Answer: B) \( 6.4 \times 10^{10} \, \text{J} \)
Step 1: Escape energy for launching the spaceship
The energy required to launch an object of mass \( m \) from the earth's surface into free space is equal to the work done against gravitational potential energy:
\[
E = \frac{GMm}{R}
\]
We can relate this to the acceleration due to gravity \( g \) using the relation:
\[
GM = gR^2
\]
Thus, the required energy becomes:
\[
E = \frac{g m R}{1}
\]
Step 2: Substituting the given values
Given:
- Mass of the spaceship \( m = 1000 \, \text{kg} \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)
- Radius of earth \( R = 6400 \, \text{km} = 6400 \times 1000 = 6.4 \times 10^6 \, \text{m} \)
Substitute these values into the equation \( E = g m R \):
\[
E = 10 \times 1000 \times 6.4 \times 10^6
\]
Multiply the terms:
\[
E = 6.4 \times 10^{10} \, \text{J}
\]
Step 3: Final Answer
The required energy to launch the spaceship is:
\[
\boxed{6.4 \times 10^{10} \, \text{J}}
\]
Correct Option: B) \( 6.4 \times 10^{10} \, \text{J} \)
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