NASA plans to launch a mission called DART to kinetically impact an asteroid, primarily for the defense of Earth. One thing that I've been thinking about is the same mission could be used potentially to very accurately measure the fundamental constant G.
For reference, G is the constant that says how much gravity an object will have for a given amount of mass. It is currently only known to about 4 significant digits, making it one of, if not the biggest unknown among fundamental constants. The reason why it is so hard to accurately know is it is very difficult to accurately measure the mass of an object. We know to a very high value what the gravitational force is of Earth, which directly corresponds to G times the mass of Earth. But we don't accurately know the mass of Earth, and thus can't accurately know G.
DART will take an impactor with a fixed mass, and transfer its momentum to an asteroid. If we can take in to account any mass lost, it seems like it might be a reasonably good way to measure G. In addition, DART's target of Didymos’ moonlet can have its Gravitational field accurately determined, as it is orbiting Didymos.
Has any study been done to see if this is even possible, and if not, is it?