A spaceship approaches the Moon (mass `= M` and radius `= R` along a parabolic path which is almost tangential to its surface. At the moment of the maximum approach, the brake rocket is fired to convert the spaceship into a satellite of the Moon. Find the change in speed.

Explanation

To solve the problem of finding the change in speed of a spaceship that approaches the Moon and then becomes a satellite, we can follow these steps:

### Step 1: Understand the Initial Conditions
The spaceship is initially moving along a parabolic path, which means it is at the escape velocity from the Moon. The escape velocity \( v_e \) from the Moon is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the Moon, and \( R \) is the radius of the Moon.

### Step 2: Determine the Final Conditions
When the spaceship fires its brake rocket, it converts into a satellite of the Moon. The orbital velocity \( v_o \) for a circular orbit around the Moon is given by:
\[ v_o = \sqrt{\frac{GM}{R}} \]

### Step 3: Calculate the Change in Speed
The change in speed \( \Delta v \) is the difference between the initial speed (escape velocity) and the final speed (orbital velocity):
\[ \Delta v = v_o - v_e \]
Substituting the expressions for \( v_o \) and \( v_e \):
\[ \Delta v = \sqrt{\frac{GM}{R}} - \sqrt{\frac{2GM}{R}} \]

### Step 4: Factor Out Common Terms
We can factor out \( \sqrt{\frac{GM}{R}} \):
\[ \Delta v = \sqrt{\frac{GM}{R}} \left(1 - \sqrt{2}\right) \]

### Step 5: Simplify the Expression
We know that \( \sqrt{2} \approx 1.414 \), so:
\[ 1 - \sqrt{2} \approx 1 - 1.414 = -0.414 \]
Thus, we can express the change in speed as:
\[ \Delta v \approx -0.414 \sqrt{\frac{GM}{R}} \]

### Final Result
The change in speed required to convert the spaceship into a satellite of the Moon is:
\[ \Delta v = \sqrt{\frac{GM}{R}} (1 - \sqrt{2}) \]