My guess is that the answer lies in linear momentum and impulse force.

Linier momentum p = mv  and so the units will be kg m/s.  Impulse is force applied over a period of time, so  Impulse = Ft.  which will give a formula P(final) - P(initial) = Ft.  Where P(final) is the linear momentum at point of impact and P(initial) = 0.

If you know the height of the first item and its mass, you can calculate its velocity at the point in time that it hits the second object.

If you know the contact time, then it is possible to work out the average force applied over the period of time (an integral equation).  If the second object is made of a softer material the amount of time the two objects are in contact increases which means the force imparted becomes larger.  If the contact surface area of the second item is large but the surface area on the opposite side is smaller, then the force per unit of area will be increased.  (Pressure is Force / Area or in a falling object,  Pressure = Weight / Area).

That's my guess.