Unmanned spacecraft would have to be one of the Voyager probes that slingshotted around the sun (and a few planets?) on its way out of the solar system. Speed is amazing - dozens or hundreds of miles per second. I'll give it to you, Kiwi, as you're close enough: It's ORBITING the Sun!
Railed vehicle - the japanese have fast trains but I'll say the french Tres Grande Vitesse (TGV). Unless it's a rocket sled like Widget suggested. It's a rocket sled, like Widget suggested. Whose?
Manned air breathing craft - some experimental plane of some sort like Chuck Yeager flew (X-1 or X-5?)
When you say air-breathing, do you mean something terrestrial, or something that literally has an engine that uses air/oxygen? So a jet as opposed to a rocket? See next quoted post....
Kerr geometry.
I think the Blackbird is the fastest plane - or was - so must be in one of the categories you described.
Yup! It's the SR-71 Blackbird awright! (Fastest manned air-gulping)
That gives us:
1) Unmanned Spacecraft | Helios 2 Solar Probe | BFM_Kiwi |
2) Manned Spacecraft | Apollo 10 | BFM_Lucky |
3) Unmanned Air-Breathing Craft | ? | |
4) Railed Vehicle | ? | ·WídgêT·? |
5) Manned Air-Breathing Craft | SR-71 Blackbird | BFM_three60 |
If you can come up with just whose rocket sled (general terms, not the specific name) we're talking about here, I'll give you that one.
That would make it a four-way tie.
The first one of you four (Widget has extra work, tho, too) who comes up with the answer to #3 goes next!...
(The suspense is killing you, right??)