Imagine that you are in a spacecraft orbiting around the earth in a circle of radius `7000 km` (from the centre of the earth). If you decrease the magnitude of mechanical energy of the spacecraft `-` earth system by `10%` by firing the rockets, then what is the greatest height you can take your spacecraft above the surface of the earth? [`R_(e) = 6400 km`]

Options

A. (a)`6400 km`
B. (b)`540km`
C. (c)`2140 km`
D. (d)`3000 km`

Explanation

To solve the problem, we need to determine the greatest height a spacecraft can reach above the Earth's surface after decreasing its mechanical energy by 10%.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of Earth, \( R_e = 6400 \, \text{km} \)
- Radius of orbit, \( r = 7000 \, \text{km} \)
- Decrease in mechanical energy = 10%

2. **Calculate the Initial Mechanical Energy:**
The mechanical energy \( E \) of a spacecraft in orbit is given by:
\[ E = -\frac{G M m}{2r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the spacecraft.

3. **Calculate the New Mechanical Energy:**
If the mechanical energy is decreased by 10%, the new energy \( E' \) is:
\[ E' = E - 0.1E = 0.9E \]
Substituting the expression for \( E \):
\[ E' = 0.9 \left(-\frac{G M m}{2r}\right) = -\frac{0.9 G M m}{2r} \]

4. **Set Up the Equation for the New Orbit:**
The new mechanical energy in a different orbit (with radius \( r' \)) can be expressed as:
\[ E' = -\frac{G M m}{2r'} \]
Setting the two expressions for \( E' \) equal gives:
\[ -\frac{0.9 G M m}{2r} = -\frac{G M m}{2r'} \]

5. **Simplify the Equation:**
Canceling out common terms:
\[ \frac{0.9}{r} = \frac{1}{r'} \]
Rearranging gives:
\[ r' = \frac{r}{0.9} = \frac{7000 \, \text{km}}{0.9} \approx 7777.78 \, \text{km} \]

6. **Calculate the Greatest Height Above the Earth's Surface:**
The height \( h \) above the Earth's surface is given by:
\[ h = r' - R_e \]
Substituting the values:
\[ h = 7777.78 \, \text{km} - 6400 \, \text{km} \approx 1377.78 \, \text{km} \]

### Final Answer:
The greatest height the spacecraft can reach above the surface of the Earth is approximately **1377.78 km**.