To find the required energy for launching the spaceship from the Earth's surface into free space, we can use the concept of gravitational potential energy. The work done to move the spaceship from the Earth's surface to a point at infinity can be calculated as follows:
### Step-by-Step Solution:
1. **Identify the given values:**
- Mass of the spaceship, \( m = 1000 \, \text{kg} \)
- Gravitational acceleration at the Earth's surface, \( g = 10 \, \text{m/s}^2 \)
- Radius of the Earth, \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m} \)
2. **Understand the potential energy change:**
- The gravitational potential energy \( U \) at a distance \( r \) from the center of the Earth is given by:
\[
U = -\frac{GMm}{r}
\]
- At the Earth's surface, \( r = R \), and at infinity, \( r \to \infty \), where the potential energy \( U \) approaches 0.
3. **Calculate the gravitational potential energy at the surface:**
- The gravitational potential energy at the Earth's surface is:
\[
U_{\text{initial}} = -\frac{GMm}{R}
\]
- Here, \( G \) (the gravitational constant) can be related to \( g \) as:
\[
g = \frac{GM}{R^2} \implies GM = gR^2
\]
4. **Substituting the values:**
- Substitute \( GM \) into the potential energy equation:
\[
U_{\text{initial}} = -\frac{gR^2 m}{R} = -gRm
\]
5. **Calculate the work done (energy required):**
- The work done by the external agent to move the spaceship to infinity is:
\[
W = U_{\text{final}} - U_{\text{initial}} = 0 - (-gRm) = gRm
\]
- Now substitute the values:
\[
W = 10 \, \text{m/s}^2 \times 6.4 \times 10^6 \, \text{m} \times 1000 \, \text{kg}
\]
6. **Perform the multiplication:**
- Calculate \( W \):
\[
W = 10 \times 6.4 \times 10^6 \times 1000 = 64 \times 10^6 \, \text{J} = 6.4 \times 10^7 \, \text{J}
\]
### Final Answer:
The required energy for launching the spaceship into free space is \( 6.4 \times 10^7 \, \text{J} \).
