I believe part of the cleverness of the Kalman Filter is that it works out the degree to which your measurements are correlated for you. I haven’t looked at it in a while, though.

Not your measurements. That correlation must be specified. It works out the correlations of your state (the thing you are estimating).

In the above example, their measurements are noisy mechanical states (position and momentum). However your measurements can be any (linear plus noise) function of the state, but you need the covariance of your sensor noise.

I meant speeds derived from previous position measurements. Using those speeds is not getting something from nothing. It means using previous measurements to estimate where the object might be now then combine that estimate with current measurement to have the "best" estimate for its current position, for some definition of best.