Consider a charged object moving at high speed (nearing c) toward an object with the same charge, but on a path to pass "beside it" (it doesn't really matter by how much, so long as the one will not hit the other); treat the second object as a stationary reference frame, to make this conceptually easier.
At any given time, there is force of repulsion between the two objects. Normally (relative speed much, much less than c), this force can be approximated as an r-squared force acting along the line separating the two objects.
So far so good. No problems with conservation of anything here, all the vectors in the various force diagrams add up just fine.
But the objects are moving relative to each other at speed nearing the speed of light, and the repulsive force is transmitted at the speed of light...
...which means that at any given instant, the force exerted on one particle by the other is parallel not to the line drawn from one to the other at the moment the force is exerted (at the moment it arrives) but parallel to that line from a few moments ago.
At any given time, when the objects are close enough to one another, the forces acting on each object are an angle to the line drawn from one the other.
In order to have conservation of anything, one has to make a vector integration over a sufficiently long time frame ("long enough" before and "long enough" after the collision, that is, the moment the incoming object passes by or begins moving away from the other).
Otherwise, things just don't add up.
Like I said, I am casting my mind back to a brain-popping moment from 30 years ago in which we sat there slack jawed, eyes agape, and a little aghast, starting at the blackboard while the prof stood there with a Cheshire Cat grin tossing his chalk up and down, waiting for us to accept - or at least merely acknowledge - that we weren't in Kansas anymore, that the universe was rabbit holes all the way down, and that we'd been lied to since junior high.
Fun times.