Find out energy required (in Giga Joule) to escape a space shuttle of 1000 kg mass from surface of earth. `(R_(e)=6400 km)`

Options

A. 20
B. 64
C. 50
D. 28

Answer

To find the energy required for a space shuttle of mass 1000 kg to escape from the surface of the Earth, we will use the concept of gravitational potential energy. The formula for gravitational potential energy (U) when an object is at a height (h) from the surface of the Earth is given by:

\[ U = m \cdot g \cdot h \]

Where:
- \( m \) = mass of the object (in kg)
- \( g \) = acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))
- \( h \) = height from the surface of the Earth (in meters)

Given:
- Mass of the space shuttle, \( m = 1000 \, \text{kg} \)
- Radius of the Earth, \( R_e = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

### Step 1: Convert the radius of the Earth to meters
\[ R_e = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m} \]

### Step 2: Substitute the values into the potential energy formula
\[ U = m \cdot g \cdot h \]
\[ U = 1000 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 6.4 \times 10^6 \, \text{m} \]

### Step 3: Calculate the potential energy
\[ U = 1000 \cdot 9.8 \cdot 6.4 \times 10^6 \]
\[ U = 62720000 \, \text{J} \]

### Step 4: Convert Joules to GigaJoules
Since \( 1 \, \text{GJ} = 10^9 \, \text{J} \):
\[ U = \frac{62720000 \, \text{J}}{10^9} = 0.06272 \, \text{GJ} \]

### Step 5: Final conversion to GigaJoules
To express this in GigaJoules:
\[ U = 62.72 \, \text{GJ} \]

### Conclusion
The energy required for the space shuttle to escape from the surface of the Earth is approximately:
\[ U \approx 62.72 \, \text{GJ} \]