Step 5: Substitute into the Energy Equation
Substituting the potential energies into the conservation of energy equation:
\[ KE_1 - \frac{GMm}{r_1} = KE_2 - \frac{GMm}{r_2} \]

### Step 6: Rearrange to Find Change in Kinetic Energy
Rearranging the equation to find the change in kinetic energy:
\[ KE_2 - KE_1 = \frac{GMm}{r_1} - \frac{GMm}{r_2} \]

### Step 7: Factor out Common Terms
Factoring out \( GMm \):
\[ KE_2 - KE_1 = GMm \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]

### Step 8: Final Expression for Increase in Kinetic Energy
Thus, the increase in kinetic energy as the spaceship moves from position 1 to position 2 is:
\[ \Delta KE = KE_2 - KE_1 = GMm \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]