I really dn't want to argue, so this is going to be my last post on the subject.
dmytryp
Even if the radar is constantly scanning, it's impossible to target an RKV. It'll tell you where the RKV was, but it won't tell you where it IS. Space is such a huge place, that as long as my RKV keeps moving around a bit, you're never going to stop it. If it's a big object, even if you hit it, it'll fragment and still destroy the target.
You really should stop moving the goal post. Either these are inanimate objects (like rocks) hurtled at great speed. And then they are smaller, harder to detect, need less energy to get to said speed, but can't chnage their trajectory. Or these are ships (don't have to be manned ships), which are larger, have propoltion systems and can change their course. Me personally, I'd go with rocks. Change corrections at these speeds would be painfully slow anyway (remember time dilation) and energy demanding (due to the increased mass). So, basically there is no "shifting" around. They run on the trajectory they were sent.
Second, relativistic speed aren't some magical number (not a number at all as there is no definite limit) that suddenly renders radar useless. If you scan continuously (like any radar does), you are tracking a moving object and getting a trajectory. We are talking space, so, even 0.5c isn't terribly fast in terms of distance covering. If you can calculate the trajectory, you can calculate where it is going to be.
Third, you don't have to destroy it completely. Either you push it out of the course (which would be somewhat energy consuming since we are talking increased mass), or you fragment it enough to burn away in atmosphere/plasma shield/whatever.
How are you going to hit it with a laser? You can try to lase where you think it'll be, but the chances of you actually hitting such a tiny object, that is shifting around is astronomical. Your missing the point, because even if I know a course and heading, if the time lag is significant between where it was and where it is, I can't target it.
But this is excatly how Nautilus or any other anti-missile system works. It calculates the trajectory of the missile and shoots where it will be when the anti-missile weapon arrives. The idea that because your weapon moves at 0.5c you for some reason can't do this has no legs. Light (laser/emp or whatever) is still faster. Do you think the Patriot missile is signifficantly faster then the missiles it intercepts? It doesn't even have to catch the projectile. Only intercept it.
The counter to an RKV is simple. MOVE. As long as you are on something that isn't a relatively static object, or predictable in motion you're safe. Unfortunately on a planet, that's impossible but on a space station, or something like that, it can easily be done.
It's like trying to target an airplane with sonar(like a high subsonic airplane). It's not going to work. The only way a bat pulls it off, is because relative to the speed of sound insects are SLOOW. Ever try playing a first person shooter, with a huge amount of lag? It doesn't work too well, even when I try to extrapolate where an opponent is going.
As I said, this weapon would be useless in ship-to-ship combat. As for shooting some planet from enough distance -- maybe, but again you'd have enough of a variety of defenses -- alert stations, orbital sattelites etc.
As for why you can't do it -- computational capabilities. As you said, you
try to extrapolate, but your brain lacks the needed computational accuracy and speed. Targeting computers on the other hand... And even with our poor human brains there are people who are able to shoot down rapidly moving enemies.
EDIT: I didn't get your point at first, so this is an addition. The reason why sonar with high subsonic plane wouldn't work isn't speed differences, but distance. Once you get the relevant info, your plane is out of range. Not so in space. Also, there wouldn't be any lag. The incident beams are sent continuosly. You'd only be getting the info from hours/days/whatever ago, but this would happen anyway with cosmic distances. Let's examine a scenario -- you have alert stations around Pluto that see an object eading to earth at 0.5c. They pass the info about the trajectory to orbital stations and those have
hours until the object gets their to target it.
Thanks for the discussion