Gravity Assist

Gravity assist – How to increase the speed of a spacecraft using the gravity of the moon or Celestial Body

Gravity assist is the use of the relative movement (e.g. orbit around the Sun) and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft. ‘Gravity Assist’ is also called ‘gravitational slingshot’ or ‘swing-by’. All the same.
Using this gravity assist to increase the speed of a spacecraft is economical because you do not have to use the spacecraft’s own energy.
Scientists believe that using gravity assist can make space travel possible with less energy.

In the movie “Armageddon,” there is a scene where spacecraft turns around the moon to get enough speed.

If the moon is stopped and the spacecraft approaches with constant velocity v,

The spacecraft will proceed to an elliptical orbit that focuses on the moon.
There is no change in the speed of the spacecraft that went around the moon.
If you look at this from afar, a tiny spacecraft appears to have a perfectly elastic collision with the moon’s surface.

If the moon moves with a constant speed V,

The spacecraft gets close to the moon.
From the moon’s perspective, the ship will appear to approach at the spacecraft of V + v, and the speed of the returning ship will also appear as V + v.
In addition, when observing from the outside, the moon’s own speed V must be added, so the speed of the spacecraft is 2V + v.