It is not possible to simultaneously know the exact position and momentum of a particle as a consequence of non-commutativity of the position and momentum operators. But what if I consider a simple two-particle collision, where I make a simultaneous measurement of the position of one particle and momentum of the other particle?
Suppose I first measure the momenta of both particles and find particle A to be at rest and particle B moving with some specified initial momentum. If I later again measure the momentum of particle A, which used to be at rest, and find it now to move with a certain momentum, I would know that it had scattered with the other, initially moving particle. Because of conservation of momentum, I now know the exact momentum of particle B. If I hence simultaneously measure the position of particle B when I am measuring the momentum of particle A, should this give me a complete description of particle B in violation of the uncertainty principle?