Unfortunately, if you can't catch it, you don't need it for that, as you'll be traveling faster that it to be able to intercept it and would just burn fuel trying to break enough to match its speed.
Might make a good radiation shield, though.
Raise the orbit of satellites in LEO which are affected by atmospheric drag. Or re-boost things like the international space station, proposed future Chinese space station, etc.
Extend the lifetime of geostationary satellites which are still electrically functional, but out of station keeping propellant. One such thing docked for the very first time with a satellite earlier this year. https://en.wikipedia.org/wiki/Mission_Extension_Vehicle
On unmanned missions, slowly move cargo from low earth orbit to destinations at the Moon or Mars. If you can use ion and hall effect type thrusters for your missions to move cargo around, you can establish a logistics supply chain for essential supplies consisting of unmanned craft.
An interesting example of using ion engines to maintain low earth orbit, through long continual thrust was this mission:
It boils down to the fact that in space you can, with a very good degree of approximation, determine the trajectory of an object not under thrust, knowing only these two things: it's position at a given time, and it's velocity at that same time. These two vectors give you a single trajectory through space.
It follows from two physical laws that you may remember from high school:
- Newton's second law, or F=ma
- Newton's law of universal gravitation, or F=GMm/r^2
If you put them against each other, you get ma=GMm/r^2, or consequently a=GM/r^2. The acceleration (and thus, future position and velocity) of an inert object in space is independent of its mass. So all you need to tell where it will go is to know where it is, and what is it's current speed and direction of movement. And, of course, what other bodies influence it with their gravity.
Of course, in practice, there are other considerations in which the mass may become relevant. For instance, solar radiation acts on the surface, impacting force that's scaled by object's mass. The same is true for collisions with various stray atoms, especially prevalent near planets with atmospheres.
But discounting above factors (and, again, you can go pretty far just ignoring them), if you have two objects very close to each other and not moving relative to each other, they'll just go together along the same path for a long, long time.
1. Tug gets in same position and velocity as its target, and docks.
2. Tug performs some kinda of burn to put both it and its target on the new trajectory.
3. Tug undocks and performs a burn to head off to where ever else it's needed.
If I understand correctly, it's really just a way to avoid putting engines and fuel tanks on things that don't need to change trajectory much.
But landing on it and using the free large fuel tank of hydrogen it has now became you can 1) accelerate more 2) decelerate at your destination 3) steer (at least a bit).
Seems like a real hard mechanical problem, but not fundamentally impossible.
If you ever wanted to send a probe to Proxima Centauri, and stop when it got there, accelerating the fuel you'd want to have available to decelerate up to solar escape velocity would be a tremendous energy expenditure, even if you can mine it in the asteroid belt.
If, instead, you got lucky and jumped your probe (fuel tanks dry) on an interstellar wanderer that happened to be going approximately in the direction you wanted, you could have hundreds of tons of rocket fuel moving at interstellar velocities.
I'd always imagined that an alien intelligence might have seeded every solar system in the galaxy with a probe to monitor for evolving intelligence - but after Oumuamua, I realized it would be far more efficient and reliable to have swarms of them coasting through systems periodically to check on them. ...or maybe both.
This is of course well beyond our current capabilities, but it's not something that's impossible to consider in the medium-distant future. Plus, even if you expend a colossal amount of energy accelerating you're still talking about thousands of years for the journey where you'll need to keep those reactors running for the entire trip (there is no useful solar collection in deep space) having a literal mountain of fuel to start with is important.
About the only good thing it could be used for is reaction mass. But you can use almost anything as reaction mass as long as you have a source of energy, you can exhaust any particles to propel yourself forward, doesn't have to be hydrogen.
Well, not today, but we can't reach it today either.
Once you dock with Oumuamu, you can run it’s material through at an 890 ISP and gradually accelerate to much higher speeds.
First you have to add something like 40 km/sec to go from Earth orbit to escape into interstellar space, and then you have to add another 20 km/sec to match speeds with Oumuanu. That's 60 km/sec, which is 3 times the speed and takes 9 times the energy.
Pioneer and Voyager got to interstellar space with gravity assists. But those work best when you barely get to a planet's orbit, then it picks you up and you go close to 180 around it. (Thereby adding 2x its velocity to your own.) Once you start getting to the velocities needed for Oumuamu, you just shoot by the planet and don't get much of an assist. (Assuming, that is, that there is an alignment between the planets and a random interstellar visitor. Which is extremely unlikely.)
I stand by my statement. We do not have good enough rockets for this maneuver.
How big is our solar system? I always assumed it referred to everything between the sun and Pluto, but 10,000 years suggests it's far larger.
It’s curious to note that the proposed outer limit of the Oort cloud (3.2 ly) is more than half the distance to the closest star system (4.4 ly)  though I’m unclear whether the cloud would be roughly spherical or have some sort of directionality.
-  https://en.wikipedia.org/wiki/Hill_sphere
Can you aerobrake around a star?
Unfortunately, no. The boost you can steal from a powered flyby is only of the same order as the escape velocity of the massive object. For the sun, this is 617 km/s at its surface, or 0.002 c. The term you're interested in is "Oberth effect".
To get relativistic speeds on the table, you'd need exotic, compact objects whose escape velocities are relativistic. I.e. white dwarfs, neutron stars & black holes.
No one asked, but: Freeman Dyson pointed out  some astounding properties of a short-period white dwarf binary. (Particularly: there are two stars; both are extremely dense; and they are orbiting around each other at an extremely high relative velocity). If you could perform a close slingshot maneuver around one of these stars, you would be accelerated to relativistic speeds in mere seconds. What's more, this would be an entirely unpowered, propellantless manuevuer (stealing orbital momentum from the binary system). And finally, as an incredible example of the equivalence principle, human passengers wouldn't even notice the insane ~10^4 g's of acceleration: it's equivalent to free-fall.
 This is a PDF https://www.ifa.hawaii.edu/~barnes/ast242_s14/Dyson_Machines... (1963)
Not a physicist, but it seems like you should be able to do something similar to aerobraking (not one-to-one since the Sun's "atmosphere" is different from a planet's). Maybe using a solar sail of some sort directed at the right angle?
If I repeat this long enough, my engine will melt, but that's a minor matter of materials science. Maybe I can win my drag race before it melts. Is that the limiting factor or is there some other reason I'm missing that we need a heat gradient?
If you do draw it with enough detail, you'll notice that if you do not heat it more at each turn, you won't be able to get any pressure difference at your "explosion".
If you can keep your cool, why not? Matter is matter.
What's better, you could technically radiation-brake around a star. I don't know how big of a solar sail you'd have to present to get a meaningful force from it.
You can only slingshot around the sun if you were already in a hyperbolic orbit w.r.t the sun. An object in a closed helio centric orbit cannot gain any momentum by "slingshotting" around the sun.
But this isn't my field of expertise.
I love that this sentence exists. Because it's absurd and enlightening at the same time.
It already slows down when you are moving too fast.