To launch a spaceship from Earth's surface into free space, we need to provide energy equal to the work done against Earth's gravitational force. This energy must overcome the gravitational potential energy binding the spaceship to Earth.

The gravitational potential energy (U) at a distance r from Earth's center is given by:

where G is the gravitational constant, M is Earth's mass, m is the spaceship's mass, and r is the distance from the center.

At Earth's surface (r = R), the potential energy is:

In free space, far away from Earth (r → ∞), the potential energy approaches zero:

The energy required is the difference in potential energy between free space and the surface:

We know that the acceleration due to gravity at Earth's surface, g, is given by:

Therefore,

Substituting this into the energy equation:

Now, plug in the given values:

m = 1000 kg, g = 10 m/s², R = 6400 km = 6,400,000 m = 6.4 × 10⁶ m

So,

Calculate step by step:

10 × 1000 = 10,000 = 10⁴

10⁴ × 6.4 × 10⁶ = 6.4 × 10¹⁰

Therefore, the required energy is 6.4 × 10¹⁰ Joules.

Related Topics

Gravitational Potential Energy: The energy an object possesses due to its position in a gravitational field. For Earth, it is negative at the surface and zero at infinity.

Escape Velocity: The minimum speed needed for an object to escape a planet's gravitational pull without further propulsion. The energy we calculated is related to the kinetic energy needed for escape velocity.

Formulae

Gravitational Potential Energy at distance r: $U=-\frac{GMm}{r}$

Relation between g, G, M, and R: $g=\frac{GM}{R^2}$

Energy to escape to infinity: $E=\frac{GMm}{R}=mgR$