Given,
Mass of spaceship (m) = 1000 kg
Acceleration due to gravity (g) = 10 m/s²
Radius of Earth (R) = 6400 km
We need to find out the required energy to launch the spaceship from Earth's surface to free space.

Potential Energy Calculation
When we lift an object against the gravitational force, the work done is stored as potential energy. The potential energy of an object at a height 'h' above the surface of the Earth is given by:
PE = mgh
where m is the mass of the object, g is the acceleration due to gravity and h is the height.

- Calculate the height attained by the spaceship
The spaceship is launched from the Earth's surface to free space. Therefore, the height attained by the spaceship is equal to the distance between the Earth's surface and the edge of the atmosphere.

- Calculate the radius of the atmosphere
The edge of the atmosphere is not clearly defined. However, the Kármán line is considered to be the boundary between the Earth's atmosphere and outer space. The Kármán line is located at a height of 100 km above the Earth's surface.

- Calculate the height of the spaceship
Height of the spaceship = Height of Kármán line - Radius of the Earth
Height of the spaceship = 100 km - 6400 km
Height of the spaceship = -6300 km (negative sign indicates that the spaceship is below the Earth's surface)

- Calculate the potential energy of the spaceship
PE = mgh
PE = 1000 kg x 10 m/s² x (-6300000 m)
PE = -6.3 x 10¹⁰ J (negative sign indicates that the potential energy is negative since the spaceship is below the Earth's surface)

Kinetic Energy Calculation
The kinetic energy of an object is given by:
KE = ½mv²
where m is the mass of the object and v is its velocity.

- Calculate the escape velocity of the spaceship
The escape velocity is the minimum velocity required to escape the gravitational field of the Earth. It is given by:
Vesc = √(2GM/R)
where G is the gravitational constant, M is the mass of the Earth and R is the distance between the spaceship and the center of the Earth.

- Calculate the distance between the spaceship and the center of the Earth
The distance between the spaceship and the center of the Earth is equal to the sum of the radius of the Earth and the height of the spaceship.

Distance between spaceship and center of Earth = Radius of Earth + Height of spaceship
Distance between spaceship and center of Earth = 6400 km - 6300 km
Distance between spaceship and center of Earth = 100 km

- Calculate the escape velocity of the spaceship
Vesc = √(2GM/R)
Vesc = √(2 x 6.67 x 10⁻¹¹ Nm²/kg² x 5.97 x 10²⁴ kg / 6.38 x 10⁶ m)
Vesc = 11.2 km/s

- Calculate the kinetic energy of the spaceship
KE = ½mv²
KE = ½ x 1000 kg x (11.2 km/s)²
KE = 6.2 x 10¹⁰ J

Total Energy Calculation
The total energy required