Directly below her $10\text{ m}$ tall tower, Alokananda has piled up a heap of sand $1\text{ m}$ high.

When she drops a ball of density $\rho$ from the top of the tower, the ball does reach the bottom of the sand (whose density, by the way, is $\sigma$) pit, but by then, it is completely exhausted of its kinetic energy.

Modelling the heap of sand as a pool of fluid and assuming that the Archimedes' principle is the only reason for the resistance provided by the sand, what is the value of $\dfrac{\sigma}{\rho}$?

Assume that the ball has a positive radius negligible compared to the height of fall. Further, also assume that the ball immerses completely and no energy is lost in the air-sand transition.