  We introduce and study a simple model capturing the main features of
unbalanced optimal transport. It is based on equipping the conical extension of
the group of all diffeomorphisms with a natural metric, which allows a
Riemannian submersion to the space of volume forms of arbitrary total mass. We
describe its finite-dimensional version and present a concise comparison study
of the geometry, Hamiltonian features, and geodesics for this and other
extensions. One of the corollaries of this approach is that along any geodesic
the total mass evolves with constant acceleration, as an object's height in a
constant buoyancy field.
