According to quantum mechanics you can't measure both the position and momentum of a particle at the same time. A particle has an associated wave function. If you want to measure the momentum of a particle precisely, you have to make two precise measurements of position and time.
If we measure the position of one particle precisely, the wave function collapses to a sharp peak in space and is thus not the same wave function as the one you start with. A second measurement of position will thus yield a position that corresponds to the particle with the modified wave function.
So we actually need two particles with identical wave functions. You measure the position of one particle and subsequently you measure the position of the second. That way you know the momentum of a particle with the associated particle.
But if the wave function corresponds to a sharply defined momentum it is widely spread in space. So if we make a precise first position measurement of the first particle, and after short time of the second you can get wildly varying measurements of the position differences, and thus of the momentum. Which suggests that we should use the same particle. But this has again the problem I mentioned earlier.
Or is that no problem at all? If we make a precise position measurement, the position is sharply defined and evolves by dispersing very quickly while traveling along as a (dispersive) wave function. Should we then make a second measurement very fast, so the dispersion hasn't proceeded that much? What about the extra momentum the particle got from the first mesurement?
How can we measure the momentum of a quantum particle best, assuming a Gaussian wavepacket as its wave fubction?
