This may very well be a question for Earth Science SE, but as it is very hypothetical, maybe pure physics is more appropriate...
Imagine we have have a wedge that tapers down into an infinitesimally small point. We can balance the wedge so that this point is orthogonal to the Earth's surface.
How heavy would the wedge need to be, so that it pushes its way (sinks) into the Earth's core? Is there even a weight (theoretically or otherwise) that could make this happen?
Would impulse be necessary in any case? Analogous to hitting a nail with a hammer.
- the incompressibility of a solid is not considered
- We can add more force to the motion of the wedge (e.g. with n rocket engines) than it gets alone from its gravitational relationship with the earth.