Thanks for reading and for your question! What you’re talking about is the Heisenberg Uncertainty Principle: the idea that you can’t perfectly know both the position and momentum of a particle at exactly the same time. The math itself is fairly simple and is contained in one short equation. The product of the uncertainty in position and the uncertainty in momentum for a particle must be greater than Planck’s constant divided by two. This means that it is impossible for both uncertainties to be equal to zero at the same time.
To answer your question, there is nothing to stop a scientist from measuring first the position and then the momentum of the same particle. In fact, a scientist could measure both position and momentum at once. Those measurements will not give you the “exact” position and momentum (if you believe such things even exist). As I said above, the product of those numbers must be greater than or equal to Planck’s constant divided by two. To get a better idea of what is happening with the particle in a question, a scientist in my field will prepare an identical particle many, many times and make the same position and momentum measurements with each identical particle.
After repeating the position and momentum measurements, our scientist would find a distribution of answers. The standard deviation in the distribution of position measurements multiplied by the standard deviation in the distribution of momentum measurements will always obey the Heisenberg Uncertainty Principle. This is true even in a perfect experiment with no noise sources.
This is what happens. It is scientific fact. The interpretations of what these results mean are not clear, however. It is common for physicists to get caught up in arguments of the interpretation of quantum mechanics, but this is more a question for philosophers. At the core of the field of physics is the math and the data. The math works. It can predict how particles in our universe behave. We have confirmed this with many experiments. I know it isn’t intuitive, but I hope this is a satisfactory answer for now. If you’re curious, read some more about the double-slit experiment. It should be confusing, but maybe it will give you more insight into the wacky nature of quantum mechanics! I’ll also add this to the list of things to discuss in future articles.