Cool. Or rather hot, as it seems to come pretty close to the sun.

Here is more about the project: https://en.m.wikipedia.org/wiki/Project_Lyra

And TIL about the Oberth effect: https://en.m.wikipedia.org/wiki/Oberth_effect

If those objects happen to be more frequent than expected, I think we should envision a way to send small probes with the ability to slowly mine their resources and build things like larger communication antennas or telescope mirrors using in-situ materials, taking advantage of their great mass and speed to create much bigger structures than we could reasonably send out there

A back-of-napkin calculation says this maneuver can also be used to send a probe to Proxima Centauri in ... 128 days? Why are we not doing that?

Not sure if you’re sarcastic or not but Promixa is 4+ light years away.

So unless we have FTL tech, this is impossible.

Nope, just bad at both math and basic reasoning yesterday.

But Oumuamua will engage it's cloaking device or warp speed. After all it's an alien ship that will be divulged in the latest round IAP disclosures by the US government ;)

This looks like scenario B from this paper: https://arxiv.org/pdf/1902.04935.pdf

Mission E-DSM-E-J-6SR-1I (15.3 km/s of delta v required):

  (2030-06-09) Launch from Earth
  (2031-11-05) Deep Space Maneuver at 3.2 AU
  (2033-04-17) Earth return
  (2034-07-12) Jupiter arrival
  (2036-02-24) 6 solar radii
  (2052-07-29) 'Oumuamua arrival

A Solar Oberth mission requires a heat shield, which takes up weight that can be used for other payload. It's also riskier. There are also proposals for other mission profiles that only use the Jupiter Oberth, but these would either require more delta v or more inner planet assists, and so could take longer. Example: https://arxiv.org/pdf/2201.04240.pdf

Mission V-E-DSM-E-J with 2028 launch date, 26 years flight duration, 15.8 km/s of delta v required. Animation: https://www.youtube.com/watch?v=qrCUsAGZjUw

Note that delta v does not necessarily imply anything about the final spacecraft velocity. Rather, it's indicative how much thrust you need throughout the mission. So how much delta v you have at your disposal is a function of what kind of rocket(s) you have and how much propellant you have.

And how far you can get with a certain delta v budget depends on how smartly you use it. To get the most bang for your buck, you fire your rockets at strategic locations in your orbit.

This is where the physics behind Oberth maneuver come in: It's most efficient to gain kinetic energy at periapsis, when you are travelling the fastest. This is because kinetic energy is 1/2 mv^2. The greater kinetic energy will take you farther out of the gravity well than if you fired your engines at a lower velocity.

How fast is it going when it arrives at Oumuamua?

Or to put it another way... how long will it have to gather data?

In the two examples above, the Solar Oberth Maneuver would have it pass 1I with a relative speed of ~30 km/s. The Jupiter-only maneuver passes 1I with a relative speed of ~18 km/s.