How much energy does it take to
send a spaceship far from the Earth?
Let's find out.
We'll use a simple model for a spaceship:
a big bullet of mass **M = 1 kg **
shot out of a big gun, like this one:

Illustration from Jules Verne's book *From the Earth to the Moon. *

The bullet starts at the Earth's surface, a distanceR1 = 6.37 x 10meters from the center of the Earth, and then flies to a final distance of^{6}R2 = 2 x 6.37 x 10meters, where it stops momentarily.^{6}

- How much gravitational potential energy has it gained?
- What was the bullet's kinetic energy when it left the gun?
- What was the bullet's speed when it left the gun?

The bullet starts at the Earth's surface, a distanceR1 = 6.37 x 10meters from the center of the Earth, and then flies to a final distance of^{6}R2 = 11 x 6.37 x 10meters, where it stops momentarily.^{6}

- How much gravitational potential energy has it gained?
- What was the bullet's kinetic energy when it left the gun?
- What was the bullet's speed when it left the gun?

The bullet starts at the Earth's surface, a distanceR1 = 6.37 x 10meters from the center of the Earth, and then flies to a final distance of^{6}R2 = 101 x 6.37 x 10meters, where it stops momentarily.^{6}

- How much gravitational potential energy has it gained?
- What was the bullet's kinetic energy when it left the gun?
- What was the bullet's speed when it left the gun?

(Yes, we are ignoring the effect of the Moon's gravitational force ...)

The bullet starts at the Earth's surface, a distanceR1 = 6.37 x 10meters from the center of the Earth, and then flies to a final distance of^{6}R2 = 1,001 x 6.37 x 10meters, where it stops momentarily.^{6}

- How much gravitational potential energy has it gained?
- What was the bullet's kinetic energy when it left the gun?
- What was the bullet's speed when it left the gun?

The bullet starts at the Earth's surface, a distanceR1 = 6.37 x 10meters from the center of the Earth, and then flies to a final distance of^{6}R2 = 10,001 x 6.37 x 10meters, where it stops momentarily.^{6}

- How much gravitational potential energy has it gained?
- What was the bullet's kinetic energy when it left the gun?
- What was the bullet's speed when it left the gun?

(Yes, we are ignoring the effect of the Sun's gravitational force, and that of the other planets ....)

The bullet starts at the Earth's surface, a distanceR1 = 6.37 x 10meters from the center of the Earth, and then flies to a final distance of^{6}R2 = 100,001 x 6.37 x 10meters, where it stops momentarily.^{6}

- How much gravitational potential energy has it gained?
- What was the bullet's kinetic energy when it left the gun?
- What was the bullet's speed when it left the gun?

(Yes, we are ignoring the effect of the Sun's gravitational force, and that of the other planets ....)

The bullet starts at the Earth's surface, a distanceR1 = 6.37 x 10meters from the center of the Earth, and then flies to a final distance of^{6}R2 = 1,000,001 x 6.37 x 10meters, where it stops momentarily.^{6}

- How much gravitational potential energy has it gained?
- What was the bullet's kinetic energy when it left the gun?
- What was the bullet's speed when it left the gun?

(Yes, we are ignoring the effect of the Sun's gravitational force, and that of the other planets ....)

R1 (km) R2 (km) gained GPE (J) initial speed (m/s) ---------------------------------------------------------------------- 6.37x10^6 2 x 6.37x10^6 31.3 million 7,913 11 x 6.37x10^6 56.9 million 10,670 101 x 6.37x10^6 62.0 million 11,135 1,001 x 6.37x10^6 62.55 million 11,185 10,001 x 6.37x10^6 62.61 million 11,190 100,001 x 6.37x10^6 62.616 million 11,190.7 1,000,001 x 6.37x10^6 62.61626 million 11,190.73 ----------------------------------------------------------------------

There's a clear pattern: at some point,
the bullet gains a VERY large
extra distance for just a small increase in initial
velocity.
Suppose that we make the final distance
**R2 = infinity**.
We can use the conservation of energy
to find the speed required to keep the bullet
going forever:

If we ask the bullet to reach a final distance
**R _{f} = ∞**
with a final speed

That allows us to define a special
initial speed, the
**escape velocity from the Earth**:

We can finish our table now ....

R1 (km) R2 (km) gained GPE (J) initial speed (m/s) ---------------------------------------------------------------------- 100,001 x 6.37x10^6 62.616 million 11,190.7 1,000,001 x 6.37x10^6 62.61626 million 11,190.73 infinity 62.61633 million 11,190.74 ----------------------------------------------------------------------

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.