... in 2-D.
Yeah, orbits always follow a 2D plane. Why can't they be represented in 2D?
We are discussing an alleged gravity assist
kick of a small space craft at a certain speed/direction that encounters at small distance a big planet at another speed/direction.
It is suggested that, if the encounter is almost head on (missing at a certain distance) in the same plane (a 2-D encounter - speed vectors are in same plane), the space craft with speed original v is kicked around 180° by the planet that comes in the opposite direction with speed U. The space craft then has speed 2U+v in the opposite direction, i.e. it has turned 180° while remaining of course in the 2-D plane.
However, space is 3-D and it is very difficult to arrange an encounter between a space craft travelling at a certain speed/direction(orbit) in one plane and a planet travelling at another speed/direction (orbit) in another plane.
But if it is possible (which I doubt) it is suggested that the big planet will kick the small space craft in another direction at, I assume, an increased speed.
I have only tested the proposed math solution in 2-D with planet speed U = 0. Then the
kick is a 180° turn of the space craft at constant speed v and constant altitude above the planet. No increase of speed. The
kick is just a 180° turn around the planet.
Evidently it is not possible. The gravity force of the planet will attract the space craft, so the altitude cannot be constant.
NASA has suggested that it is easy to demonstrate above by sending a space craft towards the Moon and that the space craft with a certain arrival speed v at the Moon will do a 180° turn around the Moon and then return with a departure speed v in the opposite direction. It is evidently nonsense.
Reason is that there is no arrival speed v. As soon as the space craft is in the vicinity of the Moon it goes faster and faster attracted by the gravity of the Moon and will crash (unless it misses altogether - no gravity attraction).
Part of the Heiwa Challenge 2 (the Moon trip) (topic) is to explain how you brake your space craft approaching the Moon to avoid a crash. You can e.g. try to first fly into a Moon orbit (Apollo 1960's) and later land with a another module, or do it Tintin Belgian style 1953, i.e. brake all the time until you arrive at Moon surface at speed 0 - touch down.
Both possibilites are impossible ... like any gravity assist
kick in space. Only silly, religious fools believe in Father Christmas, Moses getting stone tablets from the sky, Jesus walking on water, Muhammed hearing voices in a cave, people flying in space, etc, etc.
They do not understand the God made such things impossible.