Question: I know that our planetary probes often use planetary gravity to boost their speeds. That only works for prograde speeds, right? Because you're subtracting a miniscule amount of orbital speed from the planet and adding it to your spacecraft's speed. You couldn't whip around a planet and somehow use that to give you retrograde speeds, could you? (Presumably an airless planet, like Mercury.) Or what about using a large moon during the retrograde (relative to the planet's motion around the sun) part of its orbit?
Problem is another. To got additional speed from other body, you need to move very close to it and in perigee use some powerful acceleration to increase rotation speed fast, so could not use slow acceleration engines, like ion engines.
Idea of gravity-assist acceleration, mechanically is just rotation of pair tightly tied bodies (and cut tie in right moment, so one body got acceleration and other got deceleration), but as it is impractical to tie for example to Moon with rope, used gravity force.
What also interesting, gravity-assist could use not only orbital speed of large body, but also got some acceleration from rotation of large body, as for gravitation, large planet is not just one material point, but system of few smaller (sub)bodies, and closer (sub)bodies give more acceleration than others.
I’m not an expert on orbital mechanics, I just want to provide some data points.
The voyager craft, which not only had very good acceleration early on (the best we could do, really), combined with exceptional gravity slingshots and a lot of time - and are by all accounts some of the fastest man made objects ever - are going 15.4 km/s and 17 km/s relative to the sun.
3I/ATLAS is going so much faster than these objects they might as well be stationary.
Even ignoring the limited amount of time we have to intercept, catching up to 3I/ATLAS would be incredibly difficult to do. Perhaps impossible with our current technology*. Like catching up to a semi-truck going full speed on a highway with a bicycle. After it’s already passed us and is a couple miles down the road.
*barring theoretical (and kind of insane and dangerous) tech like Orion drives.
When you achieve speed in space, after acceleration, the speed won’t change forever unless you encounter some other force, like a celestial body gravity to change it. So if you achieve interestelar comet’s speed, you can shutdown the rocket and just travel at that speed for eternity like the comet does.
Even better: you can forget the comet, accelerate, keep accelerating until there is no more power or even a working motor while also extending a big sail to let solar wind accelerate you a little more.
